Faradays first dynamo A retrospectiveGlenn S Smith

Citation American Journal of Physics 81 907 (2013) doi 10111914825232 View online httpdxdoiorg10111914825232 View Table of Contents httpscitationaiporgcontentaaptjournalajp8112ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Hands-on Experiments on Faradays Law Phys Teach 43 228 (2005) 10111911888083 An experimental observation of Faradayrsquos law of induction Am J Phys 70 595 (2002) 10111911405504 Unconventional Dynamo Phys Teach 40 220 (2002) 10111911474144 Only the integral form of the law of electromagnetic induction explains the dynamo Am J Phys 66 543 (1998) 101119118903 Some rigorous results for the kinematic dynamo problem with general boundary conditions J Math Phys 38 1583 (1997) 1010631531817

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

Faradayrsquos first dynamo A retrospective

Glenn S Smitha)

School of Electrical and Computer Engineering Georgia Institute of Technology AtlantaGeorgia 30332-0250

(Received 28 June 2013 accepted 2 October 2013)

In the early 1830s Michael Faraday performed his seminal experimental research on

electromagnetic induction in which he created the first electric dynamomdasha machine for

continuously converting rotational mechanical energy into electrical energy His machine was a

conducting disc rotating between the poles of a permanent magnet with the voltagecurrent

obtained from brushes contacting the disc In his first dynamo the magnetic field was asymmetric

with respect to the axis of the disc This is to be contrasted with some of his later symmetric

designs which are the ones almost invariably discussed in textbooks on electromagnetism In this

paper a theoretical analysis is developed for Faradayrsquos first dynamo From this analysis the eddy

currents in the disc and the open-circuit voltage for arbitrary positioning of the brushes are

determined The approximate analysis is verified by comparing theoretical results with

measurements made on an experimental recreation of the dynamo Quantitative results from the

analysis are used to elucidate Faradayrsquos qualitative observations from which he learned so much

about electromagnetic induction For the asymmetric design the eddy currents in the disc

dissipate energy that makes the dynamo inefficient prohibiting its use as a practical generator of

electric power Faradayrsquos experiments with his first dynamo provided valuable insight into

electromagnetic induction and this insight was quickly used by others to design practical

generators VC 2013 American Association of Physics Teachers

[httpdxdoiorg10111914825232]

I INTRODUCTION

In 1831 Michael Faraday (Fig 1) embarked on one of themost productive periods of his career during which heexperimented with electromagnetism and among otherthings discovered electromagnetic induction Faraday wasaware of research that had been done earlier on theContinent in particular the experiments on magnetism of theFrench scientist Francois Arago In 1824 Arago discovered aphenomenon that would become known as ldquoAragorsquosrotationsrdquo1 Arago suspended a magnetized needle parallel toand above a copper disc When he rotated the disc about itsaxis he noticed that the needle also rotated in the samedirection as the disc At the time there was much speculationas to the cause of these rotations In his ExperimentalResearches in Electricity Faraday commented that ldquohellipIhope to make the experiment of Arago a new source of elec-tricityhelliprdquo and this he did in his subsequent research2

Faraday invented the first electric dynamosmdashmachines forcontinuously converting rotational mechanical energy intoelectrical energy

Figure 2(a) shows a sketch of Faradayrsquos first dynamo Thissketch is based on a drawing from Faradayrsquos diary datedOctober 28 183134 Probably because of his familiarity withAragorsquos experiment Faraday chose to use a copper discmounted on an axle in his experiment He placed the discbetween the poles of a strong permanent magnet and heused mercury-coated sliding contacts to connect the disc toan external circuit containing a simple but sensitive galva-nometer Faraday called these contacts ldquoelectric collectorsrdquobut we would now call them ldquobrushesrdquo As Faraday rotatedthe disc he observed a continuous deflection of the galva-nometer Faraday performed many experiments with thisdynamo in an effort to understand the physics of electromag-netic induction We will have more to say about these experi-ments later in this paper

After studying his first dynamo Faraday examined otherconfigurations for the dynamo one of which is shown in Fig2(b) This sketch is based on a drawing from SilvanusThompsonrsquos biography of Faraday and the dynamo is dis-cussed in an entry in Faradayrsquos diary dated December 1618315ndash7 Faraday cut a circular hole of slightly bigger

Fig 1 Portrait of Michael Faraday (ca 1840s) taken sometime after the

publication of the first volume of his Experimental Researches inElectricity in which he describes his early dynamos Smithsonian Institution

Archives Image SIA2012-1087

907 Am J Phys 81 (12) December 2013 httpaaptorgajp VC 2013 American Association of Physics Teachers 907

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

diameter than his disc in a lead sheet then he placed the cop-per disc in the hole and filled the space between the disc andsheet with mercury He fastened one lead from his galva-nometer to the sheet and placed the other lead in a mercury-filled cup atop the axle of the disc The magnetic field wasthe Earthrsquos which has a significant component normal to thesurface of the Earth in London hence normal to the horizon-tal disc As Faraday rotated the disc he observed a deflectionof the galvanometer Notice that unlike Faradayrsquos firstdynamo all of the components of this dynamo (the disc thebrush formed by the mercury-filled trough and the compo-nent of the magnetic field normal to the disc) are rotationallysymmetric about the axis of the disc

Most textbooks on electromagnetism contain a discussion orproblem involving a symmetrical dynamo which in almost allrespects is the same as Faradayrsquos design in Fig 2(b) Sometimesit is referred to as a ldquounipolar generatorrdquo or a ldquohomopolar gener-atorrdquo Because of the simplicity of the symmetrical structure asolution for the open-circuit voltage V at the terminals A-B iseasily found by a few different methods to be

V frac14 xB0A2

2 (1)

where A and x are the radius and the angular frequency forthe disc and B0 is the component of the applied magneticfield that is normal to the disc Experimental results that ver-ify Eq (1) have been presented in a number of papers8ndash12

and more detailed discussions of the symmetrical dynamoare in the books of Woodson and Melcher and Van Bladeland in the papers of Van Bladel and Montgomery13ndash16

Faradayrsquos first dynamo the asymmetric one shown in Fig2(a) has received much less attention There is no mentionof it in most textbooks on electromagnetism In a paper from1942 and later in the third edition of his graduate-level text-book William Smythe analyzed a geometry similar toFaradayrsquos but he did not discuss it in the context of adynamo for example he did not determine the open-circuitvoltage1718

The purpose of this paper is to present a detailed discussionof Faradayrsquos first dynamo supported by new analytical and ex-perimental results This material particularly the graphicscan be used to present the historically important story ofFaradayrsquos accomplishments to students in undergraduate andgraduate courses in electromagnetism In Sec II we describean analysis of the dynamo in which Smythersquos approximateresult for the magnetic scalar potential is used to obtain thecurrent in the disc and we extend this result to include theopen-circuit voltage for all points on the disc In Sec III wepresent quantitative numerical results from the analysis andshow how these results are related to Faradayrsquos qualitativeobservations In Sec IV we describe an experiment that issimilar to Faradayrsquos but based on modern materials and instru-mentation Measurements from this experiment validate theapproximate analysis Finally Sec V contains a discussion inwhich we address the question ldquoWhat ever became ofFaradayrsquos first dynamordquo or stated differently ldquoHow is hismachine used todayrdquo

II ANALYSIS OF FARADAYrsquoS FIRST DYNAMO

The analysis for the dynamo is based on the model shownin Fig 3(a) the coordinates and dimensions are on the planview of the disc in Fig 3(b) The circular disc of radius Aand thickness d has electrical conductivity r permeabilityl0 and rotates with angular frequency x The applied axialmagnetic field ~Ba frac14 B0z is uniform over the circular area ofradius a located at a distance c from the axis of the disc(gray area in the figure) The external circuit formed fromperfectly conducting wire makes contact with the disc at thebrush on the axis (C) and at the brush at the point ethqbubTHORN onthe disc (D) The open-circuit voltage V is determined at theterminals A-B in the external circuit The analysis is per-formed in the laboratory frame which is the frame in whichthe magnet and the external circuit are at rest The electro-magnetic field is time-invariant in this frame because of therotational symmetry of the disc To see this consider whathappens when a hole is cut into the disc destroying the rota-tional symmetry The electromagnetic field in the laboratoryframe is then a periodic function of time

The analysis is a quasi-static one in which magneticeffects are assumed to be dominant and all dimensions areelectrically small so that any retardation in time for the fieldcan be ignored19 In Maxwellrsquos equations the displacementcurrent ~D=t and the volume charge density q are ignoredThe latter condition means that any convection current dueto the motion of free charge is also ignored In addition tothe applied magnetic field ~Ba there is the magnetic field ~Bi

Fig 2 (a) Faradayrsquos first dynamo based on a drawing from Faradayrsquos di-

ary3 (b) One of Faradayrsquos rotationally symmetric configurations for a

dynamo based on a drawing by Thompson5

908 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 908

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

due to the current induced in the disc In the source-freeregion outside the disc the two Maxwellrsquos equations thatinvolve the magnetic field ~Bi are

r ~Bi frac14 0 (2)

and

r ~Bi 0 (3)

The volume density for the conduction current ~J is assumedto be uniform throughout the thickness of the disc with anegligible axial component Thus it can be approximated bya surface- or sheet-current density ~Js ~Jd20

The boundary condition that relates the surface currentdensity to the magnetic field is

~JsethquTHORN frac141

l0

z frac12~Biethqu z frac14 0thornTHORN ~Biethqu z frac14 0THORN

frac14 2

l0

z ~Biethqu z frac14 0thornTHORN (4)

where for the last term we have used the fact that the compo-nent of the magnetic field tangential to the surface of the dischas odd symmetry about the disc as shown in Fig 4(a)

A Magnetic scalar potential

We can represent the irrotational magnetic field Bi by ascalar magnetic potential U via21

~Bi frac14 rU (5)

This is inserted into Eq (2) to obtain Laplacersquos equation forthe potential in a source-free region

r2U frac14 0 (6)

For describing the potential at the top surface of the disc(zfrac14 0thorn) we will define the three regions on the disc shownin Fig 5 The two radial lines at ua frac14 6sin1etha=cTHORN are tan-gent to the boundary of the circular region containing theapplied magnetic field Region I (light gray) is bounded bythese lines (ua u ua) and is above the applied mag-netic field qo lt q A with qo frac14 cfrac12cosu

thornffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffietha=cTHORN2 sin2u

q Region II (dark gray) is within the

applied magnetic field ua u ua and qi q qo

with qi frac14 cfrac12cosuffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffietha=cTHORN2 sin2u

q Region III (white) is

everywhere else on the disc Using this notation the appliedmagnetic field can be expressed as

~BaethquTHORN frac14 BaethquTHORNzfrac14 B0fUfrac12q qiethuTHORN Ufrac12q qoethuTHORNgfrac12Uethuthorn uaTHORN Uethu uaTHORNz (7)

in which U(n) is the Heaviside unit-step function

Fig 3 (a) Model used in the analysis of the dynamo (b) Plan view of the

disc showing the coordinates and dimensions

Fig 4 Details for the surface current in the disc (a) Boundary condition

relating the magnetic field to the surface current density (b) Tube of surface

current showing coordinates

909 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 909

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

Smythe gives approximate expressions for the magnetic scalar potential in these regions as171822

UIethqnu z frac14 0thornTHORN frac14 UIIIethqnu z frac14 0thornTHORN frac14 xB0Acrd

2a2

nqnsinu1

q2n thorn c2

n 2cnqncosu 1

c2nq

2n thorn 1 2cnqncosu

(8)

and

UIIethqnu z frac14 0thornTHORN

frac14 xB0Acrd

2qnsinu 1 a2

n

c2nq

2n thorn 1 2cnqncosu

(9)

in which we have normalized the dimensions to the radius ofthe disc that is qn frac14 q=A an frac14 a=A etc

Figure 6 is a contour plot of the scalar potential normal-ized so that maxjUnj frac14 10 for the case aAfrac14 0170 andcAfrac14 0667 These parameters are roughly the same as thoseused for the experiments described later All of the contoursare spaced by the increment Un frac14 01 except the incom-plete contours for Un frac14 6001 and 6 005 Notice that thepotential has odd symmetry about the line u frac14 0 180

(positive on the left and negative on the right) and the poten-tial is zero along that line and along the edge of the discwhere qfrac14A

B Surface current density

We are generally more used to interpreting the surfacecurrent density than the magnetic scalar potential and theformer is easily obtained from the latter Inserting Eq (5)into Eq (4) we obtain

~JsethquTHORN frac14 2

l0

z r2Uethqu z frac14 0thornTHORN (10)

where we have use r2 to indicate that the gradient is onlywith respect to the two variables q u

The equipotential lines (Ufrac14 constant) such as those inFig 6 are also streamlines of the surface current densitythat is at every point along one of these lines the surface cur-rent is tangential to the line To show this we first note thatr2U is normal to the equipotential lines The operation

r2U ~Js frac14 2

lo

r2U ethz r2UTHORN

frac14 2

lo

z ethr2Ur2UTHORN frac14 0 (11)

shows that the surface current density is normal to r2UHence the surface current is parallel or tangential to theequipotential lines

Two adjacent equipotential lines such as the lines for U1

and U2 in Fig 4(b) bound a two-dimensional tube of surfacecurrent23 At any cross section of the tube the total current Ithrough the tube is the same To show this we first recallfrom Eq (3) that the magnetic field is irrotational so in anyregion in which the field is well defined the line integral of~Bi between two points such as from P1 to P2 in Fig 4(b) isindependent of the path of integration21 Thus

ethP2

P1

~Bi t dlsquo frac14 ethP2

P1

r2U t dlsquo frac14 ethU2 U1THORN (12)

Fig 5 Regions and coordinates used in the description of the scalar mag-

netic potential U at the top surface (zfrac14 0thorn) of the disc

Fig 6 Contour plot for the magnetic scalar potential Unethqu z frac14 0thornTHORN for

the case aAfrac14 0170 and cAfrac14 0667 The potential is normalized so that

maxjUnj frac14 10 and the contours are spaced by the increment Un frac14 01

except the incomplete contours for Unfrac146001 and 6005 The equipoten-

tial lines are also streamlines of the surface current density and the arrows

on these lines show the direction of the current Two adjacent equipotential

lines bound a tube of surface current through which the total current I is the

same at any cross section

910 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 910

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

where t is a unit vector tangent to the path After taking frac12zEq (4) t and rearranging terms we obtain

~Bi t frac14 l0

2ethz ~JsTHORN t (13)

Substituting Eq (13) into the left-hand side of Eq (12) gives

l0

2

ethP2

P1

ethz ~JsTHORN t dlsquo frac14 l0

2

ethP2

P1

~Js s dlsquo frac14 ethU2 U1THORN

(14)

in which the unit vector s frac14 z t is in the plane of the discand normal to the path as shown in Fig 4(b) Now from Eq(14) we see that the total current through any cross sectionof the tube is

I frac14ethP2

P1

~Js s dlsquo frac14 2

l0

ethU2 U1THORN (15)

which is clearly a constantIf two different tubes of current are bounded by equipoten-

tial lines separated by the same increment U the two tubeshave the same total current Thus in Fig 6 all of the tubeswith Un frac14 01 have the same total current From this ob-servation we see that the current density is largest under thepole of the magnet and smallest near the edge of the disc atangles near u frac14 180 Notice that the tubes of current areclosed and the radial component of the surface current iszero at the edge of the disc The closed tubes of current areanalogous to eddies that form around an object in flowingwater hence the name ldquoeddy currentsrdquo (also ldquoFoucaultcurrentsrdquo) Because of the finite conductivity of the disc theeddy currents dissipate energy This energy must be suppliedby the mechanical source turning the axle of the disc

C Faradayrsquos law and the open-circuit voltage

Faradayrsquos law in integral form isthornCethtTHORN

frac12~Eeth~r tTHORN thorn~veth~r tTHORN ~Beth~r tTHORN d~lsquo frac14 d

dt

eth ethSethtTHORN

~Beth~r tTHORN d~S

(16)

where we explicitly indicate that the closed contour Cbounding the open surface S can vary with time and ~v isthe velocity for a point on the contour24 We will apply

Eq (16) to the contour ABCDA in Fig 3(a) shown inmore detail in Fig 7

For this contour in the limit u frac14 xt 0 the expres-sion on the left-hand side of Eq (16) becomes

VethqbubTHORN thornethqb

qfrac140

frac12~Eethqub 0THORN thorn~vethqub 0THORN

~Bethqub 0THORN q dq (17)

The integral in Eq (17) is for the portion of the contourCD that is in the disc where the constitutive equation forthe surface current density in the laboratory frame applieswhich is

~JsethqubTHORN ~Jethqub 0THORNd

frac14 rdfrac12~Eethqub 0THORN thorn~vethqub 0THORN ~Bethqub 0THORN(18)

When Eq (18) is substituted into Eq (17) we obtain

VethqbubTHORN thorn1

rd

ethqb

qfrac140

q ~JsethqubTHORNdq (19)

for our final result for the left-hand side of Eq (16)Now consider the expression on the right-hand-side of Eq

(16) for the contour ABCDA In the time increment t thearea S in Fig 7 increases by the increment S (gray in thefigure) We will assume that the applied magnetic field ismuch greater than axial component of the magnetic field dueto the current in the disc (j~Baj jz ~Bij) and we will ignorethe latter Then the time variation in the flux of the magneticfield through S is due to the applied field ~Ba passingthrough S or

d

dt

eth ethSethtTHORN

~Beth~r tTHORN d~S frac14 limt0

1

t

ethqb

qfrac140

~BaethqubTHORN ethzTHORNq u dq

264

375 frac14 lim

t0

ut

ethqb

qfrac140

qBaethqubTHORN dq

264

375 frac14 x

ethqb

qfrac140

qBaethqubTHORN dq

(20)

After equating the left-hand- and right-hand-sides of Eq (16) for the contour ABCDAmdashequating Eqs (19) and (20)mdashweobtain the open-circuit voltage at the terminals A-B in Fig 3(a) to be

Fig 7 Details for contour C used with Faradayrsquos law

911 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 911

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

VethqbubTHORN frac14 1

rd

ethqb

qfrac140

q ~JsethqubTHORNdqthorn xethqb

qfrac140

qBaethqubTHORN dq frac14 2

l0rd

ethqb

qfrac140

1

qUethqub z frac14 0thornTHORN

udqthorn x

ethqb

qfrac140

qBaethqubTHORN dq

(21)

where in the last line we have substituted Eq (10) for ~JsTo complete the derivation for the open-circuit voltage we must substitute for the magnetic scalar potential Eqs (8) and

(9) and perform the differentiation and integration indicated in Eq (21) For points qbub in each of the three regions shownin Fig 5 the evaluation is different for example for Region I the expression to be evaluated is

VI frac14 2

l0rd

ethqi

qfrac140

1

qUI

udqthorn

ethqo

qfrac14qi

1

qUII

udqthorn

ethqb

qo

1

qUIII

udq

264

375thorn xB0

ethqo

qfrac14qi

qdq (22)

The calculations are straightforward but tedious All of the definite integrals can be evaluated in closed form25 and final resultsfor the open-circuit voltage are

VIethqbnubTHORN frac14 xBoa2

2feth1=anTHORN2frac12q2

on q2in cnethqon qinTHORNcosub thorn Fethqin cnubTHORNthornFethqbn cnubTHORN

Fethqon cnubTHORN Fethqbn 1=cnubTHORNg (23)

VIIethqbnubTHORN frac14 xBoa2

2feth1=anTHORN2frac12q2

bn q2in cnethqbn qinTHORNcosub thorn Fethqin cnubTHORN

Fethqbn 1=cnubTHORNg (24)

and

VIIIethqbnubTHORN frac14 xBoa2

2frac12Fethqbn cnubTHORN

Fethqbn 1=cnubTHORN (25)

in which the function F is given by

Fetha bubTHORN frac14 bb acosub

a2 thorn b2 2abcosub

(26)

Figure 8 is a contour plot of the open-circuit voltage normal-ized so that maxjVnj frac14 10 for the case aAfrac14 0170 andcAfrac14 0667 This is the same case as for the magnetic scalarpotential in Fig 6 All of the contours are spaced by the incre-ment Vn frac14 01 except the incomplete contours forVnfrac14 0033 and 005 Notice that the voltage has even symme-try about the lines ub frac14 0 and 1808 It has a maximum at thetop edge of the magnetic pole (qb frac14 cthorn aub frac14 0) and hassmall negative values in the area just above the center of thedisc A comparison of the lines of constant voltage in Fig 8with the streamlines for the current in Fig 6 shows that thetwo sets of lines are orthogonal in Regions I and III wherethere is no applied magnetic field This is the relationship weexpect for an electrically-thin conductor in the quasi-staticlimit We will have more to say about this plot when we dis-cuss Faradayrsquos observations in the next section of the paper

III FARADAYrsquoS OBSERVATIONS

Faraday spent a considerable amount of time experimentingwith his first dynamo A measure of this can be seen in his

diary where he devoted five pages including 23 figures to hisfirst dynamo By comparison he devoted two short paragraphswithout figures to the dynamo shown in Fig 2(b)36 Faradayrsquosmeasurements were made with a galvanometer that hereferred to as ldquoroughly made yet sufficiently delicate in itsindicationsrdquo The instrument could detect a very small currentbut the measurements were qualitative in the sense that

Fig 8 Contour plot for the open-circuit voltage VnethqbubTHORN for the case

aAfrac14 0170 and cAfrac14 0667 The voltage is normalized so that

maxjVnj frac14 10 and the contours are spaced by the increment Vn frac14 01

except the incomplete contours for Vnfrac14 0033 and 005

912 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 912

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

Faraday recorded the deflection of the galvanometer usingterms such as ldquomost stronglyrdquo and ldquoless stronglyrdquo Faradayvaried the arrangement of the elements (brushes magnet etc)in his dynamo to determine the characteristics of the currenthe thought was produced in the disc In this section using theanalysis just discussed we will examine Faradayrsquos observa-tions and his interpretation for them We will assume that ouropen-circuit voltage is proportional to the current that causedthe deflection of Faradayrsquos galvanometer26

Faraday reversed both the direction of rotation of the discand the direction of the applied magnetic field and he sawthat a reversal of either changed the sense of the deflectionof the galvanometer He reduced the applied magnetic fieldby moving one of the poles of the magnet away from thedisc and he saw the deflection decrease These observationsare in agreement with the product xBo appearing in theexpressions for the open-circuit voltage in Eqs (23)ndash(25)

Faraday was able to deduce many of the key features wesee in the contour plot for the open-circuit voltage shown inFig 8 He did this by moving the brushes to different loca-tions on the disc and observing the change in the deflectionof the galvanometer He placed one brush at the center of thedisc (on the axle) and the other brush at symmetrical pointsabout the line ub frac14 0 or 1808 By noting that the deflectionof the galvanometer was the same at the two points hededuced the even symmetry we see for the pattern about thisline With one brush at the center of the disc and the other atthe edge of the disc he varied the angular position of the lat-ter and noticed that the deflection of the galvanometer wasmaximum when the brush was opposite the magnetic pole(ub frac14 0) and decreased as the brush was moved away fromthis point (ub increased) This observation is consistent withwhat we see in Fig 8 for qbfrac14A

Faraday also varied the position of the magnetic poles Forcomparison with his observations we will determine theopen-circuit voltage when the brush in Fig 3(a) is at the topedge of the disc directly above the magnetic poleqb frac14 A ub frac14 0 For this case the open-circuit voltage Eq(23) becomes simply

Vethqb frac14 Aub frac14 0THORN frac14 xB0A2

2etha=ATHORN2 1thorn c=A

1 c=A

(27)

which we have plotted in Fig 9 as a function of polersquos sizeaA and position cA

For any size pole the voltage is maximum when the poleis as far away from the center of the disc as possible whilestill being completely over the disc or when cAfrac14 1 ndash aAIn this position the velocity of the disc relative to the pole ismaximum Also the larger the pole the larger the voltagewith maximum voltage occurring when the diameter of thepole equals the radius of the disc that is aAfrac14 05 andcAfrac14 05 When Faraday lowered the magnetic poles so asto decrease cA he noted that the ldquohellipeffect was the same [aswith the poles raised] and the deflection was very sensiblerdquoThis observation can be explained by the small size ofFaradayrsquos poles They were either square or rectangular inshape and roughly equivalent to circular poles withaAfrac14 00727 Thus his results approximately correspond tothe curve aAfrac14 01 in Fig 9 which in the central region isfairly flat so the variation with cA would have been difficultto detect in his qualitative measurements

From his measurements Faraday was able to show that cur-rent was induced in the conducting disc only when the disc

moved through the magnetic field He concluded that the cur-rent went in the radial direction from the central part of thedisc past the pole to the edge of the disc where itldquodischargedrdquo or returned in the part of the disc on either sideof the pole This description is remarkably similar to what wesee for the distribution of the surface current in Fig 6

Commenting on his first dynamo Faraday said ldquoHeretherefore was demonstrated the production of a permanent[direct or continuous] current of electricity by ordinary mag-netsrdquo28 Thus he fulfilled the goal he stated earlier of makingthe experiment of Arago a new source of electricity Faradaywas confident that he had also found the explanation forAragorsquos rotations which was as follows The magnetic fieldof the needle induced current in the rotating disc this cur-rent in turn created a magnetic field that exerted a force onthe needle causing it to rotate

IV REPEATING FARADAYrsquoS EXPERIMENTS

We repeated some of the experiments Faraday made withhis first dynamo with a two-fold objective in mind First weaim to verify the analysis presented earlier which is basedon Smythersquos approximation for the magnetic scalar potentialand second we provide quantitative results that supportFaradayrsquos qualitative observations Concerning the firstobjective we followed Faradayrsquos advice that ldquoNothing is asgood as an experiment which whilst it sets error right givesan absolute advancement in knowledgerdquo29

