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1 Maxwell's Equations I

Version 1.0.2, Aug 26, 2006. Copyright © 2006 Rafael López-Mobilia. Please send comments, suggestions, and/or errata to [email protected]. Thank you.

1.1 Maxwell's Equations in Integral Form

I will start with a brief overview of the fundamental equations that underlie all electro-magnetic phenomena: Maxwell's equations. I will try not to drift too much into histori-cal details, even if they are fascinating and often throw light on interpretational issues.Rather, I will assume that you already have studied the basics of electromagnetismfrom an introductory physics course and that you know about Coulomb's law, some ofthe historical underpinnings, and so on. I will not waste time either on reviewingvector calculus in detail, as one of the prerequisites for this course is MathematicalPhysics I, in which you are supposed to have studied the topic.

Most likely you have seen before Maxwell's equations in integral form,

(1)® E ◊ da =QinÅÅÅÅÅÅÅÅÅÅÅe0

,

(2)® B ◊ da = 0 ,

(3)® E ◊ dr = -d

ÅÅÅÅÅÅÅdt

‡ B ◊ da ,

(4)® B ◊ dr = m0 I + m0 e0 d

ÅÅÅÅÅÅÅdt

‡ E ◊ da .

Eq. (1) is Gauss's law, which says that the flux of the electric field through anyclosed surface is proportional to the net charge Qin inside. (See fig. 1.) So, it tells usthat charges are sources of electric fields.

Eq. (2) is Gauss's law for magnetic fields: the magnetic flux through an arbitraryclosed surface is zero. Every magnetic field line that enters the region bounded by thesurface must leave eventually. In other words, the field lines don't start or end at any

point inside the region and since is arbitrary, magnetic field lines don't start or end atany point in space, period. Thus, as far as anyone knows, there are no magneticcharges (also know as magnetic monopoles) in nature. (Fig. 2.)

Figure 1

Eq. (3) is the integral form of Faraday's law: the line integral of the electric fieldalong a closed curve (that is, the induced emf if the curve is a wire) equals the nega-tive of the time rate of change of the magnetic flux (FB ª Ÿ B ◊ da) through a surface

whose boundary is . (See fig. 3. below.) This means that changing magnetic fluxesare also sources of electric fields.

Figure 2

Finally, equation 4 is Ampère-Maxwell's law: the piece ò B ◊ d = m0 I is Ampère'slaw, which basically tells us that electric currents are sources of magnetic fields. Thesecond term on the right-hand side, m0 Id , where Id ª m0 e0 dÅÅÅÅÅÅdt Ÿ E ◊ da is Maxwell'sdisplacement current accounts for the fact that changing electric fluxes are alsosources of magnetic fields.

2 01_Maxwell's Equations I.nb

Figure 3

To these, we need to add the Lorentz's force law, which specifies how the charges areaffected by the fields,

(5)F = qHE + v μ BL .

Figure 4

That's it! All of classical electromagnetism is contained in these equations. They con-tain the dynamics of the fields, how charge distributions and electric currents produceelectric and magnetic fields and, in turn, how these fields affect the motion of charges.The rest of this course (and the next) will be an elucidation of the consequences ofthese equations under a variety of conditions, some very interesting, some very practi-cal, some fascinating, and some... well, not so much.

The first step into new–and possibly unknown to you–territory is to recast Max-well's equations in differential form. This makes them more manageable for manyapplications (in fact, they are simpler) and provides us with extra physical insight. Butbefore we do it, let me remind you of some important definitions and theorems fromvector calculus.

01_Maxwell's Equations I.nb 3

1.2 A Vector Calculus Interlude

You must have seen (and surely remember!) the definitions of the vector differentialoperations gradient, divergence and curl. Most likely what stuck in your memory istheir Cartesian expressions using the — ("del" or "nabla") operator

— ª x ∑ ê ∑ x + y ∑ ê ∑ y + z ∑ ê ∑ z .

