CI I " t | -*{ ~~~~~~~~~~~~~pari of spaee which is actually occupied byacuIation or t e apacitance or a another subarea Ai is essentially constant;and similarly for the potential Vii producedby qi over Ai itself.

Circular Annulus by the Method Under these assptions calculationproceeds as follows. By (1), VU produced

or Subareas over Ai by the uniform charge densityqi on Ai is proportional to q, whenceVi,=kijqi. Hence by (1) and (3) the

THOMAS JAMES HIGGINS DANIEL KINSETH REITAN total potential over Aj isMEMBER AIEE STUDENT MEMBER AIEE n n

Vj= Vij= kijqiV i=Z V i=lkiqTHE PURPOSE of this paper is two- cal problem' of determining the normal

Ifold: to advance the theory of a cer- force* on a thin annular plate moving a lin equation ingthe u ownstain approximate method which enables in a viscous fluid in the direction of its qt(i=l, p,en). Proceeding thustoformcalculation, to any desired degree of ac- nornal. Accordingly, it is most desirable the total potential over each subareacuracy, of both the charge distribution to have a method enabling computationand the capacitance of a plane area of these electrical quantities to a desired n

charged to potential V0 by a charge Q; degree of accuracy. Precisely such a Vj = Ekiiqi (j= 1,. n)to illustrate application of this theory method, hereafter termed the method ofby calculation of the charge distribution subareas, is advanced in this paper. A well-known theorem in electrostaticand capacitance of a certain annular area. Following sketch of the basic theory, it theory states that the potential is con-

Rigorous determination of the men- is applied to effect approximations to the stant, V= Vo, over a charged conductortioned electrical quantities for a plane capacitance and charge distribution of a whereon the charge is in equilibrium.area hinges on determination of the circular annulus and of a circular disk. Imposing this condition over each of thepotential function V which: Comparison of the approximate values for subareas yields

1. Satisfies Laplace's equation V IV= 0the disk with values computed from the n

everywhere; known equations evidence that the values VO = Ek,jq, (j= ., n)eeyhr

of constant value V= Vo thefor the annulus are excellent approxima- i=n=2. Is of constant value V= VO over the tions to the exact value.

plane area; a set of n linear equations in the n un-3. Vanishes at infinity except for an . knowns qi (i= 1, . . ., n). Solution of thisarbitrarily chosen constant (usually taken The Basic Theory set of equations yields a set of values foras zero).

The essential theory is to be epitomizedThe mathematical difficulties associ- as follows. Let the given area A be con-

ated with determination of V for a speci- sidered as comprised of n subareas Paper 51-174, recommended by the ATEE Basicfied plane area are of such difficulty that Ai (i= 1, ..., n) which are: Technical ProgramCo mmittee proved by the A tEEsolution has been effected to date for the AIEE Great Lakes District Meeting, Madison,only two plane areas: the elliptical disk 1. Of such small area by comparison with Wis., May 17-19, 1951. Manuscript submittedandy circlare diask (hereafiptera the given area that the charge density qi February 8, 1950; made available for printing

and heirclardis (heeafer efered is essenitially constant over each subarea March 30, 1951.to as the ellipse and the circle), the latter A THOMAS JAMES HIGGINS, Professor of Electrical

Engineering, and DANIEL KINSETH REITAN areof which is a special case of the former. 2. Of such shape that assumption of both with the University of Wisconsin, Madison,In consequence, equations for the charge uniform charge density enables simple cal- Wis.distribution and electrical capacitance culation of the potential V, produced by This paper is based in part upon a thesis supervisedare likewise known only for these two this uniformly distributed charge. to the faculty of the UnSiversittefyof ,Wisconsniareas. However, need of accurate knowl- 3. Of such dimensions and shape that if Jue 1949, in paretial fulfilltment of the require-edge of one or the other, or both, of these the subarea A, were alone in space, the trical Engineering.quantities for certain plane shapes occurs potential Vi produced by Ai over that We are indebted to onle of the reviewers for sug-in practice, particularly for the circular gesting calculation off the potefnctialofa Euntiformlannulus: for example, in determining the 5 and 6 are better suited to rapid calculation of

... * Thus, the normal force R on a thin annular Vl and Vs than are the equations in Legendreeffect of annular guard rings utilized in plate moving in the direction of its normal with polynomials given by Ramsey"° which were usedcertain precise measurements of elec- a velocity V through a liquid of viscosity ,u is in the first draft of this paper. Recently, one of

R= SgoC V, where C is the capacitance of an the discussors furnished a closed form for V, oftrical capacitance or in the hydrodynanu- annular area of the same radii as the thin plate, equation 5 similar to that of V2 of equation 6.