Figures 10(a) and 10(b) are schematic drawings for the ex-perimental model which is very similar to Faradayrsquosdynamo except that it is constructed from modern materialsThe circular copper disc of radius Afrac14 750 cm and thicknessdfrac14 086 mm was mounted on a brass axle The axle washeld in the chuck of a drill press and the speed of rotation ofthe disc which was measured with an optical tachometerwas varied by adjusting the beltstepped-pulley system of thedrill press The magnetic field was that of two neodymiumpermanent magnets (K amp J Magnetics Inc DX08) of radiusamfrac14 127 cm and length 127 cm bonded to the ends of ayoke formed from laminated steel sheets The purpose of theyoke was to direct the return field of the magnets to an areaoff the disc so that the only magnetic field applied to the discwas that between the poles Figure 10(c) is a graph of the

Fig 9 Open-circuit voltage as a function of the polersquos size aA and position

cA when the brush is at the top edge of the disc directly above the magnetic

pole at qb frac14 A ub frac14 0 The voltage is normalized so that maxjVnj frac14 10

which occurs for aAfrac14 05 cAfrac14 05 For all points on the graph the pole is

completely over the disc

913 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 913

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

magnetic field measured on the horizontal plane centeredbetween the poles of the permanent magnets The points onthis graph are averages of measurements made along threedifferent radial lines from the center of the poles The mag-netic field is seen to have a maximum value of about 108 Tat the center of the poles (ramfrac14 00) to decrease to abouthalf this value at the edges of the poles (ramfrac14 10) and torapidly decay beyond this point

The disc was connected to the external circuit by spring-loaded carbon brushes of diameter 46 mm [see Fig 10(b)]

One brush was at the center of the disc and a pair of opposingbrushes located on opposite sides of the disc and connected inparallel could be moved to various positions (qb ub) on thedisc Wires embedded in the brushes were soldered directly tothe external circuit where the open-circuit voltage V wasmeasured with a digital voltmeter The good electrical contactbetween the spring-loaded brushes and the disc resulted invery little fluctuation of the voltage as the disc rotated

One complication arose in the comparison of the theorywith the measurements The magnetic field for the theory isuniform over the area of the poles whereas the magneticfield in the experiment varies with the radial distance fromthe center of the poles To make the comparison we definedan effective radius ae for the poles used in the experimentwhich is the radius of fictitious poles with uniform magneticfield equal to Bz(rfrac14 0) that produce the same total flux as theactual poles Mathematically we express this as

pa2eBzethr frac14 0THORN frac14 2p

eth3am

rfrac140

BzethrTHORNrdr (28)

or

ae frac14 am

2

eth3am

rfrac140

BzethrTHORNrdr

a2mBzethr frac14 0THORN

26664

37775

1=2

(29)

When Eq (29) was applied to the results in Fig 10(c) thevalue aefrac14 102am was obtained A value of ae this close toam implies that the additional magnetic field outside of thepoles in Fig 10(c) just compensates for the reduction in themagnetic field under the poles For comparison of theoreticalcalculations with measurements the value ae was used for ain the theory

In all of our measurements the magnetic poles were locatedat cAfrac14 0663 and aeAfrac14 0173 These values are roughly thesame as those used in the earlier calculations For the firstmeasurement the brush was placed above the magnetic polesnear the edge of the disc (qb=A frac14 0951 ub frac14 0) and theangular speed of the disc was varied The theoretical results(curve) for the open-circuit voltage V versus angular speed xare compared with the measured results (dots) in Fig 11 The

Fig 10 Details for the experimental model which is similar to Faradayrsquos

first dynamo (a) Side view in which the brushes and external circuit are not

shown (b) Front view showing the brushes and external circuit (c) Graph of

the magnetic field measured on the horizontal plane centered between the

poles of the permanent magnets The points on the graph are the average of

measurements made along three different radial lines from the center of the

poles

Fig 11 Comparison of theoretical and measured results for the open-circuit

voltage V versus the angular frequency x with qb=A frac14 0951 ub frac14 0

(aeAfrac14 0173 and cAfrac14 0663)

914 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 914

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

theory and measurement are seen to be in good agreement andboth exhibit a linear dependence on x as expected

In the next set of measurements the angular speed of thedisc was fixed at 772 rpm and the position of the brush(qb ub) was moved over different paths on the surface ofthe disc much as Faraday did in his experimentsRepresentative results from these experiments which meas-ured the open-circuit voltage V versus position are shown inFig 12 In Fig 12(a) the radial position of the brush is fixed(qbAfrac14 0960) while the angular position ub is varied and inFig 12(b) the angular position for the brush is fixed(ub frac14 45) while the radial position qb is varied In bothcases the theory (curve) and measurement (dots) are againin good agreement Notice in Fig 12(b) that both the theoret-ical and measured results show the voltage becoming slightlynegative for qb=A lt 04

V DISCUSSION WHAT EVER BECAME OF

FARADAYrsquoS FIRST DYNAMO

In our study we wondered what became of Faradayrsquos firstdynamo and if his machine could be of use today The an-swer to this question is fairly simple As we saw from ouranalysis eddy currents are produced in the disc even when

no current is being drawn from the machine These eddy cur-rents dissipate significant energy and make the machineimpractical as a source of electric power This point is nicelyillustrated by considering the open-circuit voltage on theleft-hand side of Eq (21) to be composed of the two termson the right-hand side of this equation V frac14 Vi thorn Vs Thevoltage Vi is due to the current in the disc and can be thoughtof as the voltage produced by the surface current ~Js passingthrough the resistance of a strip of conductor of unit widthThe voltage Vs represents a source due to the applied mag-netic field traditionally called the ldquoelectromotive forcerdquo(emf) In Fig 13 the three terms V Vi and Vs (normalized sothat jVnjmax frac14 10) are shown as a function of the positionqbA along the radial line ub frac14 0 This line starts at the cen-ter of the disc passes under the pole and ends at the topedge of the disc The emf (Vsn) increases on the portion ofthe path that is under the pole and reaches a maximum ofabout Vsnfrac14 25 The voltage Vin due to the current is alwaysnegative so it subtracts from the emf making the open-circuit voltage drop to only Vnfrac14 08 at the top edge of thedisc Thus the eddy currents in the disc reduce the voltageavailable from the machine to about one-third of the emf

The inefficiency of the dynamo was also evident in themeasurements discussed in Sec IV When the disc wasrotated between the poles of the magnet a strong force resist-ing the motion was felt a consequence of Lenzrsquos law30

There was also a noticeable increase in the temperature ofthe rotating disc In a period of about one minute the tem-perature of the disc rotating at 772 rpm increased from theambient value of 246 8C (763 8F) to about 680 8C (154 8F)The energy that raises the temperature of the disc must besupplied by the mechanical source turning the axle of thedisc

Symmetric designs for the dynamo such as the one shownin Fig 2(b) do not suffer from the same problem with eddycurrents as the asymmetric design Analysis of the com-pletely symmetric structure shows that there is no conductioncurrent in the disc when the terminals are left open Andeven when current is drawn by the external circuit the cur-rent in the disc is in the radial direction that is when noeddies are formed13ndash16 Hence most attempts to make apractical unipolar or homopolar generator have used sym-metric structures These generators produce a high current at

Fig 12 Comparison of theoretical and measured results for the open-circuit

voltage V versus the position of the brush qb ub (a) Radial position fixed at

qbAfrac14 0960 while angular position is varied (b) Angular position fixed at

ub frac14 45

while radial position is varied (In both cases aeAfrac14 0173

cAfrac14 0663 and xfrac14 772 rpm)

Fig 13 The open-circuit voltage V voltage due to the current in the disc Vi

and source voltage (emf) Vs versus qbA for the radial line ub frac14 0 All of

the voltages are normalized so that maxjVnj frac14 10 and all of the parameters

are the same as those for Fig 8

915 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 915

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

a low voltage and the loss due to the current passing throughthe resistance of the brushes is of concern To lower this re-sistance some designs incorporate mercury or a liquid metal-lic alloy in the brushes as Faraday did albeit with moresophisticated designs than Faradayrsquos simple contacts31

Commercial use of even the symmetric designs has been lim-ited to very specialized applications32

The eddy-current disc brake is a structure very similar toFaradayrsquos first dynamo with the permanent magnet inFaradayrsquos design replaced by an electromagnet The conduct-ing disc is mounted on an axle that normally rotates Whenthe rotation is to be slowed or stopped the electromagnet isactivated and the kinetic energy taken from the rotationalmotion of the axle is dissipated by the eddy currents in thedisc The applications of this technology are diverse and rangefrom the brakes on the famous Japanese Shinkansen (bullettrain) to providing resistance to motion in fitness equipmentsuch as rowing machines33 The analysis for the eddy-currentdisc brake is similar to that for the Faraday disc only the em-phasis is usually on obtaining the torque on the axle versusvarious parameters such as the angular frequency and themagnetic flux34ndash37 The calculation we have used fromSmythersquos paper was originally intended for obtaining the tor-que17 In Europe linear versions of the eddy-current brake arebeing tested for railway applications38 In these devices themagnetic field is generated by a linear array of electromagnetson the car and applied to the rail The eddy currents producedin the rail slow the car and heat the rail A section of the rail isheated only as the car passes so the heating is not as severe asfor the disc brake which is heated continuously as long as thebrake is energized

As with the more common friction brakes all eddy-current brakes convert kinetic energy into heat This is unde-sirable in applications in which energy conservation is ofconcern For these applications electromagnetic regenerativebraking may be a better option Regenerative braking is usedin electric and hybrid (gasolineelectric) automobilesBatteries are used to power electric motors that propel theautomobile and when the automobile is to be slowed orstopped these motors are used as generators to charge thebatteries Thus a portion of the kinetic energy of the auto-mobile is recovered and stored for future use

The experiments Faraday made with his first dynamo gavehim insight into electromagnetic induction and among otherthings allowed him to give the first correct explanation forwhat caused Aragorsquos rotations The dynamo in its originalform was not a prototype for a practical source of electricpower However from the historical record it is clear thatFaradayrsquos first dynamo was an important step in the develop-ment of modern generators39 The fundamental principles hediscovered with his dynamo were quickly applied by othersto produce efficient generators

ACKNOWLEDGMENTS

The author would like to thank his Georgia Tech col-leagues Professors Waymond Scott Jr John Buck (bothfrom ECE) and Andrew Zangwill (Physics) for their sugges-tions for improving the manuscript

a)Electronic mail glennsmithecegatechedu1M Arago ldquoLrsquoaction que les corps aimantes et ceux qui ne le sont pas exer-

cent les uns sur les autresrdquo Ann Chim Phys 28 325 (1825) and M

Arago ldquoNote concernant les phenomenes magnetiques auxquels le

mouvement donne naissancerdquo ibid 32 213ndash223 (1826) An account in

English is in C Babbage and J F W Herschel ldquoAccount of the repetition

of M Aragorsquos experiments on magnetism manifested by various substan-

ces during the act of rotationrdquo Philos Trans Roy Soc London 115

467ndash496 (1825)2M Faraday Experimental Researches in Electricity Vol I (Taylor and

Francis London 1839 Republication Green Lion Santa Fe NM 2000)

Paragraph 83 pg 253M Faraday Faradayrsquos Diary edited by T Martin (G Bell and Sons

London 1932) Vol I Paragraphs 99ndash136 pp 381ndash3864M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraphs 83ndash100 pp 25ndash29 and 119ndash134 pp

34ndash395S P Thompson Michael Faraday His Life and Work (Cassell London

1901 Republication Elibron Classics New York 2004) pg 1236M Faraday Faradayrsquos Diary edited by T Martin (G Bell and Sons

London 1932) Vol I Paragraph 228 pg 3977M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraph 155 p 468R J Stephenson ldquoExperiments with a unipolar generator and motorrdquo

Am J Phys 5 108ndash110 (1937)9J W Then ldquoLaboratory experiments in motional electric fieldsrdquo Am J

Phys 28 557ndash559 (1960)10J W Then ldquoExperimental study of the motional electromotive forcerdquo

Am J Phys 30 411ndash415 (1962)11M J Crooks D B Litvin P W Matthews R Macaulay and J Shaw

ldquoOne-piece Faraday generator A paradoxical experiment from 1851rdquo

Am J Phys 46 729ndash731 (1978)12R D Eagleton ldquoTwo laboratory experiments involving the homopolar

generatorrdquo Am J Phys 55 621ndash623 (1987)13H H Woodson and J R Melcher Electromechanical Dynamics Part I

Discreet Systems (Wiley New York 1968) pp 286ndash28914J Van Bladel Relativity and Engineering (Springer-Verlag Berlin 1984)

pp 283ndash29115J Van Bladel ldquoRelativistic theory of rotating disksrdquo Proc IEEE 61

260ndash268 (1973)16H Montgomery ldquoCurrent flow patterns in a Faraday discrdquo Eur J Phys

25 171ndash183 (2004)17W R Smythe ldquoOn eddy currents in a rotating diskrdquo Trans AIEE 61

681ndash684 (1942)18W R Smythe Static and Dynamic Electricity 3rd ed (McGraw-Hill

New York 1968) pp 390ndash39319This type of problem is sometimes referred to as a ldquoquasi-static magnetic

field systemrdquo and it is discussed in detail in H H Woodson and J R

Melcher Electromechanical Dynamics Part I Discreet Systems (Wiley

New York 1968) pp 6ndash8 260ndash264 270ndash277 284ndash289 B19ndashB2520In the local rest frame for a point on the disc the thickness of the disc is

assumed to be much less than the skin depth in the conductor In this

frame the volume current density is then practically uniform throughout

the thickness of the disc In the quasi-static approximation the volume

current density is the same in the moving frame and in the laboratory

frame hence it is practically uniform in the laboratory frame21A I Borisenko and I E Tarapov Vector and Tensor Analysis with

Applications translated by R A Silverman (Prentice-Hall Englewood

Cliffs NJ 1968) pp 211ndash22322Smythe solves for the stream function U which is simply related to the

magnetic scalar potential UethquTHORN frac14 eth2=l0THORNUethqu z frac14 0thornTHORN His method

is briefly as follows First he considers a pole placed over an infinite con-

ducting sheet The radius of the pole is a and it is offset from the origin by

c He obtains an integral for the stream function U1 of this pole and he

evaluates this integral approximately Next he introduces a second pole

with the same total flux as the first pole but of opposite sign and he obtains

the approximate stream function U2 for this pole The radius and offset

from the origin for the second pole are ~a and ~c with ~c ~a gt A Finally

he adjusts ~a and ~c to make the superposition Ufrac14U1thornU2 satisfy the

boundary condition Ufrac14 0 at the radius qfrac14A (edge of the disc)23Normally a tube of current is a region of three-dimensional space

delineated by a side surface that is everywhere parallel to the direction of

the volume current density ~J The total current I passing through any

cross-sectional area of the tube is the samemdashthink of a pipe with the water

flowing through the pipe representing the current Here we are using the

two-dimensional version of the tube of current The current is on a surface

(on the disc) and the sides of the tube are two lines that are everywhere

parallel to the direction of the surface current density ~Js The total current

916 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 916

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

due to the current induced in the disc In the source-freeregion outside the disc the two Maxwellrsquos equations thatinvolve the magnetic field ~Bi are

r ~Bi frac14 0 (2)

and

r ~Bi 0 (3)

The volume density for the conduction current ~J is assumedto be uniform throughout the thickness of the disc with anegligible axial component Thus it can be approximated bya surface- or sheet-current density ~Js ~Jd20

The boundary condition that relates the surface currentdensity to the magnetic field is

~JsethquTHORN frac141

l0

z frac12~Biethqu z frac14 0thornTHORN ~Biethqu z frac14 0THORN

frac14 2

l0

z ~Biethqu z frac14 0thornTHORN (4)

where for the last term we have used the fact that the compo-nent of the magnetic field tangential to the surface of the dischas odd symmetry about the disc as shown in Fig 4(a)

A Magnetic scalar potential

We can represent the irrotational magnetic field Bi by ascalar magnetic potential U via21

~Bi frac14 rU (5)

This is inserted into Eq (2) to obtain Laplacersquos equation forthe potential in a source-free region

r2U frac14 0 (6)

For describing the potential at the top surface of the disc(zfrac14 0thorn) we will define the three regions on the disc shownin Fig 5 The two radial lines at ua frac14 6sin1etha=cTHORN are tan-gent to the boundary of the circular region containing theapplied magnetic field Region I (light gray) is bounded bythese lines (ua u ua) and is above the applied mag-netic field qo lt q A with qo frac14 cfrac12cosu

thornffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffietha=cTHORN2 sin2u

q Region II (dark gray) is within the

applied magnetic field ua u ua and qi q qo

with qi frac14 cfrac12cosuffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffietha=cTHORN2 sin2u

q Region III (white) is

everywhere else on the disc Using this notation the appliedmagnetic field can be expressed as

~BaethquTHORN frac14 BaethquTHORNzfrac14 B0fUfrac12q qiethuTHORN Ufrac12q qoethuTHORNgfrac12Uethuthorn uaTHORN Uethu uaTHORNz (7)

in which U(n) is the Heaviside unit-step function

Fig 3 (a) Model used in the analysis of the dynamo (b) Plan view of the

disc showing the coordinates and dimensions

Fig 4 Details for the surface current in the disc (a) Boundary condition

relating the magnetic field to the surface current density (b) Tube of surface

current showing coordinates

909 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 909

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

Smythe gives approximate expressions for the magnetic scalar potential in these regions as171822

UIethqnu z frac14 0thornTHORN frac14 UIIIethqnu z frac14 0thornTHORN frac14 xB0Acrd

2a2

nqnsinu1

q2n thorn c2

n 2cnqncosu 1

c2nq

2n thorn 1 2cnqncosu

(8)

and

UIIethqnu z frac14 0thornTHORN

frac14 xB0Acrd

2qnsinu 1 a2

n

c2nq

2n thorn 1 2cnqncosu

(9)

in which we have normalized the dimensions to the radius ofthe disc that is qn frac14 q=A an frac14 a=A etc

Figure 6 is a contour plot of the scalar potential normal-ized so that maxjUnj frac14 10 for the case aAfrac14 0170 andcAfrac14 0667 These parameters are roughly the same as thoseused for the experiments described later All of the contoursare spaced by the increment Un frac14 01 except the incom-plete contours for Un frac14 6001 and 6 005 Notice that thepotential has odd symmetry about the line u frac14 0 180

(positive on the left and negative on the right) and the poten-tial is zero along that line and along the edge of the discwhere qfrac14A

B Surface current density

We are generally more used to interpreting the surfacecurrent density than the magnetic scalar potential and theformer is easily obtained from the latter Inserting Eq (5)into Eq (4) we obtain

~JsethquTHORN frac14 2

l0

z r2Uethqu z frac14 0thornTHORN (10)

where we have use r2 to indicate that the gradient is onlywith respect to the two variables q u

The equipotential lines (Ufrac14 constant) such as those inFig 6 are also streamlines of the surface current densitythat is at every point along one of these lines the surface cur-rent is tangential to the line To show this we first note thatr2U is normal to the equipotential lines The operation

r2U ~Js frac14 2

lo

r2U ethz r2UTHORN

frac14 2

lo

z ethr2Ur2UTHORN frac14 0 (11)

shows that the surface current density is normal to r2UHence the surface current is parallel or tangential to theequipotential lines

Two adjacent equipotential lines such as the lines for U1

and U2 in Fig 4(b) bound a two-dimensional tube of surfacecurrent23 At any cross section of the tube the total current Ithrough the tube is the same To show this we first recallfrom Eq (3) that the magnetic field is irrotational so in anyregion in which the field is well defined the line integral of~Bi between two points such as from P1 to P2 in Fig 4(b) isindependent of the path of integration21 Thus

ethP2

P1

~Bi t dlsquo frac14 ethP2

P1

r2U t dlsquo frac14 ethU2 U1THORN (12)

Fig 5 Regions and coordinates used in the description of the scalar mag-

netic potential U at the top surface (zfrac14 0thorn) of the disc

Fig 6 Contour plot for the magnetic scalar potential Unethqu z frac14 0thornTHORN for

the case aAfrac14 0170 and cAfrac14 0667 The potential is normalized so that

maxjUnj frac14 10 and the contours are spaced by the increment Un frac14 01

except the incomplete contours for Unfrac146001 and 6005 The equipoten-

tial lines are also streamlines of the surface current density and the arrows

on these lines show the direction of the current Two adjacent equipotential

lines bound a tube of surface current through which the total current I is the

same at any cross section

910 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 910

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

where t is a unit vector tangent to the path After taking frac12zEq (4) t and rearranging terms we obtain

~Bi t frac14 l0

2ethz ~JsTHORN t (13)

Substituting Eq (13) into the left-hand side of Eq (12) gives

l0

2

ethP2

P1

ethz ~JsTHORN t dlsquo frac14 l0

2

ethP2

P1

~Js s dlsquo frac14 ethU2 U1THORN

(14)

in which the unit vector s frac14 z t is in the plane of the discand normal to the path as shown in Fig 4(b) Now from Eq(14) we see that the total current through any cross sectionof the tube is

I frac14ethP2

P1

~Js s dlsquo frac14 2

l0

ethU2 U1THORN (15)

which is clearly a constantIf two different tubes of current are bounded by equipoten-

tial lines separated by the same increment U the two tubeshave the same total current Thus in Fig 6 all of the tubeswith Un frac14 01 have the same total current From this ob-servation we see that the current density is largest under thepole of the magnet and smallest near the edge of the disc atangles near u frac14 180 Notice that the tubes of current areclosed and the radial component of the surface current iszero at the edge of the disc The closed tubes of current areanalogous to eddies that form around an object in flowingwater hence the name ldquoeddy currentsrdquo (also ldquoFoucaultcurrentsrdquo) Because of the finite conductivity of the disc theeddy currents dissipate energy This energy must be suppliedby the mechanical source turning the axle of the disc

C Faradayrsquos law and the open-circuit voltage

Faradayrsquos law in integral form isthornCethtTHORN

frac12~Eeth~r tTHORN thorn~veth~r tTHORN ~Beth~r tTHORN d~lsquo frac14 d

dt

eth ethSethtTHORN

~Beth~r tTHORN d~S

(16)

where we explicitly indicate that the closed contour Cbounding the open surface S can vary with time and ~v isthe velocity for a point on the contour24 We will apply

Eq (16) to the contour ABCDA in Fig 3(a) shown inmore detail in Fig 7

For this contour in the limit u frac14 xt 0 the expres-sion on the left-hand side of Eq (16) becomes

VethqbubTHORN thornethqb

qfrac140

frac12~Eethqub 0THORN thorn~vethqub 0THORN

~Bethqub 0THORN q dq (17)

The integral in Eq (17) is for the portion of the contourCD that is in the disc where the constitutive equation forthe surface current density in the laboratory frame applieswhich is

~JsethqubTHORN ~Jethqub 0THORNd

frac14 rdfrac12~Eethqub 0THORN thorn~vethqub 0THORN ~Bethqub 0THORN(18)

When Eq (18) is substituted into Eq (17) we obtain

VethqbubTHORN thorn1

rd

ethqb

qfrac140

q ~JsethqubTHORNdq (19)

for our final result for the left-hand side of Eq (16)Now consider the expression on the right-hand-side of Eq

(16) for the contour ABCDA In the time increment t thearea S in Fig 7 increases by the increment S (gray in thefigure) We will assume that the applied magnetic field ismuch greater than axial component of the magnetic field dueto the current in the disc (j~Baj jz ~Bij) and we will ignorethe latter Then the time variation in the flux of the magneticfield through S is due to the applied field ~Ba passingthrough S or

d

dt

eth ethSethtTHORN

~Beth~r tTHORN d~S frac14 limt0

1

t

ethqb

qfrac140

~BaethqubTHORN ethzTHORNq u dq

264

375 frac14 lim

t0

ut

ethqb

qfrac140

qBaethqubTHORN dq

264

375 frac14 x

ethqb

qfrac140

qBaethqubTHORN dq

(20)

After equating the left-hand- and right-hand-sides of Eq (16) for the contour ABCDAmdashequating Eqs (19) and (20)mdashweobtain the open-circuit voltage at the terminals A-B in Fig 3(a) to be

Fig 7 Details for contour C used with Faradayrsquos law

911 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 911

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

VethqbubTHORN frac14 1

rd

ethqb

qfrac140

q ~JsethqubTHORNdqthorn xethqb

qfrac140

qBaethqubTHORN dq frac14 2

l0rd

ethqb

qfrac140

1

qUethqub z frac14 0thornTHORN

udqthorn x

ethqb

qfrac140

qBaethqubTHORN dq

(21)

where in the last line we have substituted Eq (10) for ~JsTo complete the derivation for the open-circuit voltage we must substitute for the magnetic scalar potential Eqs (8) and

(9) and perform the differentiation and integration indicated in Eq (21) For points qbub in each of the three regions shownin Fig 5 the evaluation is different for example for Region I the expression to be evaluated is

VI frac14 2

l0rd

ethqi

qfrac140

1

qUI

udqthorn

ethqo

qfrac14qi

1

qUII

udqthorn

ethqb

qo

1

qUIII

udq

264

375thorn xB0

ethqo

qfrac14qi

qdq (22)

The calculations are straightforward but tedious All of the definite integrals can be evaluated in closed form25 and final resultsfor the open-circuit voltage are

VIethqbnubTHORN frac14 xBoa2

2feth1=anTHORN2frac12q2

on q2in cnethqon qinTHORNcosub thorn Fethqin cnubTHORNthornFethqbn cnubTHORN

Fethqon cnubTHORN Fethqbn 1=cnubTHORNg (23)