Let r = x x + y y + z z. If V = V HrL is a scalar field andf = f HrL = fx HrL x + fy HrL y + fz HrL z, a vector field, then, in Cartesian coordinates,

grad f = —V =∑VÅÅÅÅÅÅÅÅÅÅÅ∑ x

x +∑VÅÅÅÅÅÅÅÅÅÅÅ∑ y

y +∑VÅÅÅÅÅÅÅÅÅÅÅ∑ z

z ,

div f = — ◊ f =∑ fxÅÅÅÅÅÅÅÅÅÅÅÅ∑ x

+∑ fyÅÅÅÅÅÅÅÅÅÅÅÅ∑ y

+∑ fzÅÅÅÅÅÅÅÅÅÅÅÅ∑ z

,

curl f = — μ f = deti

k

jjjjjjjjjj

x y z∑ÅÅÅÅÅÅÅ

∑x∑ÅÅÅÅÅÅÅ

∑y∑ÅÅÅÅÅÅ

∑z

fx fy fz

y

{

zzzzzzzzzz=

ikjj ∑ fzÅÅÅÅÅÅÅÅÅÅÅÅ

∑ y-

∑ fyÅÅÅÅÅÅÅÅÅÅÅÅ∑ z

y{zz x + J ∑ fxÅÅÅÅÅÅÅÅÅÅÅÅ

∑ z-

∑ fzÅÅÅÅÅÅÅÅÅÅÅÅ∑ x

N y + ikjj ∑ fy

ÅÅÅÅÅÅÅÅÅÅÅÅ∑ x

-∑ fxÅÅÅÅÅÅÅÅÅÅÅÅ∑ y

y{zz z .

These expressions are valid only in Cartesian coordinates. You know, however, that itis perfectly all right (and often convenient and even necessary) to use other coordinatesystems, such as spherical polar coordinates, cylindrical coordinates, confocal ellipsoi-dal coordinates (I am showing off), etc. What you may not remember (or may havenever seen) is that there are coordinate-independent definitions of these operations,which provide crucial insight into their meaning. In fact, take r to be the positionvector in some arbitrary coordinate system. Given a scalar field V , its differential is

(6)dV ª grad V ◊ dr .

This defines the gradient of V , and it is valid regardless of the coordinate systememployed.

For the divergence, we define

(7)div f ª limV Ø0

1

ÅÅÅÅÅÅÅV

® f ◊ da .


And for the curl,

(8)n ◊ curl f ª limAØ0

1ÅÅÅÅÅÅA

® f ◊ dr .

(Note: It is common–although not universal–practice to use the notation —V forgrad V , — ◊ f for div f , and — μ f for curl f , even if the coordinate system is notCartesian. We will follow this convention, but you should be careful not to confusethem with their Cartesian expressions when working in other coordinate systems.)

The interpretation of definitions 6, 7, and 8 is as follows. The change of a scalarfield V from r to r + dr is dV = —V ◊ dr. Take the infinitesimal displacement dr to bealong the surface V = const. Then dV = 0 fl —V ◊ dr = 0, and therefore —V is perpen-dicular to the surface of constant V . On the other hand, for an arbitrary infinitesimaldisplacement dr, the angle q between —V and dr is given by dV = †—V § †dr§ cos q , andso dV is a "maximum" for cos q = 1, (that is, q = 0) which implies that dr is parallel to—V . Therefore, —V points in the direction that maximizes the change of V . And, ofcourse, †—V § is the rate of change of V in that direction.

As you know from your calculus courses, the differential of a scalar function V is

dV =∑VÅÅÅÅÅÅÅÅÅÅÅ∑ x

dx +∑VÅÅÅÅÅÅÅÅÅÅÅ∑ y

dy +∑VÅÅÅÅÅÅÅÅÅÅÅ∑ z

dz =

ikjjjjj

∑VÅÅÅÅÅÅÅÅÅÅÅ∑ x

x`

+∑VÅÅÅÅÅÅÅÅÅÅÅ∑ y

y +∑VÅÅÅÅÅÅÅÅÅÅÅ∑ z

zy{zzzzz ÿ Hdx x + d y y + dz zL ,

and we recover the usual Cartesian expression grad V = —V = ∑VÅÅÅÅÅÅÅÅ∑x + ∑VÅÅÅÅÅÅÅÅ