926 Hig:gins, Reitan-Calcukltion of the Capacitance of a Circular AIBE TRANSACTIONS

the unifonn charge densities qi over the Table 1. Capacitance in Micromicrofarads Table 11. Charge Distribution on the Circularsubareas. Accordingly, the approximate Versus Number of Annular Subareas Disk of Figure 6value of the total charge Q is

Subareas 2 Cm. Disk 4 X6 Annulus Loca- Rigorous Value, Approx. Value,-

Q_____=___________iA_______i____ tion Esu Esu

j=1 ...........11.1890.3.73012. ... 1.2955............ 3.8598 r=O qo =0.05066 Vo.g =4. ... 1.3519.... 3.9427 r=1/4 ..... q4=0.05106 Vs.....q4=O.05327 V.

Finally, the approximate value of the .1.4147. ............. r=3/4 .q g=O0.05464 Vo......q3=0.05715 V.capacitance follows from C=Q/Vo (the r=5/4 . 2=O.06489 V aq2=0.06559 Vs

r=7/4 ...qs=0.10463 Vo ... qi =0.14236 V..qi, and hence the Q, are expressed in terms rr=515/8.qqe.=10.V14558 Vo.qe=of V,0, which cancels out in taking the 0.655a electrostatic units (esu) = 0.727aratio). Knowledge of the approximate micromicrofarad. The best previouslydistribution of charge density over the published determination3 is 0.62211a< noting that p2=(x2+r2-2xr cos 0), we

area A is yielded by the known values of C<0.71055a esu or 0.691a<C<0.789a havethe qi. micromicrofarad. do

Obviously, the capacitance and charge dV= 2qxdx J (x)+r2-2xr cos 0)1/2density can be obtained to any desired Capacitance of a Circular Annulus 0degree of accuracy by taking subareas The potential Vat point 0 produced bysufficiently small in size. Of course, the With reference to Figure 1, the poten- a unliformly charged annulus of innerlabor involved in solving the set of n tial dV at any arbitrary point 0 in the radius b and outer radius a is to be foundlinear equations increases rapidly with plane of a uniformly charged ring of by integrating this expression for dVn. However, as evidenced by the illus- inner radius x and width dx can be found with respect to x between x = b andtrative examples of this paper, surpris- by integrating, with respect to 0, the x =a. Thusingly accurate values of capacitance and potential qxdx do/p contributed at 0 by a r decharge distribution can be obtained with the charge qxdxdo on an incremental V=2q xdx / 2-use of small n, particularly if the area area xdx do of the ring. Herein: q= Jb J=(x2+r-2Xr COS 0)1/2in question possesses symmetry. charge density* and p= distance from (2)Although the figures of this paper incremental area to point 0. Thus, Integration of equation 2 hinges on

evidence use of circular annular subareas, whether a> r> b, or r> a> b, or a> b> r..the general procedure imposes no restric- * The charge density as used here is the sum of the In consequence, these three cases neces-tion on the shape of subarea that can be charge densities on the opposite sides of the incre- sitate the following preliminary computa-

mental ring, hence is twice the actual charge densityused. In general, however, it is best to on one side. In view of the nature of the problem tion.use square subareas for plane figures under discussion it is convenient to work in unf-

rationalized centimeter-gram-second units. Ac- 1. Consider an annulus of radii xl andpossessing rectilinear symmetry and cir- cordingly, the units of all quantities mentioned r, where r> xi. In this case we have fromcular annular subareas for figures possess- are in this system except as specifically noted

ing angular symmetry. otherwise. equation 2

Finally, it is to be noted that although vr=2qf dxfr doin this paper attention is confined to ZJJ(x+ +r2-2xr cos 0)'/2plane areas, a similar procedure is appli- (3)cable to 3-dimensional configurations.In particular, the method of subareas Intoducin eshas been employed to solve the long- sin24-1 givesstanding problem of the accurate deter- fT xdx f/2mination of the capacitance of a cube 2 Figure 1 (lower left). Planar VI J=4q x I Xminaths withiles

c ithanc1p cuben co-ordinate system for calculation (x+r)Thus, within less than 1 per cent the of potential produced at an d_ _capacitance of a cube of side a is C= abitrary point O in space by a 4xr ¼

circular annulus of width dx and 1-+) sin2radius x

V I 4qj x_rdx (4)

4 ~ ~~~~~~~~ 56X(E

x

j\p X~~~~~~~

! ~~~~~~~~~~~~~~~~~~~~~~Figure2. SubannularI / ~~~~~~~~~~~~~~~~~~~~~reas of circular an-y ~~~~~~~~~~~~~~~~~~~~nulus

1951, VOLUME 70 Higgins, Reitan-Calcuk,ation of the Capacitance of a Circuktr Annulus 927

Z 34au3. wz I C-)

4~~~~~~~~~~~ A gZves-

4lipi inerloIh irs kind -ihk'( k) k Z /

4c~~~~~~~~~~~~~

4ouuk. ___e y(n^-)+...d' V <Jr dJ r2 ]/

CI)