VIIethqbnubTHORN frac14 xBoa2

2feth1=anTHORN2frac12q2

bn q2in cnethqbn qinTHORNcosub thorn Fethqin cnubTHORN

Fethqbn 1=cnubTHORNg (24)

and

VIIIethqbnubTHORN frac14 xBoa2

2frac12Fethqbn cnubTHORN

Fethqbn 1=cnubTHORN (25)

in which the function F is given by

Fetha bubTHORN frac14 bb acosub

a2 thorn b2 2abcosub

(26)

Figure 8 is a contour plot of the open-circuit voltage normal-ized so that maxjVnj frac14 10 for the case aAfrac14 0170 andcAfrac14 0667 This is the same case as for the magnetic scalarpotential in Fig 6 All of the contours are spaced by the incre-ment Vn frac14 01 except the incomplete contours forVnfrac14 0033 and 005 Notice that the voltage has even symme-try about the lines ub frac14 0 and 1808 It has a maximum at thetop edge of the magnetic pole (qb frac14 cthorn aub frac14 0) and hassmall negative values in the area just above the center of thedisc A comparison of the lines of constant voltage in Fig 8with the streamlines for the current in Fig 6 shows that thetwo sets of lines are orthogonal in Regions I and III wherethere is no applied magnetic field This is the relationship weexpect for an electrically-thin conductor in the quasi-staticlimit We will have more to say about this plot when we dis-cuss Faradayrsquos observations in the next section of the paper

III FARADAYrsquoS OBSERVATIONS

Faraday spent a considerable amount of time experimentingwith his first dynamo A measure of this can be seen in his

diary where he devoted five pages including 23 figures to hisfirst dynamo By comparison he devoted two short paragraphswithout figures to the dynamo shown in Fig 2(b)36 Faradayrsquosmeasurements were made with a galvanometer that hereferred to as ldquoroughly made yet sufficiently delicate in itsindicationsrdquo The instrument could detect a very small currentbut the measurements were qualitative in the sense that

Fig 8 Contour plot for the open-circuit voltage VnethqbubTHORN for the case

aAfrac14 0170 and cAfrac14 0667 The voltage is normalized so that

maxjVnj frac14 10 and the contours are spaced by the increment Vn frac14 01

except the incomplete contours for Vnfrac14 0033 and 005

912 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 912

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

Faraday recorded the deflection of the galvanometer usingterms such as ldquomost stronglyrdquo and ldquoless stronglyrdquo Faradayvaried the arrangement of the elements (brushes magnet etc)in his dynamo to determine the characteristics of the currenthe thought was produced in the disc In this section using theanalysis just discussed we will examine Faradayrsquos observa-tions and his interpretation for them We will assume that ouropen-circuit voltage is proportional to the current that causedthe deflection of Faradayrsquos galvanometer26

Faraday reversed both the direction of rotation of the discand the direction of the applied magnetic field and he sawthat a reversal of either changed the sense of the deflectionof the galvanometer He reduced the applied magnetic fieldby moving one of the poles of the magnet away from thedisc and he saw the deflection decrease These observationsare in agreement with the product xBo appearing in theexpressions for the open-circuit voltage in Eqs (23)ndash(25)

Faraday was able to deduce many of the key features wesee in the contour plot for the open-circuit voltage shown inFig 8 He did this by moving the brushes to different loca-tions on the disc and observing the change in the deflectionof the galvanometer He placed one brush at the center of thedisc (on the axle) and the other brush at symmetrical pointsabout the line ub frac14 0 or 1808 By noting that the deflectionof the galvanometer was the same at the two points hededuced the even symmetry we see for the pattern about thisline With one brush at the center of the disc and the other atthe edge of the disc he varied the angular position of the lat-ter and noticed that the deflection of the galvanometer wasmaximum when the brush was opposite the magnetic pole(ub frac14 0) and decreased as the brush was moved away fromthis point (ub increased) This observation is consistent withwhat we see in Fig 8 for qbfrac14A

Faraday also varied the position of the magnetic poles Forcomparison with his observations we will determine theopen-circuit voltage when the brush in Fig 3(a) is at the topedge of the disc directly above the magnetic poleqb frac14 A ub frac14 0 For this case the open-circuit voltage Eq(23) becomes simply

Vethqb frac14 Aub frac14 0THORN frac14 xB0A2

2etha=ATHORN2 1thorn c=A

1 c=A

(27)

which we have plotted in Fig 9 as a function of polersquos sizeaA and position cA

For any size pole the voltage is maximum when the poleis as far away from the center of the disc as possible whilestill being completely over the disc or when cAfrac14 1 ndash aAIn this position the velocity of the disc relative to the pole ismaximum Also the larger the pole the larger the voltagewith maximum voltage occurring when the diameter of thepole equals the radius of the disc that is aAfrac14 05 andcAfrac14 05 When Faraday lowered the magnetic poles so asto decrease cA he noted that the ldquohellipeffect was the same [aswith the poles raised] and the deflection was very sensiblerdquoThis observation can be explained by the small size ofFaradayrsquos poles They were either square or rectangular inshape and roughly equivalent to circular poles withaAfrac14 00727 Thus his results approximately correspond tothe curve aAfrac14 01 in Fig 9 which in the central region isfairly flat so the variation with cA would have been difficultto detect in his qualitative measurements

From his measurements Faraday was able to show that cur-rent was induced in the conducting disc only when the disc

moved through the magnetic field He concluded that the cur-rent went in the radial direction from the central part of thedisc past the pole to the edge of the disc where itldquodischargedrdquo or returned in the part of the disc on either sideof the pole This description is remarkably similar to what wesee for the distribution of the surface current in Fig 6

Commenting on his first dynamo Faraday said ldquoHeretherefore was demonstrated the production of a permanent[direct or continuous] current of electricity by ordinary mag-netsrdquo28 Thus he fulfilled the goal he stated earlier of makingthe experiment of Arago a new source of electricity Faradaywas confident that he had also found the explanation forAragorsquos rotations which was as follows The magnetic fieldof the needle induced current in the rotating disc this cur-rent in turn created a magnetic field that exerted a force onthe needle causing it to rotate

IV REPEATING FARADAYrsquoS EXPERIMENTS

We repeated some of the experiments Faraday made withhis first dynamo with a two-fold objective in mind First weaim to verify the analysis presented earlier which is basedon Smythersquos approximation for the magnetic scalar potentialand second we provide quantitative results that supportFaradayrsquos qualitative observations Concerning the firstobjective we followed Faradayrsquos advice that ldquoNothing is asgood as an experiment which whilst it sets error right givesan absolute advancement in knowledgerdquo29

Figures 10(a) and 10(b) are schematic drawings for the ex-perimental model which is very similar to Faradayrsquosdynamo except that it is constructed from modern materialsThe circular copper disc of radius Afrac14 750 cm and thicknessdfrac14 086 mm was mounted on a brass axle The axle washeld in the chuck of a drill press and the speed of rotation ofthe disc which was measured with an optical tachometerwas varied by adjusting the beltstepped-pulley system of thedrill press The magnetic field was that of two neodymiumpermanent magnets (K amp J Magnetics Inc DX08) of radiusamfrac14 127 cm and length 127 cm bonded to the ends of ayoke formed from laminated steel sheets The purpose of theyoke was to direct the return field of the magnets to an areaoff the disc so that the only magnetic field applied to the discwas that between the poles Figure 10(c) is a graph of the

Fig 9 Open-circuit voltage as a function of the polersquos size aA and position

cA when the brush is at the top edge of the disc directly above the magnetic

pole at qb frac14 A ub frac14 0 The voltage is normalized so that maxjVnj frac14 10

which occurs for aAfrac14 05 cAfrac14 05 For all points on the graph the pole is

completely over the disc

913 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 913

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

magnetic field measured on the horizontal plane centeredbetween the poles of the permanent magnets The points onthis graph are averages of measurements made along threedifferent radial lines from the center of the poles The mag-netic field is seen to have a maximum value of about 108 Tat the center of the poles (ramfrac14 00) to decrease to abouthalf this value at the edges of the poles (ramfrac14 10) and torapidly decay beyond this point

The disc was connected to the external circuit by spring-loaded carbon brushes of diameter 46 mm [see Fig 10(b)]

One brush was at the center of the disc and a pair of opposingbrushes located on opposite sides of the disc and connected inparallel could be moved to various positions (qb ub) on thedisc Wires embedded in the brushes were soldered directly tothe external circuit where the open-circuit voltage V wasmeasured with a digital voltmeter The good electrical contactbetween the spring-loaded brushes and the disc resulted invery little fluctuation of the voltage as the disc rotated

One complication arose in the comparison of the theorywith the measurements The magnetic field for the theory isuniform over the area of the poles whereas the magneticfield in the experiment varies with the radial distance fromthe center of the poles To make the comparison we definedan effective radius ae for the poles used in the experimentwhich is the radius of fictitious poles with uniform magneticfield equal to Bz(rfrac14 0) that produce the same total flux as theactual poles Mathematically we express this as

pa2eBzethr frac14 0THORN frac14 2p

eth3am

rfrac140

BzethrTHORNrdr (28)

or

ae frac14 am

2

eth3am

rfrac140

BzethrTHORNrdr

a2mBzethr frac14 0THORN

26664

37775

1=2

(29)

When Eq (29) was applied to the results in Fig 10(c) thevalue aefrac14 102am was obtained A value of ae this close toam implies that the additional magnetic field outside of thepoles in Fig 10(c) just compensates for the reduction in themagnetic field under the poles For comparison of theoreticalcalculations with measurements the value ae was used for ain the theory

In all of our measurements the magnetic poles were locatedat cAfrac14 0663 and aeAfrac14 0173 These values are roughly thesame as those used in the earlier calculations For the firstmeasurement the brush was placed above the magnetic polesnear the edge of the disc (qb=A frac14 0951 ub frac14 0) and theangular speed of the disc was varied The theoretical results(curve) for the open-circuit voltage V versus angular speed xare compared with the measured results (dots) in Fig 11 The

Fig 10 Details for the experimental model which is similar to Faradayrsquos

first dynamo (a) Side view in which the brushes and external circuit are not

shown (b) Front view showing the brushes and external circuit (c) Graph of

the magnetic field measured on the horizontal plane centered between the

poles of the permanent magnets The points on the graph are the average of

measurements made along three different radial lines from the center of the

poles

Fig 11 Comparison of theoretical and measured results for the open-circuit

voltage V versus the angular frequency x with qb=A frac14 0951 ub frac14 0

(aeAfrac14 0173 and cAfrac14 0663)

914 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 914

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

theory and measurement are seen to be in good agreement andboth exhibit a linear dependence on x as expected

In the next set of measurements the angular speed of thedisc was fixed at 772 rpm and the position of the brush(qb ub) was moved over different paths on the surface ofthe disc much as Faraday did in his experimentsRepresentative results from these experiments which meas-ured the open-circuit voltage V versus position are shown inFig 12 In Fig 12(a) the radial position of the brush is fixed(qbAfrac14 0960) while the angular position ub is varied and inFig 12(b) the angular position for the brush is fixed(ub frac14 45) while the radial position qb is varied In bothcases the theory (curve) and measurement (dots) are againin good agreement Notice in Fig 12(b) that both the theoret-ical and measured results show the voltage becoming slightlynegative for qb=A lt 04

V DISCUSSION WHAT EVER BECAME OF

FARADAYrsquoS FIRST DYNAMO

In our study we wondered what became of Faradayrsquos firstdynamo and if his machine could be of use today The an-swer to this question is fairly simple As we saw from ouranalysis eddy currents are produced in the disc even when

no current is being drawn from the machine These eddy cur-rents dissipate significant energy and make the machineimpractical as a source of electric power This point is nicelyillustrated by considering the open-circuit voltage on theleft-hand side of Eq (21) to be composed of the two termson the right-hand side of this equation V frac14 Vi thorn Vs Thevoltage Vi is due to the current in the disc and can be thoughtof as the voltage produced by the surface current ~Js passingthrough the resistance of a strip of conductor of unit widthThe voltage Vs represents a source due to the applied mag-netic field traditionally called the ldquoelectromotive forcerdquo(emf) In Fig 13 the three terms V Vi and Vs (normalized sothat jVnjmax frac14 10) are shown as a function of the positionqbA along the radial line ub frac14 0 This line starts at the cen-ter of the disc passes under the pole and ends at the topedge of the disc The emf (Vsn) increases on the portion ofthe path that is under the pole and reaches a maximum ofabout Vsnfrac14 25 The voltage Vin due to the current is alwaysnegative so it subtracts from the emf making the open-circuit voltage drop to only Vnfrac14 08 at the top edge of thedisc Thus the eddy currents in the disc reduce the voltageavailable from the machine to about one-third of the emf

The inefficiency of the dynamo was also evident in themeasurements discussed in Sec IV When the disc wasrotated between the poles of the magnet a strong force resist-ing the motion was felt a consequence of Lenzrsquos law30

There was also a noticeable increase in the temperature ofthe rotating disc In a period of about one minute the tem-perature of the disc rotating at 772 rpm increased from theambient value of 246 8C (763 8F) to about 680 8C (154 8F)The energy that raises the temperature of the disc must besupplied by the mechanical source turning the axle of thedisc

Symmetric designs for the dynamo such as the one shownin Fig 2(b) do not suffer from the same problem with eddycurrents as the asymmetric design Analysis of the com-pletely symmetric structure shows that there is no conductioncurrent in the disc when the terminals are left open Andeven when current is drawn by the external circuit the cur-rent in the disc is in the radial direction that is when noeddies are formed13ndash16 Hence most attempts to make apractical unipolar or homopolar generator have used sym-metric structures These generators produce a high current at

Fig 12 Comparison of theoretical and measured results for the open-circuit

voltage V versus the position of the brush qb ub (a) Radial position fixed at

qbAfrac14 0960 while angular position is varied (b) Angular position fixed at

ub frac14 45

while radial position is varied (In both cases aeAfrac14 0173

cAfrac14 0663 and xfrac14 772 rpm)

Fig 13 The open-circuit voltage V voltage due to the current in the disc Vi

and source voltage (emf) Vs versus qbA for the radial line ub frac14 0 All of

the voltages are normalized so that maxjVnj frac14 10 and all of the parameters

are the same as those for Fig 8

915 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 915

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

a low voltage and the loss due to the current passing throughthe resistance of the brushes is of concern To lower this re-sistance some designs incorporate mercury or a liquid metal-lic alloy in the brushes as Faraday did albeit with moresophisticated designs than Faradayrsquos simple contacts31

Commercial use of even the symmetric designs has been lim-ited to very specialized applications32

The eddy-current disc brake is a structure very similar toFaradayrsquos first dynamo with the permanent magnet inFaradayrsquos design replaced by an electromagnet The conduct-ing disc is mounted on an axle that normally rotates Whenthe rotation is to be slowed or stopped the electromagnet isactivated and the kinetic energy taken from the rotationalmotion of the axle is dissipated by the eddy currents in thedisc The applications of this technology are diverse and rangefrom the brakes on the famous Japanese Shinkansen (bullettrain) to providing resistance to motion in fitness equipmentsuch as rowing machines33 The analysis for the eddy-currentdisc brake is similar to that for the Faraday disc only the em-phasis is usually on obtaining the torque on the axle versusvarious parameters such as the angular frequency and themagnetic flux34ndash37 The calculation we have used fromSmythersquos paper was originally intended for obtaining the tor-que17 In Europe linear versions of the eddy-current brake arebeing tested for railway applications38 In these devices themagnetic field is generated by a linear array of electromagnetson the car and applied to the rail The eddy currents producedin the rail slow the car and heat the rail A section of the rail isheated only as the car passes so the heating is not as severe asfor the disc brake which is heated continuously as long as thebrake is energized

As with the more common friction brakes all eddy-current brakes convert kinetic energy into heat This is unde-sirable in applications in which energy conservation is ofconcern For these applications electromagnetic regenerativebraking may be a better option Regenerative braking is usedin electric and hybrid (gasolineelectric) automobilesBatteries are used to power electric motors that propel theautomobile and when the automobile is to be slowed orstopped these motors are used as generators to charge thebatteries Thus a portion of the kinetic energy of the auto-mobile is recovered and stored for future use

The experiments Faraday made with his first dynamo gavehim insight into electromagnetic induction and among otherthings allowed him to give the first correct explanation forwhat caused Aragorsquos rotations The dynamo in its originalform was not a prototype for a practical source of electricpower However from the historical record it is clear thatFaradayrsquos first dynamo was an important step in the develop-ment of modern generators39 The fundamental principles hediscovered with his dynamo were quickly applied by othersto produce efficient generators

ACKNOWLEDGMENTS

The author would like to thank his Georgia Tech col-leagues Professors Waymond Scott Jr John Buck (bothfrom ECE) and Andrew Zangwill (Physics) for their sugges-tions for improving the manuscript

a)Electronic mail glennsmithecegatechedu1M Arago ldquoLrsquoaction que les corps aimantes et ceux qui ne le sont pas exer-

cent les uns sur les autresrdquo Ann Chim Phys 28 325 (1825) and M

Arago ldquoNote concernant les phenomenes magnetiques auxquels le

mouvement donne naissancerdquo ibid 32 213ndash223 (1826) An account in

English is in C Babbage and J F W Herschel ldquoAccount of the repetition

of M Aragorsquos experiments on magnetism manifested by various substan-

ces during the act of rotationrdquo Philos Trans Roy Soc London 115

467ndash496 (1825)2M Faraday Experimental Researches in Electricity Vol I (Taylor and

Francis London 1839 Republication Green Lion Santa Fe NM 2000)

Paragraph 83 pg 253M Faraday Faradayrsquos Diary edited by T Martin (G Bell and Sons

London 1932) Vol I Paragraphs 99ndash136 pp 381ndash3864M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraphs 83ndash100 pp 25ndash29 and 119ndash134 pp

34ndash395S P Thompson Michael Faraday His Life and Work (Cassell London

1901 Republication Elibron Classics New York 2004) pg 1236M Faraday Faradayrsquos Diary edited by T Martin (G Bell and Sons

London 1932) Vol I Paragraph 228 pg 3977M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraph 155 p 468R J Stephenson ldquoExperiments with a unipolar generator and motorrdquo

Am J Phys 5 108ndash110 (1937)9J W Then ldquoLaboratory experiments in motional electric fieldsrdquo Am J

Phys 28 557ndash559 (1960)10J W Then ldquoExperimental study of the motional electromotive forcerdquo

Am J Phys 30 411ndash415 (1962)11M J Crooks D B Litvin P W Matthews R Macaulay and J Shaw

ldquoOne-piece Faraday generator A paradoxical experiment from 1851rdquo

Am J Phys 46 729ndash731 (1978)12R D Eagleton ldquoTwo laboratory experiments involving the homopolar

generatorrdquo Am J Phys 55 621ndash623 (1987)13H H Woodson and J R Melcher Electromechanical Dynamics Part I

Discreet Systems (Wiley New York 1968) pp 286ndash28914J Van Bladel Relativity and Engineering (Springer-Verlag Berlin 1984)

pp 283ndash29115J Van Bladel ldquoRelativistic theory of rotating disksrdquo Proc IEEE 61

260ndash268 (1973)16H Montgomery ldquoCurrent flow patterns in a Faraday discrdquo Eur J Phys

25 171ndash183 (2004)17W R Smythe ldquoOn eddy currents in a rotating diskrdquo Trans AIEE 61

681ndash684 (1942)18W R Smythe Static and Dynamic Electricity 3rd ed (McGraw-Hill

New York 1968) pp 390ndash39319This type of problem is sometimes referred to as a ldquoquasi-static magnetic

field systemrdquo and it is discussed in detail in H H Woodson and J R

Melcher Electromechanical Dynamics Part I Discreet Systems (Wiley

New York 1968) pp 6ndash8 260ndash264 270ndash277 284ndash289 B19ndashB2520In the local rest frame for a point on the disc the thickness of the disc is

assumed to be much less than the skin depth in the conductor In this

frame the volume current density is then practically uniform throughout

the thickness of the disc In the quasi-static approximation the volume

current density is the same in the moving frame and in the laboratory

frame hence it is practically uniform in the laboratory frame21A I Borisenko and I E Tarapov Vector and Tensor Analysis with

Applications translated by R A Silverman (Prentice-Hall Englewood

Cliffs NJ 1968) pp 211ndash22322Smythe solves for the stream function U which is simply related to the

magnetic scalar potential UethquTHORN frac14 eth2=l0THORNUethqu z frac14 0thornTHORN His method

is briefly as follows First he considers a pole placed over an infinite con-

ducting sheet The radius of the pole is a and it is offset from the origin by

c He obtains an integral for the stream function U1 of this pole and he

evaluates this integral approximately Next he introduces a second pole

with the same total flux as the first pole but of opposite sign and he obtains

the approximate stream function U2 for this pole The radius and offset

from the origin for the second pole are ~a and ~c with ~c ~a gt A Finally

he adjusts ~a and ~c to make the superposition Ufrac14U1thornU2 satisfy the

boundary condition Ufrac14 0 at the radius qfrac14A (edge of the disc)23Normally a tube of current is a region of three-dimensional space

delineated by a side surface that is everywhere parallel to the direction of

the volume current density ~J The total current I passing through any

cross-sectional area of the tube is the samemdashthink of a pipe with the water

flowing through the pipe representing the current Here we are using the

two-dimensional version of the tube of current The current is on a surface

(on the disc) and the sides of the tube are two lines that are everywhere

parallel to the direction of the surface current density ~Js The total current

916 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 916

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

Smythe gives approximate expressions for the magnetic scalar potential in these regions as171822

UIethqnu z frac14 0thornTHORN frac14 UIIIethqnu z frac14 0thornTHORN frac14 xB0Acrd

2a2

nqnsinu1

q2n thorn c2

n 2cnqncosu 1

c2nq

2n thorn 1 2cnqncosu

(8)

and

UIIethqnu z frac14 0thornTHORN

frac14 xB0Acrd

2qnsinu 1 a2

n

c2nq

2n thorn 1 2cnqncosu

(9)

in which we have normalized the dimensions to the radius ofthe disc that is qn frac14 q=A an frac14 a=A etc

Figure 6 is a contour plot of the scalar potential normal-ized so that maxjUnj frac14 10 for the case aAfrac14 0170 andcAfrac14 0667 These parameters are roughly the same as thoseused for the experiments described later All of the contoursare spaced by the increment Un frac14 01 except the incom-plete contours for Un frac14 6001 and 6 005 Notice that thepotential has odd symmetry about the line u frac14 0 180

(positive on the left and negative on the right) and the poten-tial is zero along that line and along the edge of the discwhere qfrac14A

B Surface current density

We are generally more used to interpreting the surfacecurrent density than the magnetic scalar potential and theformer is easily obtained from the latter Inserting Eq (5)into Eq (4) we obtain

~JsethquTHORN frac14 2

l0

z r2Uethqu z frac14 0thornTHORN (10)

where we have use r2 to indicate that the gradient is onlywith respect to the two variables q u

The equipotential lines (Ufrac14 constant) such as those inFig 6 are also streamlines of the surface current densitythat is at every point along one of these lines the surface cur-rent is tangential to the line To show this we first note thatr2U is normal to the equipotential lines The operation

r2U ~Js frac14 2

lo

r2U ethz r2UTHORN

frac14 2

lo

z ethr2Ur2UTHORN frac14 0 (11)

shows that the surface current density is normal to r2UHence the surface current is parallel or tangential to theequipotential lines

Two adjacent equipotential lines such as the lines for U1

and U2 in Fig 4(b) bound a two-dimensional tube of surfacecurrent23 At any cross section of the tube the total current Ithrough the tube is the same To show this we first recallfrom Eq (3) that the magnetic field is irrotational so in anyregion in which the field is well defined the line integral of~Bi between two points such as from P1 to P2 in Fig 4(b) isindependent of the path of integration21 Thus

ethP2

P1

~Bi t dlsquo frac14 ethP2

P1

r2U t dlsquo frac14 ethU2 U1THORN (12)

Fig 5 Regions and coordinates used in the description of the scalar mag-

netic potential U at the top surface (zfrac14 0thorn) of the disc

Fig 6 Contour plot for the magnetic scalar potential Unethqu z frac14 0thornTHORN for

the case aAfrac14 0170 and cAfrac14 0667 The potential is normalized so that

maxjUnj frac14 10 and the contours are spaced by the increment Un frac14 01

except the incomplete contours for Unfrac146001 and 6005 The equipoten-

tial lines are also streamlines of the surface current density and the arrows

on these lines show the direction of the current Two adjacent equipotential

lines bound a tube of surface current through which the total current I is the

same at any cross section

910 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 910

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

where t is a unit vector tangent to the path After taking frac12zEq (4) t and rearranging terms we obtain

~Bi t frac14 l0

2ethz ~JsTHORN t (13)

Substituting Eq (13) into the left-hand side of Eq (12) gives

l0

2

ethP2

P1

ethz ~JsTHORN t dlsquo frac14 l0

2

ethP2

P1

~Js s dlsquo frac14 ethU2 U1THORN

(14)

in which the unit vector s frac14 z t is in the plane of the discand normal to the path as shown in Fig 4(b) Now from Eq(14) we see that the total current through any cross sectionof the tube is

I frac14ethP2

P1

~Js s dlsquo frac14 2

l0

ethU2 U1THORN (15)

which is clearly a constantIf two different tubes of current are bounded by equipoten-

tial lines separated by the same increment U the two tubeshave the same total current Thus in Fig 6 all of the tubeswith Un frac14 01 have the same total current From this ob-servation we see that the current density is largest under thepole of the magnet and smallest near the edge of the disc atangles near u frac14 180 Notice that the tubes of current areclosed and the radial component of the surface current iszero at the edge of the disc The closed tubes of current areanalogous to eddies that form around an object in flowingwater hence the name ldquoeddy currentsrdquo (also ldquoFoucaultcurrentsrdquo) Because of the finite conductivity of the disc theeddy currents dissipate energy This energy must be suppliedby the mechanical source turning the axle of the disc