∑y + ∑VÅÅÅÅÅÅÅÅ∑z . à

Now to equation 7. The flux ò f ◊ da through the closed surface quantifies "howmuch" of the field f enters or leaves the region bounded by . Take a point P withinthat region. As we make the region shrink toward P the area of decreases and (ingeneral) so does the flux. So, generically, we would expect the integral to tend to zeroas the region shrinks to a point, since all the points on are so close to each otherwhen the region is sufficiently small that f changes very little from point to point.Dividing by the volume V of the region we are effectively computing a sort of limitingflux per unit volume, which may or may not be finite when V tends to zero. If it isfinite or zero then there is a well-defined divergence of the field at that point.

One way of visualizing the meaning of divergence is by looking at fluid flow.Choose the vector field f to be the mass flux rv of the fluid, where r = rHr, tL is themass density and v = vHr, tL, the velocity of the fluid at a given point P. (The mass fluxhas dimensions of mass per unit area per unit time, as you can easily check.) Take now


a closed surface (containing P) that the fluid is crossing. The scalar product of themass flux and an infinitesimal area element da on is the (infinitesimal) flux dF ofthe fluid through da,

dF = HrvL ◊ da ,

which measures the mass of fluid per unit time crossing da. (Check the dimensions!)So,

F = ® HrvL ◊ da

is the mass of fluid per unit time coming out of the region bounded by . Our defini-tion of divergence tells us that

— ◊ HrvL = limV Ø0

1

ÅÅÅÅÅÅÅV

® HrvL ◊ da ,

which means that — ◊ HrvL is the density (mass per unit volume) of fluid emerging fromP per unit time. If — ◊ HrvL > 0, then the fluid density is flowing outward from theinfinitesimal region around P. In this case P is a "source" of fluid. If — ◊ HrvL < 0, thefluid density is flowing into the region, in which case P is a "sink".

One more comment. The mass per unit time per unit volume of fluid going into orout of the region is Hdm êdtL êvol = d r êdt. If d r êdt > 0, then the density is increasingover time (fluid flowing into the region); d r ê dt < 0 for a net density escaping(outward flow) the region; and zero for no net change. If we shrink the region until itbecomes an infinitesimal neighborhood of P, then mass conservation (no sources orsinks present) and our interpretation above of the divergence of rv imply

d rÅÅÅÅÅÅÅÅÅÅdt

+ — ◊ HrvL = 0

This is known as a continuity equation, in this case for fluid flow. Continuity equa-tions are very important in physics as they constitute the mathematical expression ofconserved quantities.

Since electric charge is also conserved, there should be a corresponding continuityequation for it. Define the (average) magnitude of the charge current density J as thecharge flowing per unit area (normal to the current) per unit time,

(9)XJ\ ªDq

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅarea¦äDt


For charges moving in the x direction, an area patch perpendicular to the current can betaken to be Dy Dz. So,

XJ\ =Dq

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅHDy DzLä Dt=

Dq DxÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅDx HDy DzLäDt

=Dq Dx

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅvolä Dt

=DqÅÅÅÅÅÅÅÅÅÅÅvol

v = rv ,

where v is the average speed of the charges and r is the charge density. (Careful: this ris not the same as the one for the fluid!) In the limit of infinitesimal volume and time,J becomes the instantaneous current density, and we can write it in vector form as

(10)J = rv .

The continuity equation for charge conservation should then be

(11)d rÅÅÅÅÅÅÅÅÅÅdt

+ — ◊ J = 0

As a matter of fact this equation, although fundamental, doesn't need to be postulated.It is a direct consequence of Maxwell's equations! OK, don't be impatient, I will proveit to you in a little while.

Now, what about the Cartesian expression for divergence? Can we derive it fromour general definition? Yes, we can. Choose an infinitesimal rectangular box withfaces parallel to the coordinate planes and centered on P, as shown in the figure below.We want to compute the flux per unit volume of the field f through the box. Why?Because it is in the definition of divergence.