2~~~~~~~~~~~~~~~~~~~~c 0.7 ---

Fiue 3.w Approximathe campacianentro-4qcir Figur..) eromn5 heApoimategcaacitancoficir-cuaranulusask' a(functiono the numeo'= cula dis asafncino'hnme

=(1-k')/2, and___Fidx=-2r ±-/(+k2 Lx0 4 w

wherezing K(kaiirseisepnio 11 weeEr)designates the coplteV1iptic (-k)elliptk)ic em fk euto 7., =2Ll2 integral of the firstkind wit Jmod(ulusk4fmoduluk7 dfefirned )by (in ±6- V/212

k 2 34 (+r0/42V\/(+ k' / *.J - (a)2 shandintof]

Wegunow intAroducathe cpcltnemary Fi-4qg(1-ur+ . erforion approtihatecpinttonyiels

modranulusask'a(function, ofthenuk'=(r-ofcuardskasafucton te ume

--)/adds dkd'/(1+kituting')sindicatLd4 k equation of subarees12X32k,4{l4 2 2 8+ ~ ~ ~~~~~~+ (l--kl'2-)3 occur

Utilizing kth esfmiliar estexcompansi FIi4q 16 1 2 whe re2XBy e (r/ti ) designater nc wel

forlK(k)incem fk euto 7., =ql-l1-12integral of the second kind oft modulus'4X2 7/

Lher k1'2(-1/rx)2ae2 >>

pagelus k an by 4 V1=7 of|x reernc 4x2,wherex2>r. Inthis cas 2ehv

rX2r~~~t de- in4W1e14 2\c nr kd '-k1'5± du+WitheVheagnVn atihan iegratin of

modulus~~~~~~~~k= (Io k2 2r 2 Then charg denitq-iteIhoeta

=~~ ~ suanuuofchrgdenit qanddx=-2dkl(k2k'4 krUtlzigth amla sresepaso diesin 16 3 wher E11/2,deints ano ellipti

for~ ~ ~Kk) in termsofk eution77,3q n1 i i1 nega ftescn ido ouu

K(k) =ln -+- k 2 ln= -X2) 4 16 320 equation 2 reduces to appropriate corn-k' 22\k 1 /3 1 bination of V/and V82. Three cases

12X32 /4 2 2 +(k'-- k1 /+- k239-k'V4 In -1234 \4 2 occur:

22X42 k' X343119 ki14+)(5) Case 1. a>r>b

z 16-124 V=1vllIx=b + V2IX2=a (7)where kl'= (r-xl)/(r+xl. Case 2: r>a>b

2. Consider an annulus of radii r and 7 ' Vijxl=- VI xl=a (8)X2, where x2> r. In this case we have

Cs :a >

I ~~~from equation 2 Case 3:2~X a>-V2b>r 9

/X2 d'-oV"V2xaV2rb()V2 =2qf x dxI (x+'x

0 )1/2 With respect to Figure 2 let, typically,jrJO 17~~~~~~P,Q9,denote the potential at point pi,

fX2fT do due to the charge on the subannulus of

/~~~~~~~~~~~~~= q d r=1/42=0002..V eu

r2 2r \'/ charge density q,. Then the potential2

c F of point pi where of r=23/4 due to thesubannulus of charge density qIanddimensions a = 6, b = 11/2, is, from equa-