C Faradayrsquos law and the open-circuit voltage

Faradayrsquos law in integral form isthornCethtTHORN

frac12~Eeth~r tTHORN thorn~veth~r tTHORN ~Beth~r tTHORN d~lsquo frac14 d

dt

eth ethSethtTHORN

~Beth~r tTHORN d~S

(16)

where we explicitly indicate that the closed contour Cbounding the open surface S can vary with time and ~v isthe velocity for a point on the contour24 We will apply

Eq (16) to the contour ABCDA in Fig 3(a) shown inmore detail in Fig 7

For this contour in the limit u frac14 xt 0 the expres-sion on the left-hand side of Eq (16) becomes

VethqbubTHORN thornethqb

qfrac140

frac12~Eethqub 0THORN thorn~vethqub 0THORN

~Bethqub 0THORN q dq (17)

The integral in Eq (17) is for the portion of the contourCD that is in the disc where the constitutive equation forthe surface current density in the laboratory frame applieswhich is

~JsethqubTHORN ~Jethqub 0THORNd

frac14 rdfrac12~Eethqub 0THORN thorn~vethqub 0THORN ~Bethqub 0THORN(18)

When Eq (18) is substituted into Eq (17) we obtain

VethqbubTHORN thorn1

rd

ethqb

qfrac140

q ~JsethqubTHORNdq (19)

for our final result for the left-hand side of Eq (16)Now consider the expression on the right-hand-side of Eq

(16) for the contour ABCDA In the time increment t thearea S in Fig 7 increases by the increment S (gray in thefigure) We will assume that the applied magnetic field ismuch greater than axial component of the magnetic field dueto the current in the disc (j~Baj jz ~Bij) and we will ignorethe latter Then the time variation in the flux of the magneticfield through S is due to the applied field ~Ba passingthrough S or

d

dt

eth ethSethtTHORN

~Beth~r tTHORN d~S frac14 limt0

1

t

ethqb

qfrac140

~BaethqubTHORN ethzTHORNq u dq

264

375 frac14 lim

t0

ut

ethqb

qfrac140

qBaethqubTHORN dq

264

375 frac14 x

ethqb

qfrac140

qBaethqubTHORN dq

(20)

After equating the left-hand- and right-hand-sides of Eq (16) for the contour ABCDAmdashequating Eqs (19) and (20)mdashweobtain the open-circuit voltage at the terminals A-B in Fig 3(a) to be

Fig 7 Details for contour C used with Faradayrsquos law

911 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 911

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

VethqbubTHORN frac14 1

rd

ethqb

qfrac140

q ~JsethqubTHORNdqthorn xethqb

qfrac140

qBaethqubTHORN dq frac14 2

l0rd

ethqb

qfrac140

1

qUethqub z frac14 0thornTHORN

udqthorn x

ethqb

qfrac140

qBaethqubTHORN dq

(21)

where in the last line we have substituted Eq (10) for ~JsTo complete the derivation for the open-circuit voltage we must substitute for the magnetic scalar potential Eqs (8) and

(9) and perform the differentiation and integration indicated in Eq (21) For points qbub in each of the three regions shownin Fig 5 the evaluation is different for example for Region I the expression to be evaluated is

VI frac14 2

l0rd

ethqi

qfrac140

1

qUI

udqthorn

ethqo

qfrac14qi

1

qUII

udqthorn

ethqb

qo

1

qUIII

udq

264

375thorn xB0

ethqo

qfrac14qi

qdq (22)

The calculations are straightforward but tedious All of the definite integrals can be evaluated in closed form25 and final resultsfor the open-circuit voltage are

VIethqbnubTHORN frac14 xBoa2

2feth1=anTHORN2frac12q2

on q2in cnethqon qinTHORNcosub thorn Fethqin cnubTHORNthornFethqbn cnubTHORN

Fethqon cnubTHORN Fethqbn 1=cnubTHORNg (23)

VIIethqbnubTHORN frac14 xBoa2

2feth1=anTHORN2frac12q2

bn q2in cnethqbn qinTHORNcosub thorn Fethqin cnubTHORN

Fethqbn 1=cnubTHORNg (24)

and

VIIIethqbnubTHORN frac14 xBoa2

2frac12Fethqbn cnubTHORN

Fethqbn 1=cnubTHORN (25)

in which the function F is given by

Fetha bubTHORN frac14 bb acosub

a2 thorn b2 2abcosub

(26)

Figure 8 is a contour plot of the open-circuit voltage normal-ized so that maxjVnj frac14 10 for the case aAfrac14 0170 andcAfrac14 0667 This is the same case as for the magnetic scalarpotential in Fig 6 All of the contours are spaced by the incre-ment Vn frac14 01 except the incomplete contours forVnfrac14 0033 and 005 Notice that the voltage has even symme-try about the lines ub frac14 0 and 1808 It has a maximum at thetop edge of the magnetic pole (qb frac14 cthorn aub frac14 0) and hassmall negative values in the area just above the center of thedisc A comparison of the lines of constant voltage in Fig 8with the streamlines for the current in Fig 6 shows that thetwo sets of lines are orthogonal in Regions I and III wherethere is no applied magnetic field This is the relationship weexpect for an electrically-thin conductor in the quasi-staticlimit We will have more to say about this plot when we dis-cuss Faradayrsquos observations in the next section of the paper

III FARADAYrsquoS OBSERVATIONS

Faraday spent a considerable amount of time experimentingwith his first dynamo A measure of this can be seen in his

diary where he devoted five pages including 23 figures to hisfirst dynamo By comparison he devoted two short paragraphswithout figures to the dynamo shown in Fig 2(b)36 Faradayrsquosmeasurements were made with a galvanometer that hereferred to as ldquoroughly made yet sufficiently delicate in itsindicationsrdquo The instrument could detect a very small currentbut the measurements were qualitative in the sense that

Fig 8 Contour plot for the open-circuit voltage VnethqbubTHORN for the case

aAfrac14 0170 and cAfrac14 0667 The voltage is normalized so that

maxjVnj frac14 10 and the contours are spaced by the increment Vn frac14 01

except the incomplete contours for Vnfrac14 0033 and 005

912 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 912

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

Faraday recorded the deflection of the galvanometer usingterms such as ldquomost stronglyrdquo and ldquoless stronglyrdquo Faradayvaried the arrangement of the elements (brushes magnet etc)in his dynamo to determine the characteristics of the currenthe thought was produced in the disc In this section using theanalysis just discussed we will examine Faradayrsquos observa-tions and his interpretation for them We will assume that ouropen-circuit voltage is proportional to the current that causedthe deflection of Faradayrsquos galvanometer26

Faraday reversed both the direction of rotation of the discand the direction of the applied magnetic field and he sawthat a reversal of either changed the sense of the deflectionof the galvanometer He reduced the applied magnetic fieldby moving one of the poles of the magnet away from thedisc and he saw the deflection decrease These observationsare in agreement with the product xBo appearing in theexpressions for the open-circuit voltage in Eqs (23)ndash(25)

Faraday was able to deduce many of the key features wesee in the contour plot for the open-circuit voltage shown inFig 8 He did this by moving the brushes to different loca-tions on the disc and observing the change in the deflectionof the galvanometer He placed one brush at the center of thedisc (on the axle) and the other brush at symmetrical pointsabout the line ub frac14 0 or 1808 By noting that the deflectionof the galvanometer was the same at the two points hededuced the even symmetry we see for the pattern about thisline With one brush at the center of the disc and the other atthe edge of the disc he varied the angular position of the lat-ter and noticed that the deflection of the galvanometer wasmaximum when the brush was opposite the magnetic pole(ub frac14 0) and decreased as the brush was moved away fromthis point (ub increased) This observation is consistent withwhat we see in Fig 8 for qbfrac14A

Faraday also varied the position of the magnetic poles Forcomparison with his observations we will determine theopen-circuit voltage when the brush in Fig 3(a) is at the topedge of the disc directly above the magnetic poleqb frac14 A ub frac14 0 For this case the open-circuit voltage Eq(23) becomes simply

Vethqb frac14 Aub frac14 0THORN frac14 xB0A2

2etha=ATHORN2 1thorn c=A

1 c=A

(27)

which we have plotted in Fig 9 as a function of polersquos sizeaA and position cA

For any size pole the voltage is maximum when the poleis as far away from the center of the disc as possible whilestill being completely over the disc or when cAfrac14 1 ndash aAIn this position the velocity of the disc relative to the pole ismaximum Also the larger the pole the larger the voltagewith maximum voltage occurring when the diameter of thepole equals the radius of the disc that is aAfrac14 05 andcAfrac14 05 When Faraday lowered the magnetic poles so asto decrease cA he noted that the ldquohellipeffect was the same [aswith the poles raised] and the deflection was very sensiblerdquoThis observation can be explained by the small size ofFaradayrsquos poles They were either square or rectangular inshape and roughly equivalent to circular poles withaAfrac14 00727 Thus his results approximately correspond tothe curve aAfrac14 01 in Fig 9 which in the central region isfairly flat so the variation with cA would have been difficultto detect in his qualitative measurements

From his measurements Faraday was able to show that cur-rent was induced in the conducting disc only when the disc

moved through the magnetic field He concluded that the cur-rent went in the radial direction from the central part of thedisc past the pole to the edge of the disc where itldquodischargedrdquo or returned in the part of the disc on either sideof the pole This description is remarkably similar to what wesee for the distribution of the surface current in Fig 6

Commenting on his first dynamo Faraday said ldquoHeretherefore was demonstrated the production of a permanent[direct or continuous] current of electricity by ordinary mag-netsrdquo28 Thus he fulfilled the goal he stated earlier of makingthe experiment of Arago a new source of electricity Faradaywas confident that he had also found the explanation forAragorsquos rotations which was as follows The magnetic fieldof the needle induced current in the rotating disc this cur-rent in turn created a magnetic field that exerted a force onthe needle causing it to rotate

IV REPEATING FARADAYrsquoS EXPERIMENTS

We repeated some of the experiments Faraday made withhis first dynamo with a two-fold objective in mind First weaim to verify the analysis presented earlier which is basedon Smythersquos approximation for the magnetic scalar potentialand second we provide quantitative results that supportFaradayrsquos qualitative observations Concerning the firstobjective we followed Faradayrsquos advice that ldquoNothing is asgood as an experiment which whilst it sets error right givesan absolute advancement in knowledgerdquo29

Figures 10(a) and 10(b) are schematic drawings for the ex-perimental model which is very similar to Faradayrsquosdynamo except that it is constructed from modern materialsThe circular copper disc of radius Afrac14 750 cm and thicknessdfrac14 086 mm was mounted on a brass axle The axle washeld in the chuck of a drill press and the speed of rotation ofthe disc which was measured with an optical tachometerwas varied by adjusting the beltstepped-pulley system of thedrill press The magnetic field was that of two neodymiumpermanent magnets (K amp J Magnetics Inc DX08) of radiusamfrac14 127 cm and length 127 cm bonded to the ends of ayoke formed from laminated steel sheets The purpose of theyoke was to direct the return field of the magnets to an areaoff the disc so that the only magnetic field applied to the discwas that between the poles Figure 10(c) is a graph of the

Fig 9 Open-circuit voltage as a function of the polersquos size aA and position

cA when the brush is at the top edge of the disc directly above the magnetic

pole at qb frac14 A ub frac14 0 The voltage is normalized so that maxjVnj frac14 10

which occurs for aAfrac14 05 cAfrac14 05 For all points on the graph the pole is

completely over the disc

913 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 913

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

magnetic field measured on the horizontal plane centeredbetween the poles of the permanent magnets The points onthis graph are averages of measurements made along threedifferent radial lines from the center of the poles The mag-netic field is seen to have a maximum value of about 108 Tat the center of the poles (ramfrac14 00) to decrease to abouthalf this value at the edges of the poles (ramfrac14 10) and torapidly decay beyond this point

The disc was connected to the external circuit by spring-loaded carbon brushes of diameter 46 mm [see Fig 10(b)]

One brush was at the center of the disc and a pair of opposingbrushes located on opposite sides of the disc and connected inparallel could be moved to various positions (qb ub) on thedisc Wires embedded in the brushes were soldered directly tothe external circuit where the open-circuit voltage V wasmeasured with a digital voltmeter The good electrical contactbetween the spring-loaded brushes and the disc resulted invery little fluctuation of the voltage as the disc rotated

One complication arose in the comparison of the theorywith the measurements The magnetic field for the theory isuniform over the area of the poles whereas the magneticfield in the experiment varies with the radial distance fromthe center of the poles To make the comparison we definedan effective radius ae for the poles used in the experimentwhich is the radius of fictitious poles with uniform magneticfield equal to Bz(rfrac14 0) that produce the same total flux as theactual poles Mathematically we express this as

pa2eBzethr frac14 0THORN frac14 2p

eth3am

rfrac140

BzethrTHORNrdr (28)

or

ae frac14 am

2

eth3am

rfrac140

BzethrTHORNrdr

a2mBzethr frac14 0THORN

26664

37775

1=2

(29)

When Eq (29) was applied to the results in Fig 10(c) thevalue aefrac14 102am was obtained A value of ae this close toam implies that the additional magnetic field outside of thepoles in Fig 10(c) just compensates for the reduction in themagnetic field under the poles For comparison of theoreticalcalculations with measurements the value ae was used for ain the theory

In all of our measurements the magnetic poles were locatedat cAfrac14 0663 and aeAfrac14 0173 These values are roughly thesame as those used in the earlier calculations For the firstmeasurement the brush was placed above the magnetic polesnear the edge of the disc (qb=A frac14 0951 ub frac14 0) and theangular speed of the disc was varied The theoretical results(curve) for the open-circuit voltage V versus angular speed xare compared with the measured results (dots) in Fig 11 The

Fig 10 Details for the experimental model which is similar to Faradayrsquos

first dynamo (a) Side view in which the brushes and external circuit are not

shown (b) Front view showing the brushes and external circuit (c) Graph of

the magnetic field measured on the horizontal plane centered between the

poles of the permanent magnets The points on the graph are the average of

measurements made along three different radial lines from the center of the

poles

Fig 11 Comparison of theoretical and measured results for the open-circuit

voltage V versus the angular frequency x with qb=A frac14 0951 ub frac14 0

(aeAfrac14 0173 and cAfrac14 0663)

914 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 914

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

theory and measurement are seen to be in good agreement andboth exhibit a linear dependence on x as expected

In the next set of measurements the angular speed of thedisc was fixed at 772 rpm and the position of the brush(qb ub) was moved over different paths on the surface ofthe disc much as Faraday did in his experimentsRepresentative results from these experiments which meas-ured the open-circuit voltage V versus position are shown inFig 12 In Fig 12(a) the radial position of the brush is fixed(qbAfrac14 0960) while the angular position ub is varied and inFig 12(b) the angular position for the brush is fixed(ub frac14 45) while the radial position qb is varied In bothcases the theory (curve) and measurement (dots) are againin good agreement Notice in Fig 12(b) that both the theoret-ical and measured results show the voltage becoming slightlynegative for qb=A lt 04

V DISCUSSION WHAT EVER BECAME OF

FARADAYrsquoS FIRST DYNAMO

In our study we wondered what became of Faradayrsquos firstdynamo and if his machine could be of use today The an-swer to this question is fairly simple As we saw from ouranalysis eddy currents are produced in the disc even when

no current is being drawn from the machine These eddy cur-rents dissipate significant energy and make the machineimpractical as a source of electric power This point is nicelyillustrated by considering the open-circuit voltage on theleft-hand side of Eq (21) to be composed of the two termson the right-hand side of this equation V frac14 Vi thorn Vs Thevoltage Vi is due to the current in the disc and can be thoughtof as the voltage produced by the surface current ~Js passingthrough the resistance of a strip of conductor of unit widthThe voltage Vs represents a source due to the applied mag-netic field traditionally called the ldquoelectromotive forcerdquo(emf) In Fig 13 the three terms V Vi and Vs (normalized sothat jVnjmax frac14 10) are shown as a function of the positionqbA along the radial line ub frac14 0 This line starts at the cen-ter of the disc passes under the pole and ends at the topedge of the disc The emf (Vsn) increases on the portion ofthe path that is under the pole and reaches a maximum ofabout Vsnfrac14 25 The voltage Vin due to the current is alwaysnegative so it subtracts from the emf making the open-circuit voltage drop to only Vnfrac14 08 at the top edge of thedisc Thus the eddy currents in the disc reduce the voltageavailable from the machine to about one-third of the emf

The inefficiency of the dynamo was also evident in themeasurements discussed in Sec IV When the disc wasrotated between the poles of the magnet a strong force resist-ing the motion was felt a consequence of Lenzrsquos law30

There was also a noticeable increase in the temperature ofthe rotating disc In a period of about one minute the tem-perature of the disc rotating at 772 rpm increased from theambient value of 246 8C (763 8F) to about 680 8C (154 8F)The energy that raises the temperature of the disc must besupplied by the mechanical source turning the axle of thedisc

Symmetric designs for the dynamo such as the one shownin Fig 2(b) do not suffer from the same problem with eddycurrents as the asymmetric design Analysis of the com-pletely symmetric structure shows that there is no conductioncurrent in the disc when the terminals are left open Andeven when current is drawn by the external circuit the cur-rent in the disc is in the radial direction that is when noeddies are formed13ndash16 Hence most attempts to make apractical unipolar or homopolar generator have used sym-metric structures These generators produce a high current at

Fig 12 Comparison of theoretical and measured results for the open-circuit

voltage V versus the position of the brush qb ub (a) Radial position fixed at

qbAfrac14 0960 while angular position is varied (b) Angular position fixed at

ub frac14 45

while radial position is varied (In both cases aeAfrac14 0173

cAfrac14 0663 and xfrac14 772 rpm)

Fig 13 The open-circuit voltage V voltage due to the current in the disc Vi

and source voltage (emf) Vs versus qbA for the radial line ub frac14 0 All of

the voltages are normalized so that maxjVnj frac14 10 and all of the parameters

are the same as those for Fig 8

915 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 915

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

a low voltage and the loss due to the current passing throughthe resistance of the brushes is of concern To lower this re-sistance some designs incorporate mercury or a liquid metal-lic alloy in the brushes as Faraday did albeit with moresophisticated designs than Faradayrsquos simple contacts31

Commercial use of even the symmetric designs has been lim-ited to very specialized applications32

The eddy-current disc brake is a structure very similar toFaradayrsquos first dynamo with the permanent magnet inFaradayrsquos design replaced by an electromagnet The conduct-ing disc is mounted on an axle that normally rotates Whenthe rotation is to be slowed or stopped the electromagnet isactivated and the kinetic energy taken from the rotationalmotion of the axle is dissipated by the eddy currents in thedisc The applications of this technology are diverse and rangefrom the brakes on the famous Japanese Shinkansen (bullettrain) to providing resistance to motion in fitness equipmentsuch as rowing machines33 The analysis for the eddy-currentdisc brake is similar to that for the Faraday disc only the em-phasis is usually on obtaining the torque on the axle versusvarious parameters such as the angular frequency and themagnetic flux34ndash37 The calculation we have used fromSmythersquos paper was originally intended for obtaining the tor-que17 In Europe linear versions of the eddy-current brake arebeing tested for railway applications38 In these devices themagnetic field is generated by a linear array of electromagnetson the car and applied to the rail The eddy currents producedin the rail slow the car and heat the rail A section of the rail isheated only as the car passes so the heating is not as severe asfor the disc brake which is heated continuously as long as thebrake is energized

As with the more common friction brakes all eddy-current brakes convert kinetic energy into heat This is unde-sirable in applications in which energy conservation is ofconcern For these applications electromagnetic regenerativebraking may be a better option Regenerative braking is usedin electric and hybrid (gasolineelectric) automobilesBatteries are used to power electric motors that propel theautomobile and when the automobile is to be slowed orstopped these motors are used as generators to charge thebatteries Thus a portion of the kinetic energy of the auto-mobile is recovered and stored for future use

The experiments Faraday made with his first dynamo gavehim insight into electromagnetic induction and among otherthings allowed him to give the first correct explanation forwhat caused Aragorsquos rotations The dynamo in its originalform was not a prototype for a practical source of electricpower However from the historical record it is clear thatFaradayrsquos first dynamo was an important step in the develop-ment of modern generators39 The fundamental principles hediscovered with his dynamo were quickly applied by othersto produce efficient generators

ACKNOWLEDGMENTS

The author would like to thank his Georgia Tech col-leagues Professors Waymond Scott Jr John Buck (bothfrom ECE) and Andrew Zangwill (Physics) for their sugges-tions for improving the manuscript

a)Electronic mail glennsmithecegatechedu1M Arago ldquoLrsquoaction que les corps aimantes et ceux qui ne le sont pas exer-

cent les uns sur les autresrdquo Ann Chim Phys 28 325 (1825) and M

Arago ldquoNote concernant les phenomenes magnetiques auxquels le

mouvement donne naissancerdquo ibid 32 213ndash223 (1826) An account in

English is in C Babbage and J F W Herschel ldquoAccount of the repetition

of M Aragorsquos experiments on magnetism manifested by various substan-

ces during the act of rotationrdquo Philos Trans Roy Soc London 115

467ndash496 (1825)2M Faraday Experimental Researches in Electricity Vol I (Taylor and

Francis London 1839 Republication Green Lion Santa Fe NM 2000)

Paragraph 83 pg 253M Faraday Faradayrsquos Diary edited by T Martin (G Bell and Sons

London 1932) Vol I Paragraphs 99ndash136 pp 381ndash3864M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraphs 83ndash100 pp 25ndash29 and 119ndash134 pp

34ndash395S P Thompson Michael Faraday His Life and Work (Cassell London

1901 Republication Elibron Classics New York 2004) pg 1236M Faraday Faradayrsquos Diary edited by T Martin (G Bell and Sons

London 1932) Vol I Paragraph 228 pg 3977M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraph 155 p 468R J Stephenson ldquoExperiments with a unipolar generator and motorrdquo

Am J Phys 5 108ndash110 (1937)9J W Then ldquoLaboratory experiments in motional electric fieldsrdquo Am J

Phys 28 557ndash559 (1960)10J W Then ldquoExperimental study of the motional electromotive forcerdquo

Am J Phys 30 411ndash415 (1962)11M J Crooks D B Litvin P W Matthews R Macaulay and J Shaw

ldquoOne-piece Faraday generator A paradoxical experiment from 1851rdquo

Am J Phys 46 729ndash731 (1978)12R D Eagleton ldquoTwo laboratory experiments involving the homopolar

generatorrdquo Am J Phys 55 621ndash623 (1987)13H H Woodson and J R Melcher Electromechanical Dynamics Part I

Discreet Systems (Wiley New York 1968) pp 286ndash28914J Van Bladel Relativity and Engineering (Springer-Verlag Berlin 1984)

pp 283ndash29115J Van Bladel ldquoRelativistic theory of rotating disksrdquo Proc IEEE 61

260ndash268 (1973)16H Montgomery ldquoCurrent flow patterns in a Faraday discrdquo Eur J Phys

25 171ndash183 (2004)17W R Smythe ldquoOn eddy currents in a rotating diskrdquo Trans AIEE 61

681ndash684 (1942)18W R Smythe Static and Dynamic Electricity 3rd ed (McGraw-Hill

New York 1968) pp 390ndash39319This type of problem is sometimes referred to as a ldquoquasi-static magnetic

field systemrdquo and it is discussed in detail in H H Woodson and J R

Melcher Electromechanical Dynamics Part I Discreet Systems (Wiley

New York 1968) pp 6ndash8 260ndash264 270ndash277 284ndash289 B19ndashB2520In the local rest frame for a point on the disc the thickness of the disc is

assumed to be much less than the skin depth in the conductor In this

frame the volume current density is then practically uniform throughout

the thickness of the disc In the quasi-static approximation the volume

current density is the same in the moving frame and in the laboratory

frame hence it is practically uniform in the laboratory frame21A I Borisenko and I E Tarapov Vector and Tensor Analysis with

Applications translated by R A Silverman (Prentice-Hall Englewood

Cliffs NJ 1968) pp 211ndash22322Smythe solves for the stream function U which is simply related to the

magnetic scalar potential UethquTHORN frac14 eth2=l0THORNUethqu z frac14 0thornTHORN His method

is briefly as follows First he considers a pole placed over an infinite con-

ducting sheet The radius of the pole is a and it is offset from the origin by

c He obtains an integral for the stream function U1 of this pole and he

evaluates this integral approximately Next he introduces a second pole

with the same total flux as the first pole but of opposite sign and he obtains

the approximate stream function U2 for this pole The radius and offset

from the origin for the second pole are ~a and ~c with ~c ~a gt A Finally

he adjusts ~a and ~c to make the superposition Ufrac14U1thornU2 satisfy the

boundary condition Ufrac14 0 at the radius qfrac14A (edge of the disc)23Normally a tube of current is a region of three-dimensional space

delineated by a side surface that is everywhere parallel to the direction of

the volume current density ~J The total current I passing through any

cross-sectional area of the tube is the samemdashthink of a pipe with the water

flowing through the pipe representing the current Here we are using the

two-dimensional version of the tube of current The current is on a surface

(on the disc) and the sides of the tube are two lines that are everywhere

parallel to the direction of the surface current density ~Js The total current

916 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 916

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

where t is a unit vector tangent to the path After taking frac12zEq (4) t and rearranging terms we obtain

~Bi t frac14 l0

2ethz ~JsTHORN t (13)

Substituting Eq (13) into the left-hand side of Eq (12) gives

l0

2

ethP2

P1

ethz ~JsTHORN t dlsquo frac14 l0

2

ethP2

P1

~Js s dlsquo frac14 ethU2 U1THORN

(14)

in which the unit vector s frac14 z t is in the plane of the discand normal to the path as shown in Fig 4(b) Now from Eq(14) we see that the total current through any cross sectionof the tube is