Figure 5

The flux per unit volume through the faces parallel to the XZ plane is


FXZÅÅÅÅÅÅÅÅÅÅÅÅÅvol

=@ fy Hy + dyL - fyHyLD dx dzÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

dx dy dz=

@ fy Hy + dyL - fyHyLDÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

dy=

∑ fyÅÅÅÅÅÅÅÅÅÅÅÅ∑ y

Similarly, the flux per unit volume for the faces parallel to the YZ plane is ∑ fx ê ∑ x and,for the faces parallel to the XY plane, it is ∑ fz ê ∑ z. Adding them up we get the total fluxper unit volume,

∑ fxÅÅÅÅÅÅÅÅÅÅÅÅ∑ x

+∑ fyÅÅÅÅÅÅÅÅÅÅÅÅ∑ y

+∑ fzÅÅÅÅÅÅÅÅÅÅÅÅ∑ z

.

But, according to our definition (see Eq. 7) this total flux per unit volume for the infini-tesimal box is the divergence of f . So, we found the expression for — ◊ f in Cartesiancoordinates.à When the time comes, I will derive the expression for the divergence inother coordinate systems.

Finally, let us analyze the meaning of curl. Let me repeat (almost verbatim) what Isaid earlier when discussing the divergence, but suitably modified. The line integralò f ◊ dr through the closed curve (also known as the circulation of f ) quantifies theamount of "rotation"–or circulation–of the field around the two-dimensional regionbounded by . Take a point P within that region. As we make the region shrink towardP the length of decreases and (in general) so does the circulation. So, generically, wewould expect the integral to tend to zero as the region shrinks to a point, since all thepoints on are so close to each other that f changes very little from point to point.Dividing by the area A we are effectively computing a sort of limiting circulation perunit area, which may or may not be finite when A tends to zero. If it is finite (or zero)then there is a well-defined curl of the field at that point. One more thing: our defini-tion is for the component of the curl in the direction normal to the surface, n ◊ curl f .This is as it should be, since the computed circulation per unit area is a scalar, not avector.

As before, let's take look at fluid flow. The curl of the velocity field vHrL of the fluid ata given point is a measure of how much the fluid "circulates" around the point. This isknown as vorticity. If there is a vortex, a whirlpool, of fluid around P, then curl v ∫ 0.In such a case, an object floating on the region where the curl is non-zero will tend tostart rotating.

To derive the Cartesian form of the curl, choose a rectangular region on the YZ plane,centered on the point PHx = 0, y, zL. In Cartesian coordinates,

f ÿ dr = fx dx + fy dy + fz dz .

The component of the curl in the direction of the positive x axis is


H— μ f Lx = x ◊ H— μ f LTherefore, if we integrate in the direction indicated in the figure (counterclockwise asseem from the positive x axis), the normal n to the surface coincides with x and we arein business.

Figure 6

For the chosen rectangle, dx = 0, so f ÿ dr = fy dy + fz dz. There are four contributionsto the circulation, at points 1 - 4. These are

® f ÿ dr = fy J0, y, z -1ÅÅÅÅÅ2

dzN dy - fy J0, y, z +1ÅÅÅÅÅ2

dzN dy -

fz J0, y -1ÅÅÅÅÅ2

dy, zN dz + fz J0, y -1ÅÅÅÅÅ2

dy, zN dz

The negative sign in front of the second fy comes from point 4, because the integrationthere is in the negative y direction. Similar comments apply to the negative sign infront of the first fz. The partial derivative of fy respect to z is


= limDzØ0

fy Hx, y, z + DzL - fyHx, y, zLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

Dzwhich justifies writing

fy Hx, y, z + dzL = fyHx, y, zL +∑ fyÅÅÅÅÅÅÅÅÅÅÅÅ∑ z

dz

This lets us compute all the terms above in the line integral,


® f ÿ dr = A fyH0, y, zL -1ÅÅÅÅÅ2


dzE dy - A fyH0, y, zL +1ÅÅÅÅÅ2


dzE

dy + A fzH0, y, zL +1ÅÅÅÅÅ2

∑ fzÅÅÅÅÅÅÅÅÅÅÅÅ∑ y

dyE dz - A fzH0, y, zL -1ÅÅÅÅÅ2

∑ fzÅÅÅÅÅÅÅÅÅÅÅÅ∑ y

dyE dz

which, after cancellations, simplifies to

® f ÿ dr = ikjj ∑ fzÅÅÅÅÅÅÅÅÅÅÅÅ

∑ y-


y{zz dy dz

From the definition of curl, and for our rectangle,x ◊ H— μ f L = limAØ0 1ÅÅÅÅÅA ò f ◊ dr = 1ÅÅÅÅÅÅÅÅÅÅÅÅdy dz ò f ◊ dr, which implies that