x

~~~VpIq, VI IXI= ,=it112 +V2 JX2=.a= 6

0.8

0.15 0.7

0.14

0.130.12. actual0.

>0.1 I ~~~~~~approximnate>_

Q09 ~ ~ /0.5-0.9___

m~ ~ itne/fa nuu finrrdu

0.06 0.40D0.05~004

a,~~~~~~~0b, an0 ausa niatd (.87 o(8.69) ees n rtorz=15i

in0 eqaton8)(.077IV)(0.35163002 ~ ~ ~~~~~~~~~~~~~.

in 0.2

where ~~ \\,yialy I/e eigae h ____ _ Iiw\a // / 0.fannulus of inner radiusHence / j a

r ous 6wefind ~~~~~/ ///!' °IVP,=&2l59e,+&32O9e2 /\ / // 0 |2 z.5 175 2

In similar fashion, using the associatedin =4meters,cent raius r=i ta, b, and r values as indicated, (0.08673 Vo)( 18.06397)

Vp,q2=4.32009q2 (a=11/2, b=5, r=23/ (0.04042Vo)(16.49319) C1QVo=3.5484Vo/V=3.54841SUin equation 8) -+(0.03645 Vo)( 14.92241 )+ 3.9427 micromicrofarads

Vp=3.40055q8 (a=5, b=9/2, r=23/4, 3(0.05776Vo)(13.35163) Finally,in equation 8) C/r, = 0.6571 microlicrofarad

Vp =2.81083q4 (a =9/2, b =4, r=23/4, Hence, an approximation to the capac- per centimeterin equation 8)

where typically, Vp,q, designates thepotential at point pi produced by the sub..annulus of charge density q2uHence for 7oo i

VP, = Vp,q, + Vpjq2+ Vpq,q+ Vp,q4 00

we find0.07

VP, =6.21529q, 1-4.32009q2+ 0.06

3.40055q +2.81083q4 LU

Proceeding thus for Vp,, Vp,, and Vp, inCD04tuirn yields: M00

VP, =6.21529q +4.32009q2 + 0.02

3.40055q3+2.81083q4 0.01

VP, =4,72563q1±+6.13977q2+4.19496q3+3.27285q4

Vp, =4.10989q +4.64588q2+6.02205'q3+4.08899q4

VP, = 3.79474q, +4.03767q2+4.55923q3-i-5.91090q4

Table IV. Dimensionless Ratio C/r, as a associated with the assamed distribution ratio rl/ri has the same value of C/r,.Function of ry/rj of charge Q is greater than the value '" Thus, a plot of a single curve of C/r,

at equilibrium. Consequently, inasmuch versus rO/ri, where ro/r1 ranges from 1 tor0/ri C(micromicrofarads)/ro as Cexact Q2/2W', Capprox=Q2/2W and co, affords knowledge of the capacitance

W'< W, it follows that Capprox< C xact of an annulus of any desired radii.1.0213 .... 0.4525 and Capprox approaches Cexact as a limit Figure 8, plotted from the values of1. .......... ............... 0.49301.0909 .... 0.5382 as the number of subareas is increased. Table IV, comprises this universal curve.1.1250 . 0.5577 For convenience of use, C is taken in1.2000 ..................... 0.59941.2500 . 0.6197 Charge Distribution on the Circular micromicrofarads.1.5000.... 0.6571

o ....................... 0.7073 Disk and the Circular AnnulusPrevious Solutions for the Circular

In that four subannuli have been used Use of four subareas for the circuiar Annulusin obtaining this approximation, it is disk suaverage chare densitiesonappropriate to term it a fourth approxi- the four subannuli of Fgure 4 as follows Nicholson6 has attempted solution formation. Figure 3 and Table I display q =0.14236 Vo q3 = 0.05715 Vo the circular annulus by treating thethe various values of C obtained by using q2 = 0.06559 Vo q4 = 0.05327 Vo annulus as the limiting configuration ap-one, two, and four subannuli. It is proached by a toroidal ring of ellipticalevident from the curve of Figure 3 that Assuming these densities at the average cross section as the semiaxis of the ellipsethe fourth approximation of C=3.9427 radii of the corresponding subareas yields (which is perpendicular to the plane of themicromicrofarads is very near to the exact a distribution of charge over the disk as ring) approaches zero. His solutions arevalue. in Figure 6. The marked increase in invalidated by various incorrect analyti-

density toward the outer edge is in accord cal procedures. In substantiation of thisCapacitance of a Circular Disk with the theoretical fact that the charge remark we need only quote Nicholson's

density at the bounding edge of a plane comment based on his solution for theSpecific insight as to the degree of area is infinite. capacitance: "The presence of an inner

accuracy to be expected in calculating The approximate charge distribution edge much increases the capacity [overthe capacitance of annuli of various is in good agreement with the actual that of a circular disk of the same outerratios of outer to inner radius r0/ri by distribution (dashed curve) as plotted radius]." This remark is directly con-using four subareas is afforded by similar from the values of Table II which are trary to the well-known theorem in elec-calculation of the extreme special case calculated from the known equation5 trostatics that the capacitance of any(r0/r5,= co) of a circular disk of radius q = V/72(a2 - r2) /2 = V/r2a(1- r2/a2) /2, portion of a plane area, surface or volumer=2 centimeters, of which the known where r= point under consideration, a= is less than the capacitance of the whole.exact capacitance is C=2r/ir=1.273 radius of the disk, and q = charge density.* For example, we found the capacitanceesu=1.415 micromicrofarad. Figure 4 Figure 7 indicates the charge distribu- of the circular annulus considered aboveindicates the subdivision of the circular tion on the annulus of r/ri = 1.5 as plotted to be slightly greater than 3.94 micro-disk in effecting the fourth approxima- from the values of Table III, calculated microfarad; the known exact capacitancetion. The details of calculation are the in determining its capacitance. In view of a circular disk of the same externalsame as for the annulus, hence may well of the good agreement between approxi- radius is 4.24 micromicrofarad. Thus,be omitted. Figure 5 displays the values mate and exact distributions manifest in the capacitance of the annulus, a part offor the first, second, and fourth approxi- Figure 6, and having in mind the better the disk, is less than the capacitance ofmations. The value of the fourth ap- accuracy to be expected for the annulus the whole disk.proximation is C=1.2167 esu=1.3519 by virtue of the much narrower annular Lebedev7also has attempted a solutionmicromicrofarad. In this extreme case subareas, it is to be inferred that the by considering the annulus as the limit-of rl/ri = 2/0= co, the per cent error with indicated distribution of Figure 7 is a close ing case of a toroidal ring of certain ovalonily four subareas is about 4.4 per cent. approximation to the exact distribution, cross section. However, the analysis isAccordingly, for the annulus of r0/rt = extremely complicated and his end result6/4 = 1.5 a much smaller error is to be A Universal Curve is to express the capacitance in terms ofexpected, as is indicated by Figure 3. certain harmonic functions associated

It is to be noted that the curves of The primary dimensions of C in centi- with the ring which, however, are soFigures 3 and 5 evidence that as the num- meter-gram-second units are those of complicated in form that they defyber of subareas is increased the approxi- length (whence the terminology of some calculation. Similar analytic difficultiesmate values of capacitance approach the 19th century texts on electricity and mark the investigations of Poole8 andexact value as a limit from below. This magnetism in stating "a capacitance of Snow.9approach from below is a consequence of 10 centimeters"). Accordingly, the ratio In consequence, we have that the solu-a well-known theorem in electrostatics: of C to either the inner or outer radius tion by subareas, as advanced in thisthe energy associated with a charged (rO and rj) is dimensionless. In that C paper, comprises the first numerically-surface is a minimum when the charge is a function of its geometry alone, thus useful solution of the problem of deter-is in equilibrium; that is, has distributed of its two radii, and in that C/r0 (say) is mining the capacitance and chargeitself such that the surface is an equi- dimensionless, it follows that C/r0 must distribution of a circular annulus.potential surface. Inasmuch as the be a function of only the dimensionlessmethod of subareas requires assumption ratio r0/r~. Thus, any annulus of fixed Summaryof uniform charge density over each sub-area, it follows that the assumed dis- -1. The basic theory of approximatetribution of charge is not that of equilib- * It is to be recalled that the "charge density" calculation by the ulse of subareas of

under discussion is the sum of the densities on therium. Accordingly, the energy W two sides of the disk at a given point on it. the capacitance of a plane area and of the

930 Higgins, Reitan-Calc1uation of the Capacitance of a Circular Annulus AIEIE TRANSACTIONS

distribution of charge density over it is indicates that calculation of charge dis- Polya. American Mathematical Montthly (New

outlined. tribution by use of subareas affords a good 4. T.), oFuNeGRAL AND OTHeR MATHE-

4. TABLES OF INTEGRALS AND OTHER MATHE-2. The method of subareas is em- approximation to the actual distribution. MATICAL DATA (book), H. B. Dwight. The Mac-

ployed to obtain an accurate value for Accordingly, the charge distribution of millan Company, New York, N. Y., 1947.the capacitance of an annulus of ratio Figure 7 for the much narrower annulus 5. A TREATISE ON ELECTRICITY AND MAGNETISM

(book), J. C. Maxwell. Oxford University Press,of outer to inner radius of rl/ri 1.5. is to be considered as a close approxima- Oxford, England, Edition 3, 1893, volume 1, page

3. The fourth approximation to the tion to the actual distribution. 239.6. PROBLEMS RELATING TO A THIN PLANE

capacitance of a specified circular disk, 5. The universal curve of Figure 8 ANNULUS, J. W. Nicholson. Proceedings, Royalas calculated by the method of subareas, yields the capacitance of an annulus of Society of London (London, England), volume

is found to be in good agreement with any stated ratio of external to internal 7 TE FUNCTIONS ASSOCIATED WITH A RING OFthe known exact value. As a circular radii. OVAL CROSS-SECTION, N. Lebedev. Journaldisk is an annulus *ofratio of radii Technical Physics (Leningrad, USSR), volume 4,disk is an annulus of ratio of radii 1937, pages 1-24.

r/ri = co, it is to be concluded that the References 8. DIRICHLET'S PRINCIPLE FOR A FLAT RING.fourth approximation for the much E. G. C. Poole. Proceedings, London Mathematical

1. TE FLWo Visous LUIs RoND PANESociety (London, England), volume 29, 1929,

narrower annulus of ratio 1.5 is very 1. THE FLOW OFVISCOUS FLUIDSoROUND PLANE pages 342-54; volume 30, 1930, pages 174-86.OBSTACLES, R. Roscoe. Philosophical Magazinenearly the exact value. This conjecture (London, England), series 7, volume 60, 1949, 9. THE HYPERGEOMETRIC AND LEGENDRE FUNC-

pkiges 338-51. ~~~~~TIONS WITH- APPLICATIONS TO INTEGRAL EQUATIONSis substantiated by the curve of Figure 3. pages 338-51. OF POTENTIAL THEORY (book), C. Snow. National

4. Comparison in Figure 6 of the 2 CALCULATIONM O OF SUBAREAS, Thomas Bureau of Standards (Washington, D. C.), 1942,

chiarge distribution on a circular disk James Higgins, Daniel K.inseth Reitan. Journal Ofpae38

Applied Physics (New York, N. Y.), February 10. AN INTRODUCTION TO THE THEORY OF NEW-as determined both from the known 1951. TONIAN ATTRACTION (book), A. S. Ramsey. Cam-

bridge University Press, Cambridge, England,equation and by the method of subareas 3. ESTIMATING ELECTROSTATIC CAPACITY, G. 1940, page 133.

Discussion written and interesting application of a r 45 ]general method, proposed by the authors, Vs 11.092296+ -(2.654443)2for the solution of problems in electro- 529

H. B. Dwight (Massachusetts Institute of statics, where the geometry of the system is =3.07066qTechnology, Cambridge, Mass.): As men- such as to render an exact solution very This expressioil applies for annular, concen-tioned in the early part of the paper by T. difficult, if not impossible. This general tric, and coplanar subareas at all distances.J. Higgins and D. K. Reitan, there is a method is one of successive approximations, For completeness it may be mentionedformula for the capacitance of only one the accuracy attainable depending upon that, when k2'= 1 - [r/x2 J is small, theshape of finite plane area, namely, an ellipti- the number of subareas into which the con- expression for V2 may be expanded to readcal area, which includes the case of a cir- ductor is supposed to be divided. In thecular disk. There are precise calculations example given, four subareas are assumed 4 3for extremely few shapes of finite conduc- and a gratifying degree of precision is at- V2=4qrk2 log -+ 1 +-k2 Xtors. The cases of a finite cylinder, of two tainable. The greater the number chosen, k2' / 2parallel plates close together, and of groups the greater the attainable precision, but the log 4 +1 7 k2 log 4 +2of infinitely long wires require certain ratios labor of precision goes up as the square of k112/2 +4 k2 k2t 7 +to be very small. the number of subareas. By applying the

There are many other shapes for which method to the solution for the capacitance For cases where the modulus itself is small,the calculated capacitance is desired. One of a disk, where the exact formula is known, the elliptic integral expressions may beof the most common methods to be used is an upper limit to the error of the result in expanded in terms of the well-known ex-that of a Howe approximation, developed a the present problem is obtained and the use pansions of the elliptic integral in powers ofnumber of years ago by G. W. 0. Howe of of four subareas is proved to be accurate the modulus.England. Uniform charge density is as- enough for practical purposes. The authors It seems likely that the applicationi of thesumed over the metallic surfaces and the are to be congratulated on their successful general method to the finding of the capaci-calculated average potential of the surfaces method of attack for the solution of such tance of two coaxial annuli in parallelis computed. The ratio of charge to average problems. planes would yield elliptic integral ex-potential gives an approximate value of The accuracy of the solution in the pres- pressions with somewhat more complicatedcapacitance. ent case depends upon the sufficiency of equations for the moduli.

For groups of infilsitely long wires, as in the equations for the potentials VI andoverhead power circuits, the error from V2. For subareas not very far apart, theusing a Howe approximation is of the series for Vi in equation 5 will be satisfac- Ernst Weber (Polytechnic Institute oforder of only 1 per cent. This occurs when torily convergent, since k' will be small. The Brooklyn, Brooklyn, N. Y.): The method ofgeometric mean distance is used in cal- equation 6 for V2 is subject to no limitations, subareas represents, without doubt, anculatinIg capacitance in groups of wires. since the elliptic integral can be obtained from excellent contribution to the practicalFor a circular disk, the error is about 8 per tables for any modulus. This suggests the methods of computing capacitance ofcent and for a wide annulus it is several per possibility that V1 also may be expressed charge distributions. In principle, thecenlt. The method of subareas described in in elliptic integrals. This proves to be the method is an ingenious adaptation of Max-this paper can be used where greater ac- case. The equation found is well's coefficients of potential' for a systemcuracy is desired than is available by the

-of electrostatic conductors which system is

Howe approximation. The precision can V1 =4qr[1-E+k, '2K]copsdfthsuaesiowihtebe made greater and greater by lengthening where K and E are complete elliptic in- given conductor is subdivided. The evalua-the calculation. tegrals of the first and second kinds to tion of the coefficients of potential still re-

In view of the various needs for capai modulusx/r- k1'2 = 1 - (x/r)2. quires considerable detail calculation, buttance calculatsons and for flow calculationls As a check on this expression, we find for the total amount of effort is incomparablyof different kinds, such as current flow in the the case Xs =-11/2and r=23/4, that is, for smaller than the complete solution of aearth, the method of subareas should be of the first term of V1s1qi, given in the equation boundary value problem would entail.wvidespread use. immediately following equation 9. Because the coefficients of potential must

k112=1- 4=5 .8562 be directly computed in this method, it iskl'2=l-{= =0.0850662 ~necessary to choose the subareas of simple

F. W. Grover (Union College, Schenec- \23/ 529 'geometries as the authors have done for thetady, N. V.): This paper is a clearly K=2.654443, E=1.092296 circular diskand the circular annulus.

1951, VOLUME 70 Higgins, Reitan-Calculation of the Capacitance of a Circukir Annulus 931

I should like to ask a few specific ques- division by V gives for the lower limit of the as d/b decreases the agreemenit worsens, totions: capacitance, in farads, the end that the Howe approximation is in

1. How has the final value been es-2

error by about 4.8 per ceiit for the limitingtablished for Figure 3 giving the capacitance (16e/r) rO cos-(rj/ro) + (ro2-ri2)'/2 X case of the square plate, d/b = 1.of a circular annulus. Comparison with sinh-1 [ri(r02-rj2)- 1/2] In general, if a given cotnductor is char-Figure 5 would indicate that the exact where and are in meters and is 8.855 X acterized by two appropriately-chosen geo-capacitance of the annulus is 5 per cent 10-12h In the special case when ri is metrical parameters, the Howe approxima-above the value of the fourth approximation. tion is good when the value of one parameterShould one assume the same degree of ac- 0.04 meter and rO is 0.06 meter this gives is large compared to the other, and worsenscuracy for the fourth approximation of the 3.901 micromicrofarads against the 3.9427 as the two approach each other. Thus, inannulus? It would, of course, be of in- micromicrofarads by the four subarea ap- the case of the annulus, if we take r0 andproximation. Although it is 1 per cent lessterest to find the value of the eighth ap- accurateor Althoug. itS pre cent .m- (r, - ri) = t as characterizing the annulus,proximation, for example, permitting a bet- accurate for ra/r,= 1.5. its precision im- the Howe approximationl is excellent whenter approximation to the exact value. proves rapidly asrg/ri increases and whene ro/t is large and-as Professor Dwight re-

2. The method is not restricted to uni- ris zero tgilves exactly the capactance marks-worsens as r0/t -- 1, the limitingform subdivision of the conductor area. 8r ofacirculardisk case of the circular disk of radius ro. ANonuniform subdivisioin towards the edges REFERENCE similar remark applies to a cylinder ofmight lead to a very much better approxi- length d and radius r, the error being smallmation of charge densities. This could be 1. STATIC AND DYNAMIC ELECTRICITY (book). for d >>r, the case of a long wire.done without undue complication of the McGraw-Hill Book Company, New York, N. Y. On reading Professor Grover's discussioncomputations if the last subarea towards

Second edition, 1950 and comparing it with the comments ofthe edge be choseii very narrow. one of the reviewers, similarity of content re

3. For practical computations the self discussion of use of elliptic integrals indi-coefficients of potential for the circular To J Higgins andDw K Reitan: Relative cated that Professor Grover was probablyannulus can apparently be taken directly to Professor Owight's mention of Howe's the mentioned reviewer-a surmise recentlyfrom Figure 8. Would it be possible to method for approximating the capacitance confirmed by Professor Grover. The use ofsubstitute for the mutual coefficients be- of a conductor, it may be remarked that elliptic integrals, rather than the series intween subareas the coefficienits of simple this was originally used by Howe to ap- terms of Legendre polynomials used in thecircular line charges assuming the lines have proximate the capacitance of rectangular original draft of the paper, very much facili-very small but filnite diameter identical plates, in connection with some work on tates numerical computation. The writerswith the thickness of the disk. This might antennas. The authors have pending publi- are much indebted to Professor Grover forfurther simplify the detail calculations of cation a paper which cotntains a universal his valuable suggestion.the coefficients. curve for the capacitance of a rectangular It is to be remarked that the writers have

It will be interestitlig to see further ap- plate of length d and breadth b, calculated taken up preparation of a universal curveplications of this method to axially sym- by the method of subareas. Comparison for the capacitanice of a parallel-plate ca-metrically conductor surfaces for which of this curve with a similar curve stemming pacitor comprised of two identical coaxialseveral rather crude approximation methods from Howe's approximation reveals thathave been proposed in the past, as for ex- the curves are in good agreement for large Figure 1. Charge distribution on circularample by J. C. Maxwell, page 305.1 values of d/b-as is to be expected; but that annulus, using 8 subannuli

REFERENCE 0.13

1. See reference 5 of the paper. 0.12

0.11,

0.10W. R. Smythe (California Institute of 0.09Technology, Pasadena, Calif.): The methodof subareas presented in this paper is a very 0.08powerful one for attacking problems which 0.07cannot be solved rigorously by any othermeans. A striking example is the calcula- // 0.06tion by these authors of the capacitance of a / 0.05cube. In the case of the circular annulus it 0//4may be of interest to give a formula ob- 0.04tained by another method which, while not 0.03quite as accurate as the four subarea ap-proximation when r0/ri is 1.5 will be con- 0.02siderably more precise for large values of 0.01r0/ri and is exact in the limit ri = 0.

First consider a freely charged disk of4

1 1 T Tradius rO at potential V. Now calculatethe charge iliduced on an infinite sheet at l \3/potential zero coiltaining a hole of radius ri \\ \|in which there is a fixed charge distribution \\9identical in magnitude but opposite in signto that on the portion of the disk inside ri.Superposition of these two systems gives\\ /the required annulus at potential V under \ \ \\\ /// V//the influence of a small positive coplanar \\ \ \ \ / / \\ \ 4 ///charge outside it. Removal of this charge \ \\\ \, / ./ / /will decrease V and hence increase the \\\\ ////ratio Q/V so that the capacitanlce given by\\\\=_l: ,//this ratio is too low. The original charge \ \ < : / /on that part of the disk between rj and r0 \ \ 7 - //is found by integrating equation 3 on page C114 of Static and Dynamic Electricity.2 =The induced charge is found by integration = L =of the result of problem 39 page 203 of thesame book. Addition of these charges and

932 Iii gins, Reitan-Calcu¢lation of the Capacitance of a Circular Annulus AIEE TRANSACTIONS

alunuli. Professor Grover's conjecture that the slope of the curve of Figure 3 (thus, its distribution for the illustrative annulus, asmore complicated elliptic iintegral expres- rate of iucrease at this point) is much less determined through use of 8 subannuli issions would arise proves to be very true! than that of Figure 5 at the same point. shown in Figure 1 of the discussion andThe basic expression required for the men- This smaller rate of increase indicates that may be compared with the correspondingtioned problem is that for the potential of a the curve of Figure 3 is leveling off faster distribution of Figure 7 of the paper, baseduniformly charged disk, of unit charge than is the curve of Figure 5 and that the on 4 subannuli.density and of radius r, at any point in error corresponding to the value of four 3. The suggestion advanced was triedspace (a, 0, c) and this is found to be subareas is smaller for the annulus than but did not prove fruitful.

for the disk. A confirmation based on the 4. The method of subareas is applicableV=4a EI(k)-k'2 sin2 XF1(k) - suggested use of 8 annuli has been carried to the solution of the type of axially-sym-

out. The calculated value using 8 annuli metric conductors mentioned. A techni-k'2 sin X cos X - is C = 3.5875 electrostatic units = 3.9861 cally interesting axially-symmetric conduc-

-+ I F1(k) -EI(k) I X micromicrofarads. This is only about 1 per tor is the finite cylinder, the capacitance ofV/1_k'2 sin2 X 2 cent larger than the value found using 4 which has not been calculated accurately

subannuli. to date, except for the limiting cases of theF(k',X) -El(k) E(k',x) 2. The suggestion that nonuniform sub- circular disk and the very long cylinder.

divisions be used to increase accuracy of Calculation of a universal curve by thewherein approximation to the charge density is an method of subareas now is under way.

arexcellent one. Thus, in that the exact Professor Smythe's method of approxi-

k'2 =1-k2; k= . charge distribution on the disk, as indicated mating the capacitance of an annulus is ar2+0 in Figure 6, is such that the density is sub- most interesting one. He is expanding his

1 stantially constant from the center up to discussion into a paper which will appear insin2 X= a2; =r2+± about 0.5rO suggests that a better agreement The Journal of Applied Physics.

1+-r of charge distribution is to be obtained by In conclusion, it is to be emphasized that02 dividing the disk into annuli ofradii: the method of subareas is not limited to

and 0 is the positive root of effecting the capacitance and charge dis-ri =0, ro=; r=, ro=1.5; ri 1.5, tribution of conductors, but is generally

a2 c2 r, =1.75; r= 1.75, ro=2 applicable to the determination of param-r2+0 0 eters and variable quantities of numerous

This conjecture is confirmed by calculation problems in electricity, acoustics, heat, fluidCorroboratively, if the point is on the with these values. However, it is to be flow and aerodynamics which are charac-vertical axis of the disk or in its plane, this noted that a corresponding increase in the terized by a scalar potential function andexpression reduces to known expressions accuracy of the value of the capacitailce will prescribed boundary conditions. Ac-for these two special cases. not be obtained: the value of this parameter cordingly, the subarea method enables theWe answer Professor Weber's questions is relatively insensitive to such changes in rather easy solution of many important

in the same order they are advanced: the radii, being influenced primarily by the problems which have hiterto proved in-1. It is to be noted that for four subareas number of annuli used. Thus, the charge tractable to accurate computation.

1951. VOLUMF, 70 Higgins, Reitan-Calculation of the Capacitance of a- Circuktr Annulus 933