I frac14ethP2

P1

~Js s dlsquo frac14 2

l0

ethU2 U1THORN (15)

which is clearly a constantIf two different tubes of current are bounded by equipoten-

tial lines separated by the same increment U the two tubeshave the same total current Thus in Fig 6 all of the tubeswith Un frac14 01 have the same total current From this ob-servation we see that the current density is largest under thepole of the magnet and smallest near the edge of the disc atangles near u frac14 180 Notice that the tubes of current areclosed and the radial component of the surface current iszero at the edge of the disc The closed tubes of current areanalogous to eddies that form around an object in flowingwater hence the name ldquoeddy currentsrdquo (also ldquoFoucaultcurrentsrdquo) Because of the finite conductivity of the disc theeddy currents dissipate energy This energy must be suppliedby the mechanical source turning the axle of the disc

C Faradayrsquos law and the open-circuit voltage

Faradayrsquos law in integral form isthornCethtTHORN

frac12~Eeth~r tTHORN thorn~veth~r tTHORN ~Beth~r tTHORN d~lsquo frac14 d

dt

eth ethSethtTHORN

~Beth~r tTHORN d~S

(16)

where we explicitly indicate that the closed contour Cbounding the open surface S can vary with time and ~v isthe velocity for a point on the contour24 We will apply

Eq (16) to the contour ABCDA in Fig 3(a) shown inmore detail in Fig 7

For this contour in the limit u frac14 xt 0 the expres-sion on the left-hand side of Eq (16) becomes

VethqbubTHORN thornethqb

qfrac140

frac12~Eethqub 0THORN thorn~vethqub 0THORN

~Bethqub 0THORN q dq (17)

The integral in Eq (17) is for the portion of the contourCD that is in the disc where the constitutive equation forthe surface current density in the laboratory frame applieswhich is

~JsethqubTHORN ~Jethqub 0THORNd

frac14 rdfrac12~Eethqub 0THORN thorn~vethqub 0THORN ~Bethqub 0THORN(18)

When Eq (18) is substituted into Eq (17) we obtain

VethqbubTHORN thorn1

rd

ethqb

qfrac140

q ~JsethqubTHORNdq (19)

for our final result for the left-hand side of Eq (16)Now consider the expression on the right-hand-side of Eq

(16) for the contour ABCDA In the time increment t thearea S in Fig 7 increases by the increment S (gray in thefigure) We will assume that the applied magnetic field ismuch greater than axial component of the magnetic field dueto the current in the disc (j~Baj jz ~Bij) and we will ignorethe latter Then the time variation in the flux of the magneticfield through S is due to the applied field ~Ba passingthrough S or

d

dt

eth ethSethtTHORN

~Beth~r tTHORN d~S frac14 limt0

1

t

ethqb

qfrac140

~BaethqubTHORN ethzTHORNq u dq

264

375 frac14 lim

t0

ut

ethqb

qfrac140

qBaethqubTHORN dq

264

375 frac14 x

ethqb

qfrac140

qBaethqubTHORN dq

(20)

After equating the left-hand- and right-hand-sides of Eq (16) for the contour ABCDAmdashequating Eqs (19) and (20)mdashweobtain the open-circuit voltage at the terminals A-B in Fig 3(a) to be

Fig 7 Details for contour C used with Faradayrsquos law

911 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 911

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

VethqbubTHORN frac14 1

rd

ethqb

qfrac140

q ~JsethqubTHORNdqthorn xethqb

qfrac140

qBaethqubTHORN dq frac14 2

l0rd

ethqb

qfrac140

1

qUethqub z frac14 0thornTHORN

udqthorn x

ethqb

qfrac140

qBaethqubTHORN dq

(21)

where in the last line we have substituted Eq (10) for ~JsTo complete the derivation for the open-circuit voltage we must substitute for the magnetic scalar potential Eqs (8) and

(9) and perform the differentiation and integration indicated in Eq (21) For points qbub in each of the three regions shownin Fig 5 the evaluation is different for example for Region I the expression to be evaluated is

VI frac14 2

l0rd

ethqi

qfrac140

1

qUI

udqthorn

ethqo

qfrac14qi

1

qUII

udqthorn

ethqb

qo

1

qUIII

udq

264

375thorn xB0

ethqo

qfrac14qi

qdq (22)

The calculations are straightforward but tedious All of the definite integrals can be evaluated in closed form25 and final resultsfor the open-circuit voltage are

VIethqbnubTHORN frac14 xBoa2

2feth1=anTHORN2frac12q2

on q2in cnethqon qinTHORNcosub thorn Fethqin cnubTHORNthornFethqbn cnubTHORN

Fethqon cnubTHORN Fethqbn 1=cnubTHORNg (23)

VIIethqbnubTHORN frac14 xBoa2

2feth1=anTHORN2frac12q2

bn q2in cnethqbn qinTHORNcosub thorn Fethqin cnubTHORN

Fethqbn 1=cnubTHORNg (24)

and

VIIIethqbnubTHORN frac14 xBoa2

2frac12Fethqbn cnubTHORN

Fethqbn 1=cnubTHORN (25)

in which the function F is given by

Fetha bubTHORN frac14 bb acosub

a2 thorn b2 2abcosub

(26)

Figure 8 is a contour plot of the open-circuit voltage normal-ized so that maxjVnj frac14 10 for the case aAfrac14 0170 andcAfrac14 0667 This is the same case as for the magnetic scalarpotential in Fig 6 All of the contours are spaced by the incre-ment Vn frac14 01 except the incomplete contours forVnfrac14 0033 and 005 Notice that the voltage has even symme-try about the lines ub frac14 0 and 1808 It has a maximum at thetop edge of the magnetic pole (qb frac14 cthorn aub frac14 0) and hassmall negative values in the area just above the center of thedisc A comparison of the lines of constant voltage in Fig 8with the streamlines for the current in Fig 6 shows that thetwo sets of lines are orthogonal in Regions I and III wherethere is no applied magnetic field This is the relationship weexpect for an electrically-thin conductor in the quasi-staticlimit We will have more to say about this plot when we dis-cuss Faradayrsquos observations in the next section of the paper

III FARADAYrsquoS OBSERVATIONS

Faraday spent a considerable amount of time experimentingwith his first dynamo A measure of this can be seen in his

diary where he devoted five pages including 23 figures to hisfirst dynamo By comparison he devoted two short paragraphswithout figures to the dynamo shown in Fig 2(b)36 Faradayrsquosmeasurements were made with a galvanometer that hereferred to as ldquoroughly made yet sufficiently delicate in itsindicationsrdquo The instrument could detect a very small currentbut the measurements were qualitative in the sense that

Fig 8 Contour plot for the open-circuit voltage VnethqbubTHORN for the case

aAfrac14 0170 and cAfrac14 0667 The voltage is normalized so that

maxjVnj frac14 10 and the contours are spaced by the increment Vn frac14 01

except the incomplete contours for Vnfrac14 0033 and 005

912 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 912

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

Faraday recorded the deflection of the galvanometer usingterms such as ldquomost stronglyrdquo and ldquoless stronglyrdquo Faradayvaried the arrangement of the elements (brushes magnet etc)in his dynamo to determine the characteristics of the currenthe thought was produced in the disc In this section using theanalysis just discussed we will examine Faradayrsquos observa-tions and his interpretation for them We will assume that ouropen-circuit voltage is proportional to the current that causedthe deflection of Faradayrsquos galvanometer26

Faraday reversed both the direction of rotation of the discand the direction of the applied magnetic field and he sawthat a reversal of either changed the sense of the deflectionof the galvanometer He reduced the applied magnetic fieldby moving one of the poles of the magnet away from thedisc and he saw the deflection decrease These observationsare in agreement with the product xBo appearing in theexpressions for the open-circuit voltage in Eqs (23)ndash(25)

Faraday was able to deduce many of the key features wesee in the contour plot for the open-circuit voltage shown inFig 8 He did this by moving the brushes to different loca-tions on the disc and observing the change in the deflectionof the galvanometer He placed one brush at the center of thedisc (on the axle) and the other brush at symmetrical pointsabout the line ub frac14 0 or 1808 By noting that the deflectionof the galvanometer was the same at the two points hededuced the even symmetry we see for the pattern about thisline With one brush at the center of the disc and the other atthe edge of the disc he varied the angular position of the lat-ter and noticed that the deflection of the galvanometer wasmaximum when the brush was opposite the magnetic pole(ub frac14 0) and decreased as the brush was moved away fromthis point (ub increased) This observation is consistent withwhat we see in Fig 8 for qbfrac14A

Faraday also varied the position of the magnetic poles Forcomparison with his observations we will determine theopen-circuit voltage when the brush in Fig 3(a) is at the topedge of the disc directly above the magnetic poleqb frac14 A ub frac14 0 For this case the open-circuit voltage Eq(23) becomes simply

Vethqb frac14 Aub frac14 0THORN frac14 xB0A2

2etha=ATHORN2 1thorn c=A

1 c=A

(27)

which we have plotted in Fig 9 as a function of polersquos sizeaA and position cA

For any size pole the voltage is maximum when the poleis as far away from the center of the disc as possible whilestill being completely over the disc or when cAfrac14 1 ndash aAIn this position the velocity of the disc relative to the pole ismaximum Also the larger the pole the larger the voltagewith maximum voltage occurring when the diameter of thepole equals the radius of the disc that is aAfrac14 05 andcAfrac14 05 When Faraday lowered the magnetic poles so asto decrease cA he noted that the ldquohellipeffect was the same [aswith the poles raised] and the deflection was very sensiblerdquoThis observation can be explained by the small size ofFaradayrsquos poles They were either square or rectangular inshape and roughly equivalent to circular poles withaAfrac14 00727 Thus his results approximately correspond tothe curve aAfrac14 01 in Fig 9 which in the central region isfairly flat so the variation with cA would have been difficultto detect in his qualitative measurements

From his measurements Faraday was able to show that cur-rent was induced in the conducting disc only when the disc

moved through the magnetic field He concluded that the cur-rent went in the radial direction from the central part of thedisc past the pole to the edge of the disc where itldquodischargedrdquo or returned in the part of the disc on either sideof the pole This description is remarkably similar to what wesee for the distribution of the surface current in Fig 6

Commenting on his first dynamo Faraday said ldquoHeretherefore was demonstrated the production of a permanent[direct or continuous] current of electricity by ordinary mag-netsrdquo28 Thus he fulfilled the goal he stated earlier of makingthe experiment of Arago a new source of electricity Faradaywas confident that he had also found the explanation forAragorsquos rotations which was as follows The magnetic fieldof the needle induced current in the rotating disc this cur-rent in turn created a magnetic field that exerted a force onthe needle causing it to rotate

IV REPEATING FARADAYrsquoS EXPERIMENTS

We repeated some of the experiments Faraday made withhis first dynamo with a two-fold objective in mind First weaim to verify the analysis presented earlier which is basedon Smythersquos approximation for the magnetic scalar potentialand second we provide quantitative results that supportFaradayrsquos qualitative observations Concerning the firstobjective we followed Faradayrsquos advice that ldquoNothing is asgood as an experiment which whilst it sets error right givesan absolute advancement in knowledgerdquo29

Figures 10(a) and 10(b) are schematic drawings for the ex-perimental model which is very similar to Faradayrsquosdynamo except that it is constructed from modern materialsThe circular copper disc of radius Afrac14 750 cm and thicknessdfrac14 086 mm was mounted on a brass axle The axle washeld in the chuck of a drill press and the speed of rotation ofthe disc which was measured with an optical tachometerwas varied by adjusting the beltstepped-pulley system of thedrill press The magnetic field was that of two neodymiumpermanent magnets (K amp J Magnetics Inc DX08) of radiusamfrac14 127 cm and length 127 cm bonded to the ends of ayoke formed from laminated steel sheets The purpose of theyoke was to direct the return field of the magnets to an areaoff the disc so that the only magnetic field applied to the discwas that between the poles Figure 10(c) is a graph of the

Fig 9 Open-circuit voltage as a function of the polersquos size aA and position

cA when the brush is at the top edge of the disc directly above the magnetic

pole at qb frac14 A ub frac14 0 The voltage is normalized so that maxjVnj frac14 10

which occurs for aAfrac14 05 cAfrac14 05 For all points on the graph the pole is

completely over the disc

913 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 913

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

magnetic field measured on the horizontal plane centeredbetween the poles of the permanent magnets The points onthis graph are averages of measurements made along threedifferent radial lines from the center of the poles The mag-netic field is seen to have a maximum value of about 108 Tat the center of the poles (ramfrac14 00) to decrease to abouthalf this value at the edges of the poles (ramfrac14 10) and torapidly decay beyond this point

The disc was connected to the external circuit by spring-loaded carbon brushes of diameter 46 mm [see Fig 10(b)]

One brush was at the center of the disc and a pair of opposingbrushes located on opposite sides of the disc and connected inparallel could be moved to various positions (qb ub) on thedisc Wires embedded in the brushes were soldered directly tothe external circuit where the open-circuit voltage V wasmeasured with a digital voltmeter The good electrical contactbetween the spring-loaded brushes and the disc resulted invery little fluctuation of the voltage as the disc rotated

One complication arose in the comparison of the theorywith the measurements The magnetic field for the theory isuniform over the area of the poles whereas the magneticfield in the experiment varies with the radial distance fromthe center of the poles To make the comparison we definedan effective radius ae for the poles used in the experimentwhich is the radius of fictitious poles with uniform magneticfield equal to Bz(rfrac14 0) that produce the same total flux as theactual poles Mathematically we express this as

pa2eBzethr frac14 0THORN frac14 2p

eth3am

rfrac140

BzethrTHORNrdr (28)

or

ae frac14 am

2

eth3am

rfrac140

BzethrTHORNrdr

a2mBzethr frac14 0THORN

26664

37775

1=2

(29)

When Eq (29) was applied to the results in Fig 10(c) thevalue aefrac14 102am was obtained A value of ae this close toam implies that the additional magnetic field outside of thepoles in Fig 10(c) just compensates for the reduction in themagnetic field under the poles For comparison of theoreticalcalculations with measurements the value ae was used for ain the theory

In all of our measurements the magnetic poles were locatedat cAfrac14 0663 and aeAfrac14 0173 These values are roughly thesame as those used in the earlier calculations For the firstmeasurement the brush was placed above the magnetic polesnear the edge of the disc (qb=A frac14 0951 ub frac14 0) and theangular speed of the disc was varied The theoretical results(curve) for the open-circuit voltage V versus angular speed xare compared with the measured results (dots) in Fig 11 The

Fig 10 Details for the experimental model which is similar to Faradayrsquos

first dynamo (a) Side view in which the brushes and external circuit are not

shown (b) Front view showing the brushes and external circuit (c) Graph of

the magnetic field measured on the horizontal plane centered between the

poles of the permanent magnets The points on the graph are the average of

measurements made along three different radial lines from the center of the

poles

Fig 11 Comparison of theoretical and measured results for the open-circuit

voltage V versus the angular frequency x with qb=A frac14 0951 ub frac14 0

(aeAfrac14 0173 and cAfrac14 0663)

914 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 914

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

theory and measurement are seen to be in good agreement andboth exhibit a linear dependence on x as expected

In the next set of measurements the angular speed of thedisc was fixed at 772 rpm and the position of the brush(qb ub) was moved over different paths on the surface ofthe disc much as Faraday did in his experimentsRepresentative results from these experiments which meas-ured the open-circuit voltage V versus position are shown inFig 12 In Fig 12(a) the radial position of the brush is fixed(qbAfrac14 0960) while the angular position ub is varied and inFig 12(b) the angular position for the brush is fixed(ub frac14 45) while the radial position qb is varied In bothcases the theory (curve) and measurement (dots) are againin good agreement Notice in Fig 12(b) that both the theoret-ical and measured results show the voltage becoming slightlynegative for qb=A lt 04

V DISCUSSION WHAT EVER BECAME OF

FARADAYrsquoS FIRST DYNAMO

In our study we wondered what became of Faradayrsquos firstdynamo and if his machine could be of use today The an-swer to this question is fairly simple As we saw from ouranalysis eddy currents are produced in the disc even when

no current is being drawn from the machine These eddy cur-rents dissipate significant energy and make the machineimpractical as a source of electric power This point is nicelyillustrated by considering the open-circuit voltage on theleft-hand side of Eq (21) to be composed of the two termson the right-hand side of this equation V frac14 Vi thorn Vs Thevoltage Vi is due to the current in the disc and can be thoughtof as the voltage produced by the surface current ~Js passingthrough the resistance of a strip of conductor of unit widthThe voltage Vs represents a source due to the applied mag-netic field traditionally called the ldquoelectromotive forcerdquo(emf) In Fig 13 the three terms V Vi and Vs (normalized sothat jVnjmax frac14 10) are shown as a function of the positionqbA along the radial line ub frac14 0 This line starts at the cen-ter of the disc passes under the pole and ends at the topedge of the disc The emf (Vsn) increases on the portion ofthe path that is under the pole and reaches a maximum ofabout Vsnfrac14 25 The voltage Vin due to the current is alwaysnegative so it subtracts from the emf making the open-circuit voltage drop to only Vnfrac14 08 at the top edge of thedisc Thus the eddy currents in the disc reduce the voltageavailable from the machine to about one-third of the emf

The inefficiency of the dynamo was also evident in themeasurements discussed in Sec IV When the disc wasrotated between the poles of the magnet a strong force resist-ing the motion was felt a consequence of Lenzrsquos law30

There was also a noticeable increase in the temperature ofthe rotating disc In a period of about one minute the tem-perature of the disc rotating at 772 rpm increased from theambient value of 246 8C (763 8F) to about 680 8C (154 8F)The energy that raises the temperature of the disc must besupplied by the mechanical source turning the axle of thedisc

Symmetric designs for the dynamo such as the one shownin Fig 2(b) do not suffer from the same problem with eddycurrents as the asymmetric design Analysis of the com-pletely symmetric structure shows that there is no conductioncurrent in the disc when the terminals are left open Andeven when current is drawn by the external circuit the cur-rent in the disc is in the radial direction that is when noeddies are formed13ndash16 Hence most attempts to make apractical unipolar or homopolar generator have used sym-metric structures These generators produce a high current at

Fig 12 Comparison of theoretical and measured results for the open-circuit

voltage V versus the position of the brush qb ub (a) Radial position fixed at

qbAfrac14 0960 while angular position is varied (b) Angular position fixed at

ub frac14 45

while radial position is varied (In both cases aeAfrac14 0173

cAfrac14 0663 and xfrac14 772 rpm)

Fig 13 The open-circuit voltage V voltage due to the current in the disc Vi

and source voltage (emf) Vs versus qbA for the radial line ub frac14 0 All of

the voltages are normalized so that maxjVnj frac14 10 and all of the parameters

are the same as those for Fig 8

915 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 915

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

a low voltage and the loss due to the current passing throughthe resistance of the brushes is of concern To lower this re-sistance some designs incorporate mercury or a liquid metal-lic alloy in the brushes as Faraday did albeit with moresophisticated designs than Faradayrsquos simple contacts31

Commercial use of even the symmetric designs has been lim-ited to very specialized applications32

The eddy-current disc brake is a structure very similar toFaradayrsquos first dynamo with the permanent magnet inFaradayrsquos design replaced by an electromagnet The conduct-ing disc is mounted on an axle that normally rotates Whenthe rotation is to be slowed or stopped the electromagnet isactivated and the kinetic energy taken from the rotationalmotion of the axle is dissipated by the eddy currents in thedisc The applications of this technology are diverse and rangefrom the brakes on the famous Japanese Shinkansen (bullettrain) to providing resistance to motion in fitness equipmentsuch as rowing machines33 The analysis for the eddy-currentdisc brake is similar to that for the Faraday disc only the em-phasis is usually on obtaining the torque on the axle versusvarious parameters such as the angular frequency and themagnetic flux34ndash37 The calculation we have used fromSmythersquos paper was originally intended for obtaining the tor-que17 In Europe linear versions of the eddy-current brake arebeing tested for railway applications38 In these devices themagnetic field is generated by a linear array of electromagnetson the car and applied to the rail The eddy currents producedin the rail slow the car and heat the rail A section of the rail isheated only as the car passes so the heating is not as severe asfor the disc brake which is heated continuously as long as thebrake is energized

As with the more common friction brakes all eddy-current brakes convert kinetic energy into heat This is unde-sirable in applications in which energy conservation is ofconcern For these applications electromagnetic regenerativebraking may be a better option Regenerative braking is usedin electric and hybrid (gasolineelectric) automobilesBatteries are used to power electric motors that propel theautomobile and when the automobile is to be slowed orstopped these motors are used as generators to charge thebatteries Thus a portion of the kinetic energy of the auto-mobile is recovered and stored for future use

The experiments Faraday made with his first dynamo gavehim insight into electromagnetic induction and among otherthings allowed him to give the first correct explanation forwhat caused Aragorsquos rotations The dynamo in its originalform was not a prototype for a practical source of electricpower However from the historical record it is clear thatFaradayrsquos first dynamo was an important step in the develop-ment of modern generators39 The fundamental principles hediscovered with his dynamo were quickly applied by othersto produce efficient generators

ACKNOWLEDGMENTS

The author would like to thank his Georgia Tech col-leagues Professors Waymond Scott Jr John Buck (bothfrom ECE) and Andrew Zangwill (Physics) for their sugges-tions for improving the manuscript

a)Electronic mail glennsmithecegatechedu1M Arago ldquoLrsquoaction que les corps aimantes et ceux qui ne le sont pas exer-

cent les uns sur les autresrdquo Ann Chim Phys 28 325 (1825) and M

Arago ldquoNote concernant les phenomenes magnetiques auxquels le

mouvement donne naissancerdquo ibid 32 213ndash223 (1826) An account in

English is in C Babbage and J F W Herschel ldquoAccount of the repetition

of M Aragorsquos experiments on magnetism manifested by various substan-

ces during the act of rotationrdquo Philos Trans Roy Soc London 115

467ndash496 (1825)2M Faraday Experimental Researches in Electricity Vol I (Taylor and

Francis London 1839 Republication Green Lion Santa Fe NM 2000)

Paragraph 83 pg 253M Faraday Faradayrsquos Diary edited by T Martin (G Bell and Sons

London 1932) Vol I Paragraphs 99ndash136 pp 381ndash3864M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraphs 83ndash100 pp 25ndash29 and 119ndash134 pp

34ndash395S P Thompson Michael Faraday His Life and Work (Cassell London

1901 Republication Elibron Classics New York 2004) pg 1236M Faraday Faradayrsquos Diary edited by T Martin (G Bell and Sons

London 1932) Vol I Paragraph 228 pg 3977M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraph 155 p 468R J Stephenson ldquoExperiments with a unipolar generator and motorrdquo

Am J Phys 5 108ndash110 (1937)9J W Then ldquoLaboratory experiments in motional electric fieldsrdquo Am J

Phys 28 557ndash559 (1960)10J W Then ldquoExperimental study of the motional electromotive forcerdquo

Am J Phys 30 411ndash415 (1962)11M J Crooks D B Litvin P W Matthews R Macaulay and J Shaw

ldquoOne-piece Faraday generator A paradoxical experiment from 1851rdquo

Am J Phys 46 729ndash731 (1978)12R D Eagleton ldquoTwo laboratory experiments involving the homopolar

generatorrdquo Am J Phys 55 621ndash623 (1987)13H H Woodson and J R Melcher Electromechanical Dynamics Part I

Discreet Systems (Wiley New York 1968) pp 286ndash28914J Van Bladel Relativity and Engineering (Springer-Verlag Berlin 1984)

pp 283ndash29115J Van Bladel ldquoRelativistic theory of rotating disksrdquo Proc IEEE 61

260ndash268 (1973)16H Montgomery ldquoCurrent flow patterns in a Faraday discrdquo Eur J Phys

25 171ndash183 (2004)17W R Smythe ldquoOn eddy currents in a rotating diskrdquo Trans AIEE 61

681ndash684 (1942)18W R Smythe Static and Dynamic Electricity 3rd ed (McGraw-Hill

New York 1968) pp 390ndash39319This type of problem is sometimes referred to as a ldquoquasi-static magnetic

field systemrdquo and it is discussed in detail in H H Woodson and J R

Melcher Electromechanical Dynamics Part I Discreet Systems (Wiley

New York 1968) pp 6ndash8 260ndash264 270ndash277 284ndash289 B19ndashB2520In the local rest frame for a point on the disc the thickness of the disc is

assumed to be much less than the skin depth in the conductor In this

frame the volume current density is then practically uniform throughout

the thickness of the disc In the quasi-static approximation the volume

current density is the same in the moving frame and in the laboratory

frame hence it is practically uniform in the laboratory frame21A I Borisenko and I E Tarapov Vector and Tensor Analysis with

Applications translated by R A Silverman (Prentice-Hall Englewood

Cliffs NJ 1968) pp 211ndash22322Smythe solves for the stream function U which is simply related to the

magnetic scalar potential UethquTHORN frac14 eth2=l0THORNUethqu z frac14 0thornTHORN His method

is briefly as follows First he considers a pole placed over an infinite con-

ducting sheet The radius of the pole is a and it is offset from the origin by

c He obtains an integral for the stream function U1 of this pole and he

evaluates this integral approximately Next he introduces a second pole

with the same total flux as the first pole but of opposite sign and he obtains

the approximate stream function U2 for this pole The radius and offset

from the origin for the second pole are ~a and ~c with ~c ~a gt A Finally

he adjusts ~a and ~c to make the superposition Ufrac14U1thornU2 satisfy the

boundary condition Ufrac14 0 at the radius qfrac14A (edge of the disc)23Normally a tube of current is a region of three-dimensional space

delineated by a side surface that is everywhere parallel to the direction of

the volume current density ~J The total current I passing through any

cross-sectional area of the tube is the samemdashthink of a pipe with the water

flowing through the pipe representing the current Here we are using the

two-dimensional version of the tube of current The current is on a surface

(on the disc) and the sides of the tube are two lines that are everywhere

parallel to the direction of the surface current density ~Js The total current

916 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 916

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

VethqbubTHORN frac14 1

rd

ethqb

qfrac140

q ~JsethqubTHORNdqthorn xethqb

qfrac140

qBaethqubTHORN dq frac14 2

l0rd

ethqb

qfrac140

1

qUethqub z frac14 0thornTHORN

udqthorn x

ethqb

qfrac140

qBaethqubTHORN dq

(21)

where in the last line we have substituted Eq (10) for ~JsTo complete the derivation for the open-circuit voltage we must substitute for the magnetic scalar potential Eqs (8) and

(9) and perform the differentiation and integration indicated in Eq (21) For points qbub in each of the three regions shownin Fig 5 the evaluation is different for example for Region I the expression to be evaluated is

VI frac14 2

l0rd

ethqi

qfrac140

1

qUI

udqthorn

ethqo

qfrac14qi

1

qUII

udqthorn

ethqb

qo

1

qUIII

udq

264

375thorn xB0

ethqo

qfrac14qi

qdq (22)

The calculations are straightforward but tedious All of the definite integrals can be evaluated in closed form25 and final resultsfor the open-circuit voltage are

VIethqbnubTHORN frac14 xBoa2

2feth1=anTHORN2frac12q2

on q2in cnethqon qinTHORNcosub thorn Fethqin cnubTHORNthornFethqbn cnubTHORN

Fethqon cnubTHORN Fethqbn 1=cnubTHORNg (23)

VIIethqbnubTHORN frac14 xBoa2

2feth1=anTHORN2frac12q2

bn q2in cnethqbn qinTHORNcosub thorn Fethqin cnubTHORN

Fethqbn 1=cnubTHORNg (24)

and

VIIIethqbnubTHORN frac14 xBoa2

2frac12Fethqbn cnubTHORN

Fethqbn 1=cnubTHORN (25)

in which the function F is given by

Fetha bubTHORN frac14 bb acosub

a2 thorn b2 2abcosub

(26)

Figure 8 is a contour plot of the open-circuit voltage normal-ized so that maxjVnj frac14 10 for the case aAfrac14 0170 andcAfrac14 0667 This is the same case as for the magnetic scalarpotential in Fig 6 All of the contours are spaced by the incre-ment Vn frac14 01 except the incomplete contours forVnfrac14 0033 and 005 Notice that the voltage has even symme-try about the lines ub frac14 0 and 1808 It has a maximum at thetop edge of the magnetic pole (qb frac14 cthorn aub frac14 0) and hassmall negative values in the area just above the center of thedisc A comparison of the lines of constant voltage in Fig 8with the streamlines for the current in Fig 6 shows that thetwo sets of lines are orthogonal in Regions I and III wherethere is no applied magnetic field This is the relationship weexpect for an electrically-thin conductor in the quasi-staticlimit We will have more to say about this plot when we dis-cuss Faradayrsquos observations in the next section of the paper

III FARADAYrsquoS OBSERVATIONS

Faraday spent a considerable amount of time experimentingwith his first dynamo A measure of this can be seen in his

diary where he devoted five pages including 23 figures to hisfirst dynamo By comparison he devoted two short paragraphswithout figures to the dynamo shown in Fig 2(b)36 Faradayrsquosmeasurements were made with a galvanometer that hereferred to as ldquoroughly made yet sufficiently delicate in itsindicationsrdquo The instrument could detect a very small currentbut the measurements were qualitative in the sense that

Fig 8 Contour plot for the open-circuit voltage VnethqbubTHORN for the case

aAfrac14 0170 and cAfrac14 0667 The voltage is normalized so that

maxjVnj frac14 10 and the contours are spaced by the increment Vn frac14 01

except the incomplete contours for Vnfrac14 0033 and 005

912 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 912

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

Faraday recorded the deflection of the galvanometer usingterms such as ldquomost stronglyrdquo and ldquoless stronglyrdquo Faradayvaried the arrangement of the elements (brushes magnet etc)in his dynamo to determine the characteristics of the currenthe thought was produced in the disc In this section using theanalysis just discussed we will examine Faradayrsquos observa-tions and his interpretation for them We will assume that ouropen-circuit voltage is proportional to the current that causedthe deflection of Faradayrsquos galvanometer26

Faraday reversed both the direction of rotation of the discand the direction of the applied magnetic field and he sawthat a reversal of either changed the sense of the deflectionof the galvanometer He reduced the applied magnetic fieldby moving one of the poles of the magnet away from thedisc and he saw the deflection decrease These observationsare in agreement with the product xBo appearing in theexpressions for the open-circuit voltage in Eqs (23)ndash(25)

Faraday was able to deduce many of the key features wesee in the contour plot for the open-circuit voltage shown inFig 8 He did this by moving the brushes to different loca-tions on the disc and observing the change in the deflectionof the galvanometer He placed one brush at the center of thedisc (on the axle) and the other brush at symmetrical pointsabout the line ub frac14 0 or 1808 By noting that the deflectionof the galvanometer was the same at the two points hededuced the even symmetry we see for the pattern about thisline With one brush at the center of the disc and the other atthe edge of the disc he varied the angular position of the lat-ter and noticed that the deflection of the galvanometer wasmaximum when the brush was opposite the magnetic pole(ub frac14 0) and decreased as the brush was moved away fromthis point (ub increased) This observation is consistent withwhat we see in Fig 8 for qbfrac14A

Faraday also varied the position of the magnetic poles Forcomparison with his observations we will determine theopen-circuit voltage when the brush in Fig 3(a) is at the topedge of the disc directly above the magnetic poleqb frac14 A ub frac14 0 For this case the open-circuit voltage Eq(23) becomes simply

Vethqb frac14 Aub frac14 0THORN frac14 xB0A2

2etha=ATHORN2 1thorn c=A

1 c=A

(27)

which we have plotted in Fig 9 as a function of polersquos sizeaA and position cA

For any size pole the voltage is maximum when the poleis as far away from the center of the disc as possible whilestill being completely over the disc or when cAfrac14 1 ndash aAIn this position the velocity of the disc relative to the pole ismaximum Also the larger the pole the larger the voltagewith maximum voltage occurring when the diameter of thepole equals the radius of the disc that is aAfrac14 05 andcAfrac14 05 When Faraday lowered the magnetic poles so asto decrease cA he noted that the ldquohellipeffect was the same [aswith the poles raised] and the deflection was very sensiblerdquoThis observation can be explained by the small size ofFaradayrsquos poles They were either square or rectangular inshape and roughly equivalent to circular poles withaAfrac14 00727 Thus his results approximately correspond tothe curve aAfrac14 01 in Fig 9 which in the central region isfairly flat so the variation with cA would have been difficultto detect in his qualitative measurements

From his measurements Faraday was able to show that cur-rent was induced in the conducting disc only when the disc

moved through the magnetic field He concluded that the cur-rent went in the radial direction from the central part of thedisc past the pole to the edge of the disc where itldquodischargedrdquo or returned in the part of the disc on either sideof the pole This description is remarkably similar to what wesee for the distribution of the surface current in Fig 6

Commenting on his first dynamo Faraday said ldquoHeretherefore was demonstrated the production of a permanent[direct or continuous] current of electricity by ordinary mag-netsrdquo28 Thus he fulfilled the goal he stated earlier of makingthe experiment of Arago a new source of electricity Faradaywas confident that he had also found the explanation forAragorsquos rotations which was as follows The magnetic fieldof the needle induced current in the rotating disc this cur-rent in turn created a magnetic field that exerted a force onthe needle causing it to rotate

IV REPEATING FARADAYrsquoS EXPERIMENTS

We repeated some of the experiments Faraday made withhis first dynamo with a two-fold objective in mind First weaim to verify the analysis presented earlier which is basedon Smythersquos approximation for the magnetic scalar potentialand second we provide quantitative results that supportFaradayrsquos qualitative observations Concerning the firstobjective we followed Faradayrsquos advice that ldquoNothing is asgood as an experiment which whilst it sets error right givesan absolute advancement in knowledgerdquo29

Figures 10(a) and 10(b) are schematic drawings for the ex-perimental model which is very similar to Faradayrsquosdynamo except that it is constructed from modern materialsThe circular copper disc of radius Afrac14 750 cm and thicknessdfrac14 086 mm was mounted on a brass axle The axle washeld in the chuck of a drill press and the speed of rotation ofthe disc which was measured with an optical tachometerwas varied by adjusting the beltstepped-pulley system of thedrill press The magnetic field was that of two neodymiumpermanent magnets (K amp J Magnetics Inc DX08) of radiusamfrac14 127 cm and length 127 cm bonded to the ends of ayoke formed from laminated steel sheets The purpose of theyoke was to direct the return field of the magnets to an areaoff the disc so that the only magnetic field applied to the discwas that between the poles Figure 10(c) is a graph of the

Fig 9 Open-circuit voltage as a function of the polersquos size aA and position

cA when the brush is at the top edge of the disc directly above the magnetic

pole at qb frac14 A ub frac14 0 The voltage is normalized so that maxjVnj frac14 10

which occurs for aAfrac14 05 cAfrac14 05 For all points on the graph the pole is

completely over the disc

913 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 913

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

magnetic field measured on the horizontal plane centeredbetween the poles of the permanent magnets The points onthis graph are averages of measurements made along threedifferent radial lines from the center of the poles The mag-netic field is seen to have a maximum value of about 108 Tat the center of the poles (ramfrac14 00) to decrease to abouthalf this value at the edges of the poles (ramfrac14 10) and torapidly decay beyond this point

The disc was connected to the external circuit by spring-loaded carbon brushes of diameter 46 mm [see Fig 10(b)]

One brush was at the center of the disc and a pair of opposingbrushes located on opposite sides of the disc and connected inparallel could be moved to various positions (qb ub) on thedisc Wires embedded in the brushes were soldered directly tothe external circuit where the open-circuit voltage V wasmeasured with a digital voltmeter The good electrical contactbetween the spring-loaded brushes and the disc resulted invery little fluctuation of the voltage as the disc rotated

One complication arose in the comparison of the theorywith the measurements The magnetic field for the theory isuniform over the area of the poles whereas the magneticfield in the experiment varies with the radial distance fromthe center of the poles To make the comparison we definedan effective radius ae for the poles used in the experimentwhich is the radius of fictitious poles with uniform magneticfield equal to Bz(rfrac14 0) that produce the same total flux as theactual poles Mathematically we express this as

pa2eBzethr frac14 0THORN frac14 2p

eth3am

rfrac140

BzethrTHORNrdr (28)

or

ae frac14 am

2

eth3am

rfrac140

BzethrTHORNrdr

a2mBzethr frac14 0THORN

26664

37775

1=2

(29)

When Eq (29) was applied to the results in Fig 10(c) thevalue aefrac14 102am was obtained A value of ae this close toam implies that the additional magnetic field outside of thepoles in Fig 10(c) just compensates for the reduction in themagnetic field under the poles For comparison of theoreticalcalculations with measurements the value ae was used for ain the theory

In all of our measurements the magnetic poles were locatedat cAfrac14 0663 and aeAfrac14 0173 These values are roughly thesame as those used in the earlier calculations For the firstmeasurement the brush was placed above the magnetic polesnear the edge of the disc (qb=A frac14 0951 ub frac14 0) and theangular speed of the disc was varied The theoretical results(curve) for the open-circuit voltage V versus angular speed xare compared with the measured results (dots) in Fig 11 The

Fig 10 Details for the experimental model which is similar to Faradayrsquos

first dynamo (a) Side view in which the brushes and external circuit are not

shown (b) Front view showing the brushes and external circuit (c) Graph of

the magnetic field measured on the horizontal plane centered between the

poles of the permanent magnets The points on the graph are the average of

measurements made along three different radial lines from the center of the

poles

Fig 11 Comparison of theoretical and measured results for the open-circuit

voltage V versus the angular frequency x with qb=A frac14 0951 ub frac14 0

(aeAfrac14 0173 and cAfrac14 0663)

914 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 914

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

theory and measurement are seen to be in good agreement andboth exhibit a linear dependence on x as expected

In the next set of measurements the angular speed of thedisc was fixed at 772 rpm and the position of the brush(qb ub) was moved over different paths on the surface ofthe disc much as Faraday did in his experimentsRepresentative results from these experiments which meas-ured the open-circuit voltage V versus position are shown inFig 12 In Fig 12(a) the radial position of the brush is fixed(qbAfrac14 0960) while the angular position ub is varied and inFig 12(b) the angular position for the brush is fixed(ub frac14 45) while the radial position qb is varied In bothcases the theory (curve) and measurement (dots) are againin good agreement Notice in Fig 12(b) that both the theoret-ical and measured results show the voltage becoming slightlynegative for qb=A lt 04

V DISCUSSION WHAT EVER BECAME OF

FARADAYrsquoS FIRST DYNAMO

In our study we wondered what became of Faradayrsquos firstdynamo and if his machine could be of use today The an-swer to this question is fairly simple As we saw from ouranalysis eddy currents are produced in the disc even when

no current is being drawn from the machine These eddy cur-rents dissipate significant energy and make the machineimpractical as a source of electric power This point is nicelyillustrated by considering the open-circuit voltage on theleft-hand side of Eq (21) to be composed of the two termson the right-hand side of this equation V frac14 Vi thorn Vs Thevoltage Vi is due to the current in the disc and can be thoughtof as the voltage produced by the surface current ~Js passingthrough the resistance of a strip of conductor of unit widthThe voltage Vs represents a source due to the applied mag-netic field traditionally called the ldquoelectromotive forcerdquo(emf) In Fig 13 the three terms V Vi and Vs (normalized sothat jVnjmax frac14 10) are shown as a function of the positionqbA along the radial line ub frac14 0 This line starts at the cen-ter of the disc passes under the pole and ends at the topedge of the disc The emf (Vsn) increases on the portion ofthe path that is under the pole and reaches a maximum ofabout Vsnfrac14 25 The voltage Vin due to the current is alwaysnegative so it subtracts from the emf making the open-circuit voltage drop to only Vnfrac14 08 at the top edge of thedisc Thus the eddy currents in the disc reduce the voltageavailable from the machine to about one-third of the emf

The inefficiency of the dynamo was also evident in themeasurements discussed in Sec IV When the disc wasrotated between the poles of the magnet a strong force resist-ing the motion was felt a consequence of Lenzrsquos law30

There was also a noticeable increase in the temperature ofthe rotating disc In a period of about one minute the tem-perature of the disc rotating at 772 rpm increased from theambient value of 246 8C (763 8F) to about 680 8C (154 8F)The energy that raises the temperature of the disc must besupplied by the mechanical source turning the axle of thedisc

Symmetric designs for the dynamo such as the one shownin Fig 2(b) do not suffer from the same problem with eddycurrents as the asymmetric design Analysis of the com-pletely symmetric structure shows that there is no conductioncurrent in the disc when the terminals are left open Andeven when current is drawn by the external circuit the cur-rent in the disc is in the radial direction that is when noeddies are formed13ndash16 Hence most attempts to make apractical unipolar or homopolar generator have used sym-metric structures These generators produce a high current at

Fig 12 Comparison of theoretical and measured results for the open-circuit

voltage V versus the position of the brush qb ub (a) Radial position fixed at

qbAfrac14 0960 while angular position is varied (b) Angular position fixed at

ub frac14 45

while radial position is varied (In both cases aeAfrac14 0173

cAfrac14 0663 and xfrac14 772 rpm)

Fig 13 The open-circuit voltage V voltage due to the current in the disc Vi

and source voltage (emf) Vs versus qbA for the radial line ub frac14 0 All of

the voltages are normalized so that maxjVnj frac14 10 and all of the parameters

are the same as those for Fig 8

915 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 915

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

a low voltage and the loss due to the current passing throughthe resistance of the brushes is of concern To lower this re-sistance some designs incorporate mercury or a liquid metal-lic alloy in the brushes as Faraday did albeit with moresophisticated designs than Faradayrsquos simple contacts31

Commercial use of even the symmetric designs has been lim-ited to very specialized applications32

The eddy-current disc brake is a structure very similar toFaradayrsquos first dynamo with the permanent magnet inFaradayrsquos design replaced by an electromagnet The conduct-ing disc is mounted on an axle that normally rotates Whenthe rotation is to be slowed or stopped the electromagnet isactivated and the kinetic energy taken from the rotationalmotion of the axle is dissipated by the eddy currents in thedisc The applications of this technology are diverse and rangefrom the brakes on the famous Japanese Shinkansen (bullettrain) to providing resistance to motion in fitness equipmentsuch as rowing machines33 The analysis for the eddy-currentdisc brake is similar to that for the Faraday disc only the em-phasis is usually on obtaining the torque on the axle versusvarious parameters such as the angular frequency and themagnetic flux34ndash37 The calculation we have used fromSmythersquos paper was originally intended for obtaining the tor-que17 In Europe linear versions of the eddy-current brake arebeing tested for railway applications38 In these devices themagnetic field is generated by a linear array of electromagnetson the car and applied to the rail The eddy currents producedin the rail slow the car and heat the rail A section of the rail isheated only as the car passes so the heating is not as severe asfor the disc brake which is heated continuously as long as thebrake is energized

As with the more common friction brakes all eddy-current brakes convert kinetic energy into heat This is unde-sirable in applications in which energy conservation is ofconcern For these applications electromagnetic regenerativebraking may be a better option Regenerative braking is usedin electric and hybrid (gasolineelectric) automobilesBatteries are used to power electric motors that propel theautomobile and when the automobile is to be slowed orstopped these motors are used as generators to charge thebatteries Thus a portion of the kinetic energy of the auto-mobile is recovered and stored for future use

The experiments Faraday made with his first dynamo gavehim insight into electromagnetic induction and among otherthings allowed him to give the first correct explanation forwhat caused Aragorsquos rotations The dynamo in its originalform was not a prototype for a practical source of electricpower However from the historical record it is clear thatFaradayrsquos first dynamo was an important step in the develop-ment of modern generators39 The fundamental principles hediscovered with his dynamo were quickly applied by othersto produce efficient generators

ACKNOWLEDGMENTS

The author would like to thank his Georgia Tech col-leagues Professors Waymond Scott Jr John Buck (bothfrom ECE) and Andrew Zangwill (Physics) for their sugges-tions for improving the manuscript

a)Electronic mail glennsmithecegatechedu1M Arago ldquoLrsquoaction que les corps aimantes et ceux qui ne le sont pas exer-

cent les uns sur les autresrdquo Ann Chim Phys 28 325 (1825) and M

Arago ldquoNote concernant les phenomenes magnetiques auxquels le

mouvement donne naissancerdquo ibid 32 213ndash223 (1826) An account in

English is in C Babbage and J F W Herschel ldquoAccount of the repetition

of M Aragorsquos experiments on magnetism manifested by various substan-

ces during the act of rotationrdquo Philos Trans Roy Soc London 115

467ndash496 (1825)2M Faraday Experimental Researches in Electricity Vol I (Taylor and

Francis London 1839 Republication Green Lion Santa Fe NM 2000)

Paragraph 83 pg 253M Faraday Faradayrsquos Diary edited by T Martin (G Bell and Sons

London 1932) Vol I Paragraphs 99ndash136 pp 381ndash3864M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraphs 83ndash100 pp 25ndash29 and 119ndash134 pp

34ndash395S P Thompson Michael Faraday His Life and Work (Cassell London

1901 Republication Elibron Classics New York 2004) pg 1236M Faraday Faradayrsquos Diary edited by T Martin (G Bell and Sons

London 1932) Vol I Paragraph 228 pg 3977M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraph 155 p 468R J Stephenson ldquoExperiments with a unipolar generator and motorrdquo

Am J Phys 5 108ndash110 (1937)9J W Then ldquoLaboratory experiments in motional electric fieldsrdquo Am J

Phys 28 557ndash559 (1960)10J W Then ldquoExperimental study of the motional electromotive forcerdquo

Am J Phys 30 411ndash415 (1962)11M J Crooks D B Litvin P W Matthews R Macaulay and J Shaw

ldquoOne-piece Faraday generator A paradoxical experiment from 1851rdquo

Am J Phys 46 729ndash731 (1978)12R D Eagleton ldquoTwo laboratory experiments involving the homopolar

generatorrdquo Am J Phys 55 621ndash623 (1987)13H H Woodson and J R Melcher Electromechanical Dynamics Part I

Discreet Systems (Wiley New York 1968) pp 286ndash28914J Van Bladel Relativity and Engineering (Springer-Verlag Berlin 1984)

pp 283ndash29115J Van Bladel ldquoRelativistic theory of rotating disksrdquo Proc IEEE 61

260ndash268 (1973)16H Montgomery ldquoCurrent flow patterns in a Faraday discrdquo Eur J Phys

25 171ndash183 (2004)17W R Smythe ldquoOn eddy currents in a rotating diskrdquo Trans AIEE 61

681ndash684 (1942)18W R Smythe Static and Dynamic Electricity 3rd ed (McGraw-Hill

New York 1968) pp 390ndash39319This type of problem is sometimes referred to as a ldquoquasi-static magnetic

field systemrdquo and it is discussed in detail in H H Woodson and J R

Melcher Electromechanical Dynamics Part I Discreet Systems (Wiley

New York 1968) pp 6ndash8 260ndash264 270ndash277 284ndash289 B19ndashB2520In the local rest frame for a point on the disc the thickness of the disc is

assumed to be much less than the skin depth in the conductor In this

frame the volume current density is then practically uniform throughout

the thickness of the disc In the quasi-static approximation the volume

current density is the same in the moving frame and in the laboratory

frame hence it is practically uniform in the laboratory frame21A I Borisenko and I E Tarapov Vector and Tensor Analysis with

Applications translated by R A Silverman (Prentice-Hall Englewood

Cliffs NJ 1968) pp 211ndash22322Smythe solves for the stream function U which is simply related to the

magnetic scalar potential UethquTHORN frac14 eth2=l0THORNUethqu z frac14 0thornTHORN His method

is briefly as follows First he considers a pole placed over an infinite con-

ducting sheet The radius of the pole is a and it is offset from the origin by

c He obtains an integral for the stream function U1 of this pole and he

evaluates this integral approximately Next he introduces a second pole

with the same total flux as the first pole but of opposite sign and he obtains

the approximate stream function U2 for this pole The radius and offset

from the origin for the second pole are ~a and ~c with ~c ~a gt A Finally

he adjusts ~a and ~c to make the superposition Ufrac14U1thornU2 satisfy the

boundary condition Ufrac14 0 at the radius qfrac14A (edge of the disc)23Normally a tube of current is a region of three-dimensional space

delineated by a side surface that is everywhere parallel to the direction of

the volume current density ~J The total current I passing through any

cross-sectional area of the tube is the samemdashthink of a pipe with the water

flowing through the pipe representing the current Here we are using the

two-dimensional version of the tube of current The current is on a surface

(on the disc) and the sides of the tube are two lines that are everywhere

parallel to the direction of the surface current density ~Js The total current

916 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 916

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

Faraday recorded the deflection of the galvanometer usingterms such as ldquomost stronglyrdquo and ldquoless stronglyrdquo Faradayvaried the arrangement of the elements (brushes magnet etc)in his dynamo to determine the characteristics of the currenthe thought was produced in the disc In this section using theanalysis just discussed we will examine Faradayrsquos observa-tions and his interpretation for them We will assume that ouropen-circuit voltage is proportional to the current that causedthe deflection of Faradayrsquos galvanometer26

Faraday reversed both the direction of rotation of the discand the direction of the applied magnetic field and he sawthat a reversal of either changed the sense of the deflectionof the galvanometer He reduced the applied magnetic fieldby moving one of the poles of the magnet away from thedisc and he saw the deflection decrease These observationsare in agreement with the product xBo appearing in theexpressions for the open-circuit voltage in Eqs (23)ndash(25)

Faraday was able to deduce many of the key features wesee in the contour plot for the open-circuit voltage shown inFig 8 He did this by moving the brushes to different loca-tions on the disc and observing the change in the deflectionof the galvanometer He placed one brush at the center of thedisc (on the axle) and the other brush at symmetrical pointsabout the line ub frac14 0 or 1808 By noting that the deflectionof the galvanometer was the same at the two points hededuced the even symmetry we see for the pattern about thisline With one brush at the center of the disc and the other atthe edge of the disc he varied the angular position of the lat-ter and noticed that the deflection of the galvanometer wasmaximum when the brush was opposite the magnetic pole(ub frac14 0) and decreased as the brush was moved away fromthis point (ub increased) This observation is consistent withwhat we see in Fig 8 for qbfrac14A

Faraday also varied the position of the magnetic poles Forcomparison with his observations we will determine theopen-circuit voltage when the brush in Fig 3(a) is at the topedge of the disc directly above the magnetic poleqb frac14 A ub frac14 0 For this case the open-circuit voltage Eq(23) becomes simply

Vethqb frac14 Aub frac14 0THORN frac14 xB0A2

2etha=ATHORN2 1thorn c=A

1 c=A

(27)

which we have plotted in Fig 9 as a function of polersquos sizeaA and position cA

For any size pole the voltage is maximum when the poleis as far away from the center of the disc as possible whilestill being completely over the disc or when cAfrac14 1 ndash aAIn this position the velocity of the disc relative to the pole ismaximum Also the larger the pole the larger the voltagewith maximum voltage occurring when the diameter of thepole equals the radius of the disc that is aAfrac14 05 andcAfrac14 05 When Faraday lowered the magnetic poles so asto decrease cA he noted that the ldquohellipeffect was the same [aswith the poles raised] and the deflection was very sensiblerdquoThis observation can be explained by the small size ofFaradayrsquos poles They were either square or rectangular inshape and roughly equivalent to circular poles withaAfrac14 00727 Thus his results approximately correspond tothe curve aAfrac14 01 in Fig 9 which in the central region isfairly flat so the variation with cA would have been difficultto detect in his qualitative measurements

From his measurements Faraday was able to show that cur-rent was induced in the conducting disc only when the disc

moved through the magnetic field He concluded that the cur-rent went in the radial direction from the central part of thedisc past the pole to the edge of the disc where itldquodischargedrdquo or returned in the part of the disc on either sideof the pole This description is remarkably similar to what wesee for the distribution of the surface current in Fig 6

Commenting on his first dynamo Faraday said ldquoHeretherefore was demonstrated the production of a permanent[direct or continuous] current of electricity by ordinary mag-netsrdquo28 Thus he fulfilled the goal he stated earlier of makingthe experiment of Arago a new source of electricity Faradaywas confident that he had also found the explanation forAragorsquos rotations which was as follows The magnetic fieldof the needle induced current in the rotating disc this cur-rent in turn created a magnetic field that exerted a force onthe needle causing it to rotate

IV REPEATING FARADAYrsquoS EXPERIMENTS

We repeated some of the experiments Faraday made withhis first dynamo with a two-fold objective in mind First weaim to verify the analysis presented earlier which is basedon Smythersquos approximation for the magnetic scalar potentialand second we provide quantitative results that supportFaradayrsquos qualitative observations Concerning the firstobjective we followed Faradayrsquos advice that ldquoNothing is asgood as an experiment which whilst it sets error right givesan absolute advancement in knowledgerdquo29

Figures 10(a) and 10(b) are schematic drawings for the ex-perimental model which is very similar to Faradayrsquosdynamo except that it is constructed from modern materialsThe circular copper disc of radius Afrac14 750 cm and thicknessdfrac14 086 mm was mounted on a brass axle The axle washeld in the chuck of a drill press and the speed of rotation ofthe disc which was measured with an optical tachometerwas varied by adjusting the beltstepped-pulley system of thedrill press The magnetic field was that of two neodymiumpermanent magnets (K amp J Magnetics Inc DX08) of radiusamfrac14 127 cm and length 127 cm bonded to the ends of ayoke formed from laminated steel sheets The purpose of theyoke was to direct the return field of the magnets to an areaoff the disc so that the only magnetic field applied to the discwas that between the poles Figure 10(c) is a graph of the

Fig 9 Open-circuit voltage as a function of the polersquos size aA and position

cA when the brush is at the top edge of the disc directly above the magnetic

pole at qb frac14 A ub frac14 0 The voltage is normalized so that maxjVnj frac14 10

which occurs for aAfrac14 05 cAfrac14 05 For all points on the graph the pole is

completely over the disc

913 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 913

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

magnetic field measured on the horizontal plane centeredbetween the poles of the permanent magnets The points onthis graph are averages of measurements made along threedifferent radial lines from the center of the poles The mag-netic field is seen to have a maximum value of about 108 Tat the center of the poles (ramfrac14 00) to decrease to abouthalf this value at the edges of the poles (ramfrac14 10) and torapidly decay beyond this point

The disc was connected to the external circuit by spring-loaded carbon brushes of diameter 46 mm [see Fig 10(b)]

One brush was at the center of the disc and a pair of opposingbrushes located on opposite sides of the disc and connected inparallel could be moved to various positions (qb ub) on thedisc Wires embedded in the brushes were soldered directly tothe external circuit where the open-circuit voltage V wasmeasured with a digital voltmeter The good electrical contactbetween the spring-loaded brushes and the disc resulted invery little fluctuation of the voltage as the disc rotated

One complication arose in the comparison of the theorywith the measurements The magnetic field for the theory isuniform over the area of the poles whereas the magneticfield in the experiment varies with the radial distance fromthe center of the poles To make the comparison we definedan effective radius ae for the poles used in the experimentwhich is the radius of fictitious poles with uniform magneticfield equal to Bz(rfrac14 0) that produce the same total flux as theactual poles Mathematically we express this as

pa2eBzethr frac14 0THORN frac14 2p

eth3am

rfrac140

BzethrTHORNrdr (28)

or

ae frac14 am

2

eth3am

rfrac140

BzethrTHORNrdr

a2mBzethr frac14 0THORN

26664

37775

1=2

(29)

When Eq (29) was applied to the results in Fig 10(c) thevalue aefrac14 102am was obtained A value of ae this close toam implies that the additional magnetic field outside of thepoles in Fig 10(c) just compensates for the reduction in themagnetic field under the poles For comparison of theoreticalcalculations with measurements the value ae was used for ain the theory

In all of our measurements the magnetic poles were locatedat cAfrac14 0663 and aeAfrac14 0173 These values are roughly thesame as those used in the earlier calculations For the firstmeasurement the brush was placed above the magnetic polesnear the edge of the disc (qb=A frac14 0951 ub frac14 0) and theangular speed of the disc was varied The theoretical results(curve) for the open-circuit voltage V versus angular speed xare compared with the measured results (dots) in Fig 11 The

Fig 10 Details for the experimental model which is similar to Faradayrsquos

first dynamo (a) Side view in which the brushes and external circuit are not

shown (b) Front view showing the brushes and external circuit (c) Graph of

the magnetic field measured on the horizontal plane centered between the

poles of the permanent magnets The points on the graph are the average of

measurements made along three different radial lines from the center of the

poles

Fig 11 Comparison of theoretical and measured results for the open-circuit

voltage V versus the angular frequency x with qb=A frac14 0951 ub frac14 0

(aeAfrac14 0173 and cAfrac14 0663)

914 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 914

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

theory and measurement are seen to be in good agreement andboth exhibit a linear dependence on x as expected

In the next set of measurements the angular speed of thedisc was fixed at 772 rpm and the position of the brush(qb ub) was moved over different paths on the surface ofthe disc much as Faraday did in his experimentsRepresentative results from these experiments which meas-ured the open-circuit voltage V versus position are shown inFig 12 In Fig 12(a) the radial position of the brush is fixed(qbAfrac14 0960) while the angular position ub is varied and inFig 12(b) the angular position for the brush is fixed(ub frac14 45) while the radial position qb is varied In bothcases the theory (curve) and measurement (dots) are againin good agreement Notice in Fig 12(b) that both the theoret-ical and measured results show the voltage becoming slightlynegative for qb=A lt 04

V DISCUSSION WHAT EVER BECAME OF

FARADAYrsquoS FIRST DYNAMO

In our study we wondered what became of Faradayrsquos firstdynamo and if his machine could be of use today The an-swer to this question is fairly simple As we saw from ouranalysis eddy currents are produced in the disc even when

no current is being drawn from the machine These eddy cur-rents dissipate significant energy and make the machineimpractical as a source of electric power This point is nicelyillustrated by considering the open-circuit voltage on theleft-hand side of Eq (21) to be composed of the two termson the right-hand side of this equation V frac14 Vi thorn Vs Thevoltage Vi is due to the current in the disc and can be thoughtof as the voltage produced by the surface current ~Js passingthrough the resistance of a strip of conductor of unit widthThe voltage Vs represents a source due to the applied mag-netic field traditionally called the ldquoelectromotive forcerdquo(emf) In Fig 13 the three terms V Vi and Vs (normalized sothat jVnjmax frac14 10) are shown as a function of the positionqbA along the radial line ub frac14 0 This line starts at the cen-ter of the disc passes under the pole and ends at the topedge of the disc The emf (Vsn) increases on the portion ofthe path that is under the pole and reaches a maximum ofabout Vsnfrac14 25 The voltage Vin due to the current is alwaysnegative so it subtracts from the emf making the open-circuit voltage drop to only Vnfrac14 08 at the top edge of thedisc Thus the eddy currents in the disc reduce the voltageavailable from the machine to about one-third of the emf

The inefficiency of the dynamo was also evident in themeasurements discussed in Sec IV When the disc wasrotated between the poles of the magnet a strong force resist-ing the motion was felt a consequence of Lenzrsquos law30

There was also a noticeable increase in the temperature ofthe rotating disc In a period of about one minute the tem-perature of the disc rotating at 772 rpm increased from theambient value of 246 8C (763 8F) to about 680 8C (154 8F)The energy that raises the temperature of the disc must besupplied by the mechanical source turning the axle of thedisc

Symmetric designs for the dynamo such as the one shownin Fig 2(b) do not suffer from the same problem with eddycurrents as the asymmetric design Analysis of the com-pletely symmetric structure shows that there is no conductioncurrent in the disc when the terminals are left open Andeven when current is drawn by the external circuit the cur-rent in the disc is in the radial direction that is when noeddies are formed13ndash16 Hence most attempts to make apractical unipolar or homopolar generator have used sym-metric structures These generators produce a high current at

Fig 12 Comparison of theoretical and measured results for the open-circuit

voltage V versus the position of the brush qb ub (a) Radial position fixed at

qbAfrac14 0960 while angular position is varied (b) Angular position fixed at

ub frac14 45

while radial position is varied (In both cases aeAfrac14 0173

cAfrac14 0663 and xfrac14 772 rpm)

Fig 13 The open-circuit voltage V voltage due to the current in the disc Vi

and source voltage (emf) Vs versus qbA for the radial line ub frac14 0 All of

the voltages are normalized so that maxjVnj frac14 10 and all of the parameters

are the same as those for Fig 8

915 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 915

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

a low voltage and the loss due to the current passing throughthe resistance of the brushes is of concern To lower this re-sistance some designs incorporate mercury or a liquid metal-lic alloy in the brushes as Faraday did albeit with moresophisticated designs than Faradayrsquos simple contacts31

Commercial use of even the symmetric designs has been lim-ited to very specialized applications32

The eddy-current disc brake is a structure very similar toFaradayrsquos first dynamo with the permanent magnet inFaradayrsquos design replaced by an electromagnet The conduct-ing disc is mounted on an axle that normally rotates Whenthe rotation is to be slowed or stopped the electromagnet isactivated and the kinetic energy taken from the rotationalmotion of the axle is dissipated by the eddy currents in thedisc The applications of this technology are diverse and rangefrom the brakes on the famous Japanese Shinkansen (bullettrain) to providing resistance to motion in fitness equipmentsuch as rowing machines33 The analysis for the eddy-currentdisc brake is similar to that for the Faraday disc only the em-phasis is usually on obtaining the torque on the axle versusvarious parameters such as the angular frequency and themagnetic flux34ndash37 The calculation we have used fromSmythersquos paper was originally intended for obtaining the tor-que17 In Europe linear versions of the eddy-current brake arebeing tested for railway applications38 In these devices themagnetic field is generated by a linear array of electromagnetson the car and applied to the rail The eddy currents producedin the rail slow the car and heat the rail A section of the rail isheated only as the car passes so the heating is not as severe asfor the disc brake which is heated continuously as long as thebrake is energized

As with the more common friction brakes all eddy-current brakes convert kinetic energy into heat This is unde-sirable in applications in which energy conservation is ofconcern For these applications electromagnetic regenerativebraking may be a better option Regenerative braking is usedin electric and hybrid (gasolineelectric) automobilesBatteries are used to power electric motors that propel theautomobile and when the automobile is to be slowed orstopped these motors are used as generators to charge thebatteries Thus a portion of the kinetic energy of the auto-mobile is recovered and stored for future use

The experiments Faraday made with his first dynamo gavehim insight into electromagnetic induction and among otherthings allowed him to give the first correct explanation forwhat caused Aragorsquos rotations The dynamo in its originalform was not a prototype for a practical source of electricpower However from the historical record it is clear thatFaradayrsquos first dynamo was an important step in the develop-ment of modern generators39 The fundamental principles hediscovered with his dynamo were quickly applied by othersto produce efficient generators

ACKNOWLEDGMENTS

The author would like to thank his Georgia Tech col-leagues Professors Waymond Scott Jr John Buck (bothfrom ECE) and Andrew Zangwill (Physics) for their sugges-tions for improving the manuscript

a)Electronic mail glennsmithecegatechedu1M Arago ldquoLrsquoaction que les corps aimantes et ceux qui ne le sont pas exer-

cent les uns sur les autresrdquo Ann Chim Phys 28 325 (1825) and M

Arago ldquoNote concernant les phenomenes magnetiques auxquels le

mouvement donne naissancerdquo ibid 32 213ndash223 (1826) An account in

English is in C Babbage and J F W Herschel ldquoAccount of the repetition

of M Aragorsquos experiments on magnetism manifested by various substan-

ces during the act of rotationrdquo Philos Trans Roy Soc London 115

467ndash496 (1825)2M Faraday Experimental Researches in Electricity Vol I (Taylor and

Francis London 1839 Republication Green Lion Santa Fe NM 2000)

Paragraph 83 pg 253M Faraday Faradayrsquos Diary edited by T Martin (G Bell and Sons

London 1932) Vol I Paragraphs 99ndash136 pp 381ndash3864M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraphs 83ndash100 pp 25ndash29 and 119ndash134 pp

34ndash395S P Thompson Michael Faraday His Life and Work (Cassell London

1901 Republication Elibron Classics New York 2004) pg 1236M Faraday Faradayrsquos Diary edited by T Martin (G Bell and Sons

London 1932) Vol I Paragraph 228 pg 3977M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraph 155 p 468R J Stephenson ldquoExperiments with a unipolar generator and motorrdquo

Am J Phys 5 108ndash110 (1937)9J W Then ldquoLaboratory experiments in motional electric fieldsrdquo Am J

Phys 28 557ndash559 (1960)10J W Then ldquoExperimental study of the motional electromotive forcerdquo

Am J Phys 30 411ndash415 (1962)11M J Crooks D B Litvin P W Matthews R Macaulay and J Shaw

ldquoOne-piece Faraday generator A paradoxical experiment from 1851rdquo

Am J Phys 46 729ndash731 (1978)12R D Eagleton ldquoTwo laboratory experiments involving the homopolar

generatorrdquo Am J Phys 55 621ndash623 (1987)13H H Woodson and J R Melcher Electromechanical Dynamics Part I

Discreet Systems (Wiley New York 1968) pp 286ndash28914J Van Bladel Relativity and Engineering (Springer-Verlag Berlin 1984)

pp 283ndash29115J Van Bladel ldquoRelativistic theory of rotating disksrdquo Proc IEEE 61

260ndash268 (1973)16H Montgomery ldquoCurrent flow patterns in a Faraday discrdquo Eur J Phys

25 171ndash183 (2004)17W R Smythe ldquoOn eddy currents in a rotating diskrdquo Trans AIEE 61

681ndash684 (1942)18W R Smythe Static and Dynamic Electricity 3rd ed (McGraw-Hill

New York 1968) pp 390ndash39319This type of problem is sometimes referred to as a ldquoquasi-static magnetic

field systemrdquo and it is discussed in detail in H H Woodson and J R

Melcher Electromechanical Dynamics Part I Discreet Systems (Wiley

New York 1968) pp 6ndash8 260ndash264 270ndash277 284ndash289 B19ndashB2520In the local rest frame for a point on the disc the thickness of the disc is

assumed to be much less than the skin depth in the conductor In this

frame the volume current density is then practically uniform throughout

the thickness of the disc In the quasi-static approximation the volume

current density is the same in the moving frame and in the laboratory

frame hence it is practically uniform in the laboratory frame21A I Borisenko and I E Tarapov Vector and Tensor Analysis with

Applications translated by R A Silverman (Prentice-Hall Englewood

Cliffs NJ 1968) pp 211ndash22322Smythe solves for the stream function U which is simply related to the

magnetic scalar potential UethquTHORN frac14 eth2=l0THORNUethqu z frac14 0thornTHORN His method

is briefly as follows First he considers a pole placed over an infinite con-

ducting sheet The radius of the pole is a and it is offset from the origin by

c He obtains an integral for the stream function U1 of this pole and he

evaluates this integral approximately Next he introduces a second pole

with the same total flux as the first pole but of opposite sign and he obtains

the approximate stream function U2 for this pole The radius and offset

from the origin for the second pole are ~a and ~c with ~c ~a gt A Finally

he adjusts ~a and ~c to make the superposition Ufrac14U1thornU2 satisfy the

boundary condition Ufrac14 0 at the radius qfrac14A (edge of the disc)23Normally a tube of current is a region of three-dimensional space

delineated by a side surface that is everywhere parallel to the direction of

the volume current density ~J The total current I passing through any

cross-sectional area of the tube is the samemdashthink of a pipe with the water

flowing through the pipe representing the current Here we are using the

two-dimensional version of the tube of current The current is on a surface

(on the disc) and the sides of the tube are two lines that are everywhere

parallel to the direction of the surface current density ~Js The total current

916 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 916

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

magnetic field measured on the horizontal plane centeredbetween the poles of the permanent magnets The points onthis graph are averages of measurements made along threedifferent radial lines from the center of the poles The mag-netic field is seen to have a maximum value of about 108 Tat the center of the poles (ramfrac14 00) to decrease to abouthalf this value at the edges of the poles (ramfrac14 10) and torapidly decay beyond this point

The disc was connected to the external circuit by spring-loaded carbon brushes of diameter 46 mm [see Fig 10(b)]

One brush was at the center of the disc and a pair of opposingbrushes located on opposite sides of the disc and connected inparallel could be moved to various positions (qb ub) on thedisc Wires embedded in the brushes were soldered directly tothe external circuit where the open-circuit voltage V wasmeasured with a digital voltmeter The good electrical contactbetween the spring-loaded brushes and the disc resulted invery little fluctuation of the voltage as the disc rotated

One complication arose in the comparison of the theorywith the measurements The magnetic field for the theory isuniform over the area of the poles whereas the magneticfield in the experiment varies with the radial distance fromthe center of the poles To make the comparison we definedan effective radius ae for the poles used in the experimentwhich is the radius of fictitious poles with uniform magneticfield equal to Bz(rfrac14 0) that produce the same total flux as theactual poles Mathematically we express this as

pa2eBzethr frac14 0THORN frac14 2p

eth3am

rfrac140

BzethrTHORNrdr (28)

or

ae frac14 am

2

eth3am

rfrac140

BzethrTHORNrdr

a2mBzethr frac14 0THORN

26664

37775

1=2

(29)

When Eq (29) was applied to the results in Fig 10(c) thevalue aefrac14 102am was obtained A value of ae this close toam implies that the additional magnetic field outside of thepoles in Fig 10(c) just compensates for the reduction in themagnetic field under the poles For comparison of theoreticalcalculations with measurements the value ae was used for ain the theory

In all of our measurements the magnetic poles were locatedat cAfrac14 0663 and aeAfrac14 0173 These values are roughly thesame as those used in the earlier calculations For the firstmeasurement the brush was placed above the magnetic polesnear the edge of the disc (qb=A frac14 0951 ub frac14 0) and theangular speed of the disc was varied The theoretical results(curve) for the open-circuit voltage V versus angular speed xare compared with the measured results (dots) in Fig 11 The

Fig 10 Details for the experimental model which is similar to Faradayrsquos

first dynamo (a) Side view in which the brushes and external circuit are not

shown (b) Front view showing the brushes and external circuit (c) Graph of

the magnetic field measured on the horizontal plane centered between the

poles of the permanent magnets The points on the graph are the average of

measurements made along three different radial lines from the center of the

poles

Fig 11 Comparison of theoretical and measured results for the open-circuit

voltage V versus the angular frequency x with qb=A frac14 0951 ub frac14 0

(aeAfrac14 0173 and cAfrac14 0663)

914 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 914

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

theory and measurement are seen to be in good agreement andboth exhibit a linear dependence on x as expected

In the next set of measurements the angular speed of thedisc was fixed at 772 rpm and the position of the brush(qb ub) was moved over different paths on the surface ofthe disc much as Faraday did in his experimentsRepresentative results from these experiments which meas-ured the open-circuit voltage V versus position are shown inFig 12 In Fig 12(a) the radial position of the brush is fixed(qbAfrac14 0960) while the angular position ub is varied and inFig 12(b) the angular position for the brush is fixed(ub frac14 45) while the radial position qb is varied In bothcases the theory (curve) and measurement (dots) are againin good agreement Notice in Fig 12(b) that both the theoret-ical and measured results show the voltage becoming slightlynegative for qb=A lt 04

V DISCUSSION WHAT EVER BECAME OF

FARADAYrsquoS FIRST DYNAMO

In our study we wondered what became of Faradayrsquos firstdynamo and if his machine could be of use today The an-swer to this question is fairly simple As we saw from ouranalysis eddy currents are produced in the disc even when

no current is being drawn from the machine These eddy cur-rents dissipate significant energy and make the machineimpractical as a source of electric power This point is nicelyillustrated by considering the open-circuit voltage on theleft-hand side of Eq (21) to be composed of the two termson the right-hand side of this equation V frac14 Vi thorn Vs Thevoltage Vi is due to the current in the disc and can be thoughtof as the voltage produced by the surface current ~Js passingthrough the resistance of a strip of conductor of unit widthThe voltage Vs represents a source due to the applied mag-netic field traditionally called the ldquoelectromotive forcerdquo(emf) In Fig 13 the three terms V Vi and Vs (normalized sothat jVnjmax frac14 10) are shown as a function of the positionqbA along the radial line ub frac14 0 This line starts at the cen-ter of the disc passes under the pole and ends at the topedge of the disc The emf (Vsn) increases on the portion ofthe path that is under the pole and reaches a maximum ofabout Vsnfrac14 25 The voltage Vin due to the current is alwaysnegative so it subtracts from the emf making the open-circuit voltage drop to only Vnfrac14 08 at the top edge of thedisc Thus the eddy currents in the disc reduce the voltageavailable from the machine to about one-third of the emf

The inefficiency of the dynamo was also evident in themeasurements discussed in Sec IV When the disc wasrotated between the poles of the magnet a strong force resist-ing the motion was felt a consequence of Lenzrsquos law30

There was also a noticeable increase in the temperature ofthe rotating disc In a period of about one minute the tem-perature of the disc rotating at 772 rpm increased from theambient value of 246 8C (763 8F) to about 680 8C (154 8F)The energy that raises the temperature of the disc must besupplied by the mechanical source turning the axle of thedisc

Symmetric designs for the dynamo such as the one shownin Fig 2(b) do not suffer from the same problem with eddycurrents as the asymmetric design Analysis of the com-pletely symmetric structure shows that there is no conductioncurrent in the disc when the terminals are left open Andeven when current is drawn by the external circuit the cur-rent in the disc is in the radial direction that is when noeddies are formed13ndash16 Hence most attempts to make apractical unipolar or homopolar generator have used sym-metric structures These generators produce a high current at

Fig 12 Comparison of theoretical and measured results for the open-circuit

voltage V versus the position of the brush qb ub (a) Radial position fixed at

qbAfrac14 0960 while angular position is varied (b) Angular position fixed at

ub frac14 45

while radial position is varied (In both cases aeAfrac14 0173

cAfrac14 0663 and xfrac14 772 rpm)

Fig 13 The open-circuit voltage V voltage due to the current in the disc Vi

and source voltage (emf) Vs versus qbA for the radial line ub frac14 0 All of

the voltages are normalized so that maxjVnj frac14 10 and all of the parameters

are the same as those for Fig 8

915 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 915

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

a low voltage and the loss due to the current passing throughthe resistance of the brushes is of concern To lower this re-sistance some designs incorporate mercury or a liquid metal-lic alloy in the brushes as Faraday did albeit with moresophisticated designs than Faradayrsquos simple contacts31

Commercial use of even the symmetric designs has been lim-ited to very specialized applications32

The eddy-current disc brake is a structure very similar toFaradayrsquos first dynamo with the permanent magnet inFaradayrsquos design replaced by an electromagnet The conduct-ing disc is mounted on an axle that normally rotates Whenthe rotation is to be slowed or stopped the electromagnet isactivated and the kinetic energy taken from the rotationalmotion of the axle is dissipated by the eddy currents in thedisc The applications of this technology are diverse and rangefrom the brakes on the famous Japanese Shinkansen (bullettrain) to providing resistance to motion in fitness equipmentsuch as rowing machines33 The analysis for the eddy-currentdisc brake is similar to that for the Faraday disc only the em-phasis is usually on obtaining the torque on the axle versusvarious parameters such as the angular frequency and themagnetic flux34ndash37 The calculation we have used fromSmythersquos paper was originally intended for obtaining the tor-que17 In Europe linear versions of the eddy-current brake arebeing tested for railway applications38 In these devices themagnetic field is generated by a linear array of electromagnetson the car and applied to the rail The eddy currents producedin the rail slow the car and heat the rail A section of the rail isheated only as the car passes so the heating is not as severe asfor the disc brake which is heated continuously as long as thebrake is energized

As with the more common friction brakes all eddy-current brakes convert kinetic energy into heat This is unde-sirable in applications in which energy conservation is ofconcern For these applications electromagnetic regenerativebraking may be a better option Regenerative braking is usedin electric and hybrid (gasolineelectric) automobilesBatteries are used to power electric motors that propel theautomobile and when the automobile is to be slowed orstopped these motors are used as generators to charge thebatteries Thus a portion of the kinetic energy of the auto-mobile is recovered and stored for future use

The experiments Faraday made with his first dynamo gavehim insight into electromagnetic induction and among otherthings allowed him to give the first correct explanation forwhat caused Aragorsquos rotations The dynamo in its originalform was not a prototype for a practical source of electricpower However from the historical record it is clear thatFaradayrsquos first dynamo was an important step in the develop-ment of modern generators39 The fundamental principles hediscovered with his dynamo were quickly applied by othersto produce efficient generators

ACKNOWLEDGMENTS

The author would like to thank his Georgia Tech col-leagues Professors Waymond Scott Jr John Buck (bothfrom ECE) and Andrew Zangwill (Physics) for their sugges-tions for improving the manuscript

a)Electronic mail glennsmithecegatechedu1M Arago ldquoLrsquoaction que les corps aimantes et ceux qui ne le sont pas exer-

cent les uns sur les autresrdquo Ann Chim Phys 28 325 (1825) and M

Arago ldquoNote concernant les phenomenes magnetiques auxquels le

mouvement donne naissancerdquo ibid 32 213ndash223 (1826) An account in

English is in C Babbage and J F W Herschel ldquoAccount of the repetition

of M Aragorsquos experiments on magnetism manifested by various substan-

ces during the act of rotationrdquo Philos Trans Roy Soc London 115

467ndash496 (1825)2M Faraday Experimental Researches in Electricity Vol I (Taylor and

Francis London 1839 Republication Green Lion Santa Fe NM 2000)

Paragraph 83 pg 253M Faraday Faradayrsquos Diary edited by T Martin (G Bell and Sons

London 1932) Vol I Paragraphs 99ndash136 pp 381ndash3864M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraphs 83ndash100 pp 25ndash29 and 119ndash134 pp

34ndash395S P Thompson Michael Faraday His Life and Work (Cassell London

1901 Republication Elibron Classics New York 2004) pg 1236M Faraday Faradayrsquos Diary edited by T Martin (G Bell and Sons

London 1932) Vol I Paragraph 228 pg 3977M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraph 155 p 468R J Stephenson ldquoExperiments with a unipolar generator and motorrdquo

Am J Phys 5 108ndash110 (1937)9J W Then ldquoLaboratory experiments in motional electric fieldsrdquo Am J

Phys 28 557ndash559 (1960)10J W Then ldquoExperimental study of the motional electromotive forcerdquo

Am J Phys 30 411ndash415 (1962)11M J Crooks D B Litvin P W Matthews R Macaulay and J Shaw

ldquoOne-piece Faraday generator A paradoxical experiment from 1851rdquo

Am J Phys 46 729ndash731 (1978)12R D Eagleton ldquoTwo laboratory experiments involving the homopolar

generatorrdquo Am J Phys 55 621ndash623 (1987)13H H Woodson and J R Melcher Electromechanical Dynamics Part I

Discreet Systems (Wiley New York 1968) pp 286ndash28914J Van Bladel Relativity and Engineering (Springer-Verlag Berlin 1984)

pp 283ndash29115J Van Bladel ldquoRelativistic theory of rotating disksrdquo Proc IEEE 61

260ndash268 (1973)16H Montgomery ldquoCurrent flow patterns in a Faraday discrdquo Eur J Phys

25 171ndash183 (2004)17W R Smythe ldquoOn eddy currents in a rotating diskrdquo Trans AIEE 61

681ndash684 (1942)18W R Smythe Static and Dynamic Electricity 3rd ed (McGraw-Hill

New York 1968) pp 390ndash39319This type of problem is sometimes referred to as a ldquoquasi-static magnetic

field systemrdquo and it is discussed in detail in H H Woodson and J R

Melcher Electromechanical Dynamics Part I Discreet Systems (Wiley

New York 1968) pp 6ndash8 260ndash264 270ndash277 284ndash289 B19ndashB2520In the local rest frame for a point on the disc the thickness of the disc is

assumed to be much less than the skin depth in the conductor In this

frame the volume current density is then practically uniform throughout

the thickness of the disc In the quasi-static approximation the volume

current density is the same in the moving frame and in the laboratory

frame hence it is practically uniform in the laboratory frame21A I Borisenko and I E Tarapov Vector and Tensor Analysis with

Applications translated by R A Silverman (Prentice-Hall Englewood

Cliffs NJ 1968) pp 211ndash22322Smythe solves for the stream function U which is simply related to the

magnetic scalar potential UethquTHORN frac14 eth2=l0THORNUethqu z frac14 0thornTHORN His method

is briefly as follows First he considers a pole placed over an infinite con-

ducting sheet The radius of the pole is a and it is offset from the origin by

c He obtains an integral for the stream function U1 of this pole and he

evaluates this integral approximately Next he introduces a second pole

with the same total flux as the first pole but of opposite sign and he obtains

the approximate stream function U2 for this pole The radius and offset

from the origin for the second pole are ~a and ~c with ~c ~a gt A Finally

he adjusts ~a and ~c to make the superposition Ufrac14U1thornU2 satisfy the

boundary condition Ufrac14 0 at the radius qfrac14A (edge of the disc)23Normally a tube of current is a region of three-dimensional space

delineated by a side surface that is everywhere parallel to the direction of

the volume current density ~J The total current I passing through any

cross-sectional area of the tube is the samemdashthink of a pipe with the water

flowing through the pipe representing the current Here we are using the

two-dimensional version of the tube of current The current is on a surface

(on the disc) and the sides of the tube are two lines that are everywhere

parallel to the direction of the surface current density ~Js The total current

916 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 916

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

theory and measurement are seen to be in good agreement andboth exhibit a linear dependence on x as expected

In the next set of measurements the angular speed of thedisc was fixed at 772 rpm and the position of the brush(qb ub) was moved over different paths on the surface ofthe disc much as Faraday did in his experimentsRepresentative results from these experiments which meas-ured the open-circuit voltage V versus position are shown inFig 12 In Fig 12(a) the radial position of the brush is fixed(qbAfrac14 0960) while the angular position ub is varied and inFig 12(b) the angular position for the brush is fixed(ub frac14 45) while the radial position qb is varied In bothcases the theory (curve) and measurement (dots) are againin good agreement Notice in Fig 12(b) that both the theoret-ical and measured results show the voltage becoming slightlynegative for qb=A lt 04

V DISCUSSION WHAT EVER BECAME OF

FARADAYrsquoS FIRST DYNAMO

In our study we wondered what became of Faradayrsquos firstdynamo and if his machine could be of use today The an-swer to this question is fairly simple As we saw from ouranalysis eddy currents are produced in the disc even when

no current is being drawn from the machine These eddy cur-rents dissipate significant energy and make the machineimpractical as a source of electric power This point is nicelyillustrated by considering the open-circuit voltage on theleft-hand side of Eq (21) to be composed of the two termson the right-hand side of this equation V frac14 Vi thorn Vs Thevoltage Vi is due to the current in the disc and can be thoughtof as the voltage produced by the surface current ~Js passingthrough the resistance of a strip of conductor of unit widthThe voltage Vs represents a source due to the applied mag-netic field traditionally called the ldquoelectromotive forcerdquo(emf) In Fig 13 the three terms V Vi and Vs (normalized sothat jVnjmax frac14 10) are shown as a function of the positionqbA along the radial line ub frac14 0 This line starts at the cen-ter of the disc passes under the pole and ends at the topedge of the disc The emf (Vsn) increases on the portion ofthe path that is under the pole and reaches a maximum ofabout Vsnfrac14 25 The voltage Vin due to the current is alwaysnegative so it subtracts from the emf making the open-circuit voltage drop to only Vnfrac14 08 at the top edge of thedisc Thus the eddy currents in the disc reduce the voltageavailable from the machine to about one-third of the emf

The inefficiency of the dynamo was also evident in themeasurements discussed in Sec IV When the disc wasrotated between the poles of the magnet a strong force resist-ing the motion was felt a consequence of Lenzrsquos law30

There was also a noticeable increase in the temperature ofthe rotating disc In a period of about one minute the tem-perature of the disc rotating at 772 rpm increased from theambient value of 246 8C (763 8F) to about 680 8C (154 8F)The energy that raises the temperature of the disc must besupplied by the mechanical source turning the axle of thedisc

Symmetric designs for the dynamo such as the one shownin Fig 2(b) do not suffer from the same problem with eddycurrents as the asymmetric design Analysis of the com-pletely symmetric structure shows that there is no conductioncurrent in the disc when the terminals are left open Andeven when current is drawn by the external circuit the cur-rent in the disc is in the radial direction that is when noeddies are formed13ndash16 Hence most attempts to make apractical unipolar or homopolar generator have used sym-metric structures These generators produce a high current at

Fig 12 Comparison of theoretical and measured results for the open-circuit

voltage V versus the position of the brush qb ub (a) Radial position fixed at

qbAfrac14 0960 while angular position is varied (b) Angular position fixed at

ub frac14 45

while radial position is varied (In both cases aeAfrac14 0173

cAfrac14 0663 and xfrac14 772 rpm)

Fig 13 The open-circuit voltage V voltage due to the current in the disc Vi

and source voltage (emf) Vs versus qbA for the radial line ub frac14 0 All of

the voltages are normalized so that maxjVnj frac14 10 and all of the parameters

are the same as those for Fig 8

915 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 915

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

a low voltage and the loss due to the current passing throughthe resistance of the brushes is of concern To lower this re-sistance some designs incorporate mercury or a liquid metal-lic alloy in the brushes as Faraday did albeit with moresophisticated designs than Faradayrsquos simple contacts31

Commercial use of even the symmetric designs has been lim-ited to very specialized applications32

The eddy-current disc brake is a structure very similar toFaradayrsquos first dynamo with the permanent magnet inFaradayrsquos design replaced by an electromagnet The conduct-ing disc is mounted on an axle that normally rotates Whenthe rotation is to be slowed or stopped the electromagnet isactivated and the kinetic energy taken from the rotationalmotion of the axle is dissipated by the eddy currents in thedisc The applications of this technology are diverse and rangefrom the brakes on the famous Japanese Shinkansen (bullettrain) to providing resistance to motion in fitness equipmentsuch as rowing machines33 The analysis for the eddy-currentdisc brake is similar to that for the Faraday disc only the em-phasis is usually on obtaining the torque on the axle versusvarious parameters such as the angular frequency and themagnetic flux34ndash37 The calculation we have used fromSmythersquos paper was originally intended for obtaining the tor-que17 In Europe linear versions of the eddy-current brake arebeing tested for railway applications38 In these devices themagnetic field is generated by a linear array of electromagnetson the car and applied to the rail The eddy currents producedin the rail slow the car and heat the rail A section of the rail isheated only as the car passes so the heating is not as severe asfor the disc brake which is heated continuously as long as thebrake is energized

As with the more common friction brakes all eddy-current brakes convert kinetic energy into heat This is unde-sirable in applications in which energy conservation is ofconcern For these applications electromagnetic regenerativebraking may be a better option Regenerative braking is usedin electric and hybrid (gasolineelectric) automobilesBatteries are used to power electric motors that propel theautomobile and when the automobile is to be slowed orstopped these motors are used as generators to charge thebatteries Thus a portion of the kinetic energy of the auto-mobile is recovered and stored for future use

The experiments Faraday made with his first dynamo gavehim insight into electromagnetic induction and among otherthings allowed him to give the first correct explanation forwhat caused Aragorsquos rotations The dynamo in its originalform was not a prototype for a practical source of electricpower However from the historical record it is clear thatFaradayrsquos first dynamo was an important step in the develop-ment of modern generators39 The fundamental principles hediscovered with his dynamo were quickly applied by othersto produce efficient generators

ACKNOWLEDGMENTS

The author would like to thank his Georgia Tech col-leagues Professors Waymond Scott Jr John Buck (bothfrom ECE) and Andrew Zangwill (Physics) for their sugges-tions for improving the manuscript

a)Electronic mail glennsmithecegatechedu1M Arago ldquoLrsquoaction que les corps aimantes et ceux qui ne le sont pas exer-

cent les uns sur les autresrdquo Ann Chim Phys 28 325 (1825) and M

Arago ldquoNote concernant les phenomenes magnetiques auxquels le

mouvement donne naissancerdquo ibid 32 213ndash223 (1826) An account in

English is in C Babbage and J F W Herschel ldquoAccount of the repetition

of M Aragorsquos experiments on magnetism manifested by various substan-

ces during the act of rotationrdquo Philos Trans Roy Soc London 115

467ndash496 (1825)2M Faraday Experimental Researches in Electricity Vol I (Taylor and

Francis London 1839 Republication Green Lion Santa Fe NM 2000)

Paragraph 83 pg 253M Faraday Faradayrsquos Diary edited by T Martin (G Bell and Sons

London 1932) Vol I Paragraphs 99ndash136 pp 381ndash3864M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraphs 83ndash100 pp 25ndash29 and 119ndash134 pp

34ndash395S P Thompson Michael Faraday His Life and Work (Cassell London

1901 Republication Elibron Classics New York 2004) pg 1236M Faraday Faradayrsquos Diary edited by T Martin (G Bell and Sons

London 1932) Vol I Paragraph 228 pg 3977M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraph 155 p 468R J Stephenson ldquoExperiments with a unipolar generator and motorrdquo

Am J Phys 5 108ndash110 (1937)9J W Then ldquoLaboratory experiments in motional electric fieldsrdquo Am J

Phys 28 557ndash559 (1960)10J W Then ldquoExperimental study of the motional electromotive forcerdquo

Am J Phys 30 411ndash415 (1962)11M J Crooks D B Litvin P W Matthews R Macaulay and J Shaw

ldquoOne-piece Faraday generator A paradoxical experiment from 1851rdquo

Am J Phys 46 729ndash731 (1978)12R D Eagleton ldquoTwo laboratory experiments involving the homopolar

generatorrdquo Am J Phys 55 621ndash623 (1987)13H H Woodson and J R Melcher Electromechanical Dynamics Part I

Discreet Systems (Wiley New York 1968) pp 286ndash28914J Van Bladel Relativity and Engineering (Springer-Verlag Berlin 1984)

pp 283ndash29115J Van Bladel ldquoRelativistic theory of rotating disksrdquo Proc IEEE 61

260ndash268 (1973)16H Montgomery ldquoCurrent flow patterns in a Faraday discrdquo Eur J Phys

25 171ndash183 (2004)17W R Smythe ldquoOn eddy currents in a rotating diskrdquo Trans AIEE 61

681ndash684 (1942)18W R Smythe Static and Dynamic Electricity 3rd ed (McGraw-Hill

New York 1968) pp 390ndash39319This type of problem is sometimes referred to as a ldquoquasi-static magnetic

field systemrdquo and it is discussed in detail in H H Woodson and J R

Melcher Electromechanical Dynamics Part I Discreet Systems (Wiley

New York 1968) pp 6ndash8 260ndash264 270ndash277 284ndash289 B19ndashB2520In the local rest frame for a point on the disc the thickness of the disc is

assumed to be much less than the skin depth in the conductor In this

frame the volume current density is then practically uniform throughout

the thickness of the disc In the quasi-static approximation the volume

current density is the same in the moving frame and in the laboratory

frame hence it is practically uniform in the laboratory frame21A I Borisenko and I E Tarapov Vector and Tensor Analysis with

Applications translated by R A Silverman (Prentice-Hall Englewood

Cliffs NJ 1968) pp 211ndash22322Smythe solves for the stream function U which is simply related to the

magnetic scalar potential UethquTHORN frac14 eth2=l0THORNUethqu z frac14 0thornTHORN His method

is briefly as follows First he considers a pole placed over an infinite con-

ducting sheet The radius of the pole is a and it is offset from the origin by

c He obtains an integral for the stream function U1 of this pole and he

evaluates this integral approximately Next he introduces a second pole

with the same total flux as the first pole but of opposite sign and he obtains

the approximate stream function U2 for this pole The radius and offset

from the origin for the second pole are ~a and ~c with ~c ~a gt A Finally

he adjusts ~a and ~c to make the superposition Ufrac14U1thornU2 satisfy the

boundary condition Ufrac14 0 at the radius qfrac14A (edge of the disc)23Normally a tube of current is a region of three-dimensional space

delineated by a side surface that is everywhere parallel to the direction of

the volume current density ~J The total current I passing through any

cross-sectional area of the tube is the samemdashthink of a pipe with the water

flowing through the pipe representing the current Here we are using the

two-dimensional version of the tube of current The current is on a surface

(on the disc) and the sides of the tube are two lines that are everywhere

parallel to the direction of the surface current density ~Js The total current

916 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 916

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

a low voltage and the loss due to the current passing throughthe resistance of the brushes is of concern To lower this re-sistance some designs incorporate mercury or a liquid metal-lic alloy in the brushes as Faraday did albeit with moresophisticated designs than Faradayrsquos simple contacts31

Commercial use of even the symmetric designs has been lim-ited to very specialized applications32

The eddy-current disc brake is a structure very similar toFaradayrsquos first dynamo with the permanent magnet inFaradayrsquos design replaced by an electromagnet The conduct-ing disc is mounted on an axle that normally rotates Whenthe rotation is to be slowed or stopped the electromagnet isactivated and the kinetic energy taken from the rotationalmotion of the axle is dissipated by the eddy currents in thedisc The applications of this technology are diverse and rangefrom the brakes on the famous Japanese Shinkansen (bullettrain) to providing resistance to motion in fitness equipmentsuch as rowing machines33 The analysis for the eddy-currentdisc brake is similar to that for the Faraday disc only the em-phasis is usually on obtaining the torque on the axle versusvarious parameters such as the angular frequency and themagnetic flux34ndash37 The calculation we have used fromSmythersquos paper was originally intended for obtaining the tor-que17 In Europe linear versions of the eddy-current brake arebeing tested for railway applications38 In these devices themagnetic field is generated by a linear array of electromagnetson the car and applied to the rail The eddy currents producedin the rail slow the car and heat the rail A section of the rail isheated only as the car passes so the heating is not as severe asfor the disc brake which is heated continuously as long as thebrake is energized

As with the more common friction brakes all eddy-current brakes convert kinetic energy into heat This is unde-sirable in applications in which energy conservation is ofconcern For these applications electromagnetic regenerativebraking may be a better option Regenerative braking is usedin electric and hybrid (gasolineelectric) automobilesBatteries are used to power electric motors that propel theautomobile and when the automobile is to be slowed orstopped these motors are used as generators to charge thebatteries Thus a portion of the kinetic energy of the auto-mobile is recovered and stored for future use

The experiments Faraday made with his first dynamo gavehim insight into electromagnetic induction and among otherthings allowed him to give the first correct explanation forwhat caused Aragorsquos rotations The dynamo in its originalform was not a prototype for a practical source of electricpower However from the historical record it is clear thatFaradayrsquos first dynamo was an important step in the develop-ment of modern generators39 The fundamental principles hediscovered with his dynamo were quickly applied by othersto produce efficient generators

ACKNOWLEDGMENTS

The author would like to thank his Georgia Tech col-leagues Professors Waymond Scott Jr John Buck (bothfrom ECE) and Andrew Zangwill (Physics) for their sugges-tions for improving the manuscript

a)Electronic mail glennsmithecegatechedu1M Arago ldquoLrsquoaction que les corps aimantes et ceux qui ne le sont pas exer-

cent les uns sur les autresrdquo Ann Chim Phys 28 325 (1825) and M

Arago ldquoNote concernant les phenomenes magnetiques auxquels le

mouvement donne naissancerdquo ibid 32 213ndash223 (1826) An account in

English is in C Babbage and J F W Herschel ldquoAccount of the repetition

of M Aragorsquos experiments on magnetism manifested by various substan-

ces during the act of rotationrdquo Philos Trans Roy Soc London 115

467ndash496 (1825)2M Faraday Experimental Researches in Electricity Vol I (Taylor and

Francis London 1839 Republication Green Lion Santa Fe NM 2000)

Paragraph 83 pg 253M Faraday Faradayrsquos Diary edited by T Martin (G Bell and Sons

London 1932) Vol I Paragraphs 99ndash136 pp 381ndash3864M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraphs 83ndash100 pp 25ndash29 and 119ndash134 pp

34ndash395S P Thompson Michael Faraday His Life and Work (Cassell London

1901 Republication Elibron Classics New York 2004) pg 1236M Faraday Faradayrsquos Diary edited by T Martin (G Bell and Sons

London 1932) Vol I Paragraph 228 pg 3977M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraph 155 p 468R J Stephenson ldquoExperiments with a unipolar generator and motorrdquo

Am J Phys 5 108ndash110 (1937)9J W Then ldquoLaboratory experiments in motional electric fieldsrdquo Am J

Phys 28 557ndash559 (1960)10J W Then ldquoExperimental study of the motional electromotive forcerdquo

Am J Phys 30 411ndash415 (1962)11M J Crooks D B Litvin P W Matthews R Macaulay and J Shaw

ldquoOne-piece Faraday generator A paradoxical experiment from 1851rdquo

Am J Phys 46 729ndash731 (1978)12R D Eagleton ldquoTwo laboratory experiments involving the homopolar

generatorrdquo Am J Phys 55 621ndash623 (1987)13H H Woodson and J R Melcher Electromechanical Dynamics Part I

Discreet Systems (Wiley New York 1968) pp 286ndash28914J Van Bladel Relativity and Engineering (Springer-Verlag Berlin 1984)

pp 283ndash29115J Van Bladel ldquoRelativistic theory of rotating disksrdquo Proc IEEE 61

260ndash268 (1973)16H Montgomery ldquoCurrent flow patterns in a Faraday discrdquo Eur J Phys

25 171ndash183 (2004)17W R Smythe ldquoOn eddy currents in a rotating diskrdquo Trans AIEE 61

681ndash684 (1942)18W R Smythe Static and Dynamic Electricity 3rd ed (McGraw-Hill

New York 1968) pp 390ndash39319This type of problem is sometimes referred to as a ldquoquasi-static magnetic

field systemrdquo and it is discussed in detail in H H Woodson and J R

Melcher Electromechanical Dynamics Part I Discreet Systems (Wiley

New York 1968) pp 6ndash8 260ndash264 270ndash277 284ndash289 B19ndashB2520In the local rest frame for a point on the disc the thickness of the disc is

assumed to be much less than the skin depth in the conductor In this

frame the volume current density is then practically uniform throughout

the thickness of the disc In the quasi-static approximation the volume

current density is the same in the moving frame and in the laboratory

frame hence it is practically uniform in the laboratory frame21A I Borisenko and I E Tarapov Vector and Tensor Analysis with

Applications translated by R A Silverman (Prentice-Hall Englewood

Cliffs NJ 1968) pp 211ndash22322Smythe solves for the stream function U which is simply related to the

magnetic scalar potential UethquTHORN frac14 eth2=l0THORNUethqu z frac14 0thornTHORN His method

is briefly as follows First he considers a pole placed over an infinite con-

ducting sheet The radius of the pole is a and it is offset from the origin by

c He obtains an integral for the stream function U1 of this pole and he

evaluates this integral approximately Next he introduces a second pole

with the same total flux as the first pole but of opposite sign and he obtains

the approximate stream function U2 for this pole The radius and offset

from the origin for the second pole are ~a and ~c with ~c ~a gt A Finally

he adjusts ~a and ~c to make the superposition Ufrac14U1thornU2 satisfy the

boundary condition Ufrac14 0 at the radius qfrac14A (edge of the disc)23Normally a tube of current is a region of three-dimensional space

delineated by a side surface that is everywhere parallel to the direction of

the volume current density ~J The total current I passing through any

cross-sectional area of the tube is the samemdashthink of a pipe with the water

flowing through the pipe representing the current Here we are using the

two-dimensional version of the tube of current The current is on a surface

(on the disc) and the sides of the tube are two lines that are everywhere

parallel to the direction of the surface current density ~Js The total current

916 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 916

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302

I passing through any cross-sectional contour joining these lines is the

same24H Gelman ldquoFaradayrsquos law for relativistic and deformed motion of a

circuitrdquo Eur J Phys 12 230ndash233 (1991)25I S Gradshteyn and I M Ryzhik Tables of Integrals Series and

Products 7th ed (Academic New York 2007)26The deflection of the galvanometer is a function of the current through it

If the internal resistance of the disc is always small compared to the resist-

ance of the external circuit this current exhibits the same proportionality

to the open-circuit voltage for all positions of the brush (qb ub)27This equivalence is based on equating the area for a circular pole to that

for the square or rectangular pole28M Faraday Experimental Researches in Electricity (Taylor and Francis

London 1839) Vol I Paragraph 90 p 2729S P Thompson Michael Faraday His Life and Work (Cassell London

1901) pg 27630W M Saslow ldquoMaxwellrsquos theory of eddy currents in thin conducting

sheets and applications to electromagnetic shielding and MAGLEVrdquo Am

J Phys 60 693ndash711 (1992)31D A Watt ldquoThe development and operation of a 10-kW homopolar gen-

erator with mercury brushesrdquo Proc IEE (London) 105A 233ndash240 (1958)

32A I Miller ldquoEssay 3 Unipolar Induction a Case Study of the

Interaction between Science and Technologyrdquo in A I Miller Frontiersof Physics 1900ndash1911 Selected Essays (Birkheuroauser Boston 1986) pp

153ndash18733A Mochizuki ldquoA new series of the Shinkansen EMU train made its

debutrdquo Jpn Railw Eng 25 6ndash11 (1985)34H D Wiederick N Gauthier D A Campbell and P Rochon ldquoMagnetic

braking Simple theory and experimentrdquo Am J Phys 55 500ndash503

(1987)35M A Heald ldquoMagnetic braking improved theoryrdquo Am J Phys 56

521ndash522 (1988)36M Marcuso R Gass D Jones and C Rowlett ldquoMagnetic drag in the

quasi-static limit Experimental data and analysisrdquo Am J Phys 59

1123ndash1129 (1991)37J M Aguirregabiria A Hernandez and M Rivas ldquoMagnetic braking

revisitedrdquo Am J Phys 65 851ndash856 (1997)38J Schykowski ldquoEddy-current braking A long road to successrdquo http

wwwrailwaygazettecomnewssingle-viewvieweddy-current-braking-a-

long-road-to-successhtml (2008)39S P Thompson Polyphase Electric Currents and Alternate-Current

Motors 2nd ed (Spon and Chamberlain New York 1900) pp 421ndash471

917 Am J Phys Vol 81 No 12 December 2013 Glenn S Smith 917

This article is copyrighted as indicated in the article Reuse of AAPT content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

13023920174 On Sun 24 Aug 2014 212302