H— μ f Lx =∑ fzÅÅÅÅÅÅÅÅÅÅÅÅ∑ y

-∑ fyÅÅÅÅÅÅÅÅÅÅÅÅ∑ z

.

A completely analogous derivation with rectangles sitting on the XZ plane and the XYplane gives, respectively,

H— μ f Ly =∑ fxÅÅÅÅÅÅÅÅÅÅÅÅ∑ z

-∑ fzÅÅÅÅÅÅÅÅÅÅÅÅ∑ x

and H— μ f Lz =∑ fyÅÅÅÅÅÅÅÅÅÅÅÅ∑ x

-∑ fxÅÅÅÅÅÅÅÅÅÅÅÅ∑ y

.

That's it. (Wasn't that fun?!)

Moving on, let me remind you of a couple of very important theorems of vector calcu-lus. First is the divergence (or Gauss's) theorem:

(12)‡ — ◊ f dt = ® f ◊ da

Figure 7


As you can see, a volume integral over some region is related to an integral over thesurface boundary of the region.

Then there is Stokes' theorem:

(13)‡ H“ μ f L ◊ da = ® f ◊ dr

Figure 8

In this case an integral over a surface is equated to an integral over the boundary of the surface.

Given our definitions of curl and div, the proof of the theorems are almost trivial. In aninfinitesimal region of volume dt the (infinitesimal) flux of f through the boundary ofthe region is (see Eq. 7),

®d

f ◊ da = — ◊ f dt .

(Note: The symbol d is to remind you that this is an infinitesimal area, and that theintegral is also infinitesimal.) This looks almost like the divergence theorem, exceptthat we need to convert it from an infinitesimal to a finite volume. We do it by noticingthat any finite region can be decomposed into an infinite number of infinitesimalblocks, as shown in figure 9. The (finite) volume integral is just the sum of the contribu-tions of every block, Ÿ — ◊ f dt. For the surface integral, there is a subtlety. Look atthe adjacent blocks 1 and 2. The flux through the right face of block 1 is equal inmagnitude but opposite in sign to the flux through the left face of block 2. (For block 1the field is coming out of the block, while for 2 it is coming into the block.) So, whenadding the flux contributions, adjacent faces cancel out. This is true for every pair ofadjacent faces on adjacent blocks, so the net contribution to the flux comes only fromthose faces that are not paired up and their fluxes canceled, such as the right face of


block 2 in the figure. These faces correspond to the outer surface of the finite region,and so the total flux is just what's expected, ò f ◊ da. This proves the theorem. à

Figure 9

To prove Stokes' theorem, we can do something similar. We take the surface bounded by the curve , and subdivide it into infinitesimal rectangular cells (figure 10).

Figure 10

From the definition of curl, the (infinitesimal) circulation around one infinitesimal cellis

®d

f ◊ dr = Hn ◊ — μ f L da = H— μ f L ◊ da

When adding the circulations (closed line integrals) from all the cells, the contributionsfrom adjacent sides of adjacent cells cancel each other out. This is so because thedirection of integration for the adjacent sides is opposite, as is evident from the figure.What remains then is the contributions from unpaired sides, which correspond to thecurve , and the sum equals the circulation ò f ◊ dr. For the curl piece, the addition is


even simpler: we just add all the scalar products to get the surface integralŸ H“ μ f L ◊ da. à

1.3 Maxwell's Equations in Differential Form

With the divergence and Stokes' theorems in hand, we can now rewrite Maxwell'sequations. Start from Gauss's law and apply the divergence theorem:

QinÅÅÅÅÅÅÅÅÅÅÅe0

= ® E ◊ da = ‡ — ◊ E dt .

But Qin = Ÿ r dt, where r = rHrL is the charge density. So,

‡ J 1ÅÅÅÅÅÅÅe0

r - — ◊ EN dt = 0

which must be valid for any region . The only way for this to be true is if the inte-grand is itself zero, which means that

(14)— ◊ E =1

ÅÅÅÅÅÅÅÅ¶0

r .

An analogous derivation starting from ò B ◊ da = 0 tells us that

(15)— ◊ B = 0 .

Let's work now on Faraday's law. Assuming that the surface area remains constant, wecan bring the time derivative inside the integral,

-d

ÅÅÅÅÅÅÅdt

‡ B ◊ da = ‡ -dBÅÅÅÅÅÅÅÅÅÅÅdt

◊ da .

But Stoke's theorem tells us that

® E ◊ dr = ‡ H— μ EL ◊ da .

Thus, from Faraday's law (Eq. 3),

‡ H— μ EL ◊ da = ‡ -dBÅÅÅÅÅÅÅÅÅÅÅdt

◊ da ,

which has to be valid for any surface . Therefore,


(16)— μ E = -∑BÅÅÅÅÅÅÅÅÅÅÅ∑ t

.

Finally, Ampère-Maxwell's law. The current I is related to the current density Jthrough I = Ÿ J ÿ da. Following steps similar to those used with Faraday's law we find(do it!)

(17)— μ B = m0 J + m0 ¶0 ∑EÅÅÅÅÅÅÅÅÅÅÅ∑ t

.

Equations 14-17 are Maxwell's equations in differential form. The interpretation ofthese equations is facilitated by the meaning of div and curl we discussed above. Eq.(14) tells us that electric fields have a scalar source: electric charges (charge densitiesor charge distributions). Eq. (15) similarly tells us that magnetic fields have no scalarsource, they never start and/or end at any point. So, no magnetic monopoles in nature.Eq. (16) implies that electric fields can also have a vector source, a changing magneticfield, in which case their curl is non-zero, a property that gives the field a differentcharacter than in electrostatics. (It is non-conservative. More about this later.) Finally,Eq. (17) provides vector sources for magnetic fields. This come in two flavors: electriccurrents (current density J) and/or changing electric fields (also known as the displace-ment current term).

I promised before that I would prove the conservation of electric charge (continuityequation) from Maxwell's equations. It goes like this. Start by taking the time deriva-tive of Gauss's law in differential form,

(18)d

ÅÅÅÅÅÅÅdt

— ◊ E =1

ÅÅÅÅÅÅÅÅ¶0


Reversing the order of the operators d ê dt and — and using eq. (17),

(19)

1ÅÅÅÅÅÅÅÅ¶0


= — ◊ J dEÅÅÅÅÅÅÅÅÅÅÅdt

N =

1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅm0 ¶0

— ◊ H— μ B - m0 JL ïd rÅÅÅÅÅÅÅÅÅÅdt

=1

ÅÅÅÅÅÅÅÅÅm0

— ◊ H— μ BL - — ◊ J .

But — ◊ H— μ f L = 0, for any vector field f . So, d r ê dt = -— ◊ J. à


1.4 A Note on Units

Unlike in classical mechanics, the equations of electromagnetic theory look slightlydifferent depending on the system of units employed. There are three systems in useand none can really be said to be superior in all respects to the others. It depends on theapplications. It is then desirable for you to have at least a passing acquaintance with allthree systems. We will use the Système International d'Unités (International System ofUnits, international abbreviation SI, http://www.bipm.fr/en/si/; see alsophysics.nist.gov/cuu/Units/), because it is the one used by most introductory textbooks,and recommended by the International Union of Pure and Applied Physics(http://www.iupap.org/index.html). The base units of this system are the meter(length), kilogram (mass), second (time), ampere (electric current), kelvin(temperature), mole (amount of substance), and candela (luminous intensity). In the SIsystem the (derived) unit of charge is the coulomb.

The other two systems, Gaussian and Heaviside, are also widely used. Please, seeAppendix C of Griffiths' book (or check any other textbook) for an explanation andconversion factors between SI and Gaussian and Heaviside.

1.5 Electric-Magnetic Duality

Let's rewrite Maxwell's equations in vacuum (r = 0, J = 0) and in Gaussian units tomake them look more symmetrical:

(20)— ◊ E = 0, — μ E = -∑B ê ∑ t— ◊ B = 0, — μ B = ∑E ê ∑ t.

Written like this we see that there is a remarkable symmetry between electric andmagnetic fields. In fact, if we make the duality transformation, E Ø B and B Ø -E,the equations remain invariant. (The upper set becomes the lower set, and vice versa.)This is known as electric-magnetic duality which, as we can see, is a property ofMaxwell's equations in vacuum. However, this doesn't work if charges and currents arepresent. (Try it!) In order to restore the duality symmetry we would need to includemagnetic charge densities rm associated to magnetic charges qm (monopoles), and


magnetic currents Jm. The generalized Maxwell's equations would then look like this(in Gaussian units, again):

(21)— ◊ E = 4 pre, — μ E = -H4 p ê cL Jm - H1 ê cL ∑B ê ∑ t— ◊ B = 4 prm, — μ B = +H4 p ê cL Je + H1 ê cL ∑E ê ∑ t.

where re and Je are the electric charge density and current, respectively, and rm andJm are the magnetic charge density and current. The full duality transformations thatleave these equations invariant are

(22)E Ø B, B Ø -E,

re Ø rm, rm Ø - re ,Je Ø Jm, Jm Ø -Je .

You should check that, in fact, equations (21) remain invariant under the transforma-tion (22). A direct consequence of eqs. (21) is the conservation of electric and mag-netic charge, expressed in their corresponding continuity equations

(23)d r ê dt = -— ◊ J

where J = Je or Jm. Why did nature choose Maxwell's equations without magneticcharges rather than their extended counterpart (21) is still a mystery. Some (for nowhypothetical) recent extensions of our well-established theories of the fundamentalinteractions in nature predict the existence of magnetic monopoles, but no experimen-tal evidence has ever been found for them. Even the notion of duality has becomeimportant in recent speculations regarding the basic laws of nature. Time will tell.

1.6 Maxwell's Equations for Static Fields

Maxwell's equations when the fields don't change over time are

(24)— ◊ E =1

ÅÅÅÅÅÅÅÅ¶0

r,

(25)— ◊ B = 0,

(26)— μ E = 0,

(27)— μ B = m0 J.

Equations (20) and (22) are Maxwell's equations for electrostatics. If we make theassumption that the electric field at an infinite distance from all sources is zero, theyare equivalent to Coulomb's law. (In fact, we will prove this statement later.) Similarly,


eqs. (21) and (23) are Maxwell's equations for magnetostatics, and they are equivalentto the Biot-Savart law under the assumption that the magnetic field vanishes at aninfinite distance from any currents. A major portion of this course will be devoted to astudy of the solutions to these equations, since they have important physicalapplications.

The fact that, in electrostatics, — μ E = 0 implies that the electric field can bewritten as the gradient of a scalar function

(28)E = -—V .

[This is so on account of the identity — μ H— fL = 0, valid for any scalar field f.] Thescalar field V is known as the electric potential (or just potential). Dimensionally,electric potential is potential energy per unit charge.

The fact that the curl of the electrostatic field E is zero is equivalent to the state-ment that the line integral of E over any closed path is zero: ò E ÿ dr = 0. You mayrecall that the work of any conservative over a closed path is always zero. Well, this isprecisely what I am saying here. The work done per unit charge done by an electro-static field is zero for any closed path. In other words, the electrostatic field is conserva-tive. The very existence of a potential function V tells us the same. Whenever a vectorfield is conservative, there is an associated scalar potential, and vice versa.

As I explain in more detail in section 1.7, there is an ambiguity in the definition of thepotential. In the case of electrostatic fields, this ambiguity means that we can add anyconstant to the potential without changing the value of the electric field. (This is obvi-ous, since E is the gradient of V.) In more elementary treatments we say that what wemeasure are only differences in potential energy, and that we need always to fix a "zerolevel" of potential energy. (The standard convention is to make V = 0 at infinity.)

The advantage of using V instead of E comes from the fact that, in general, it iseasier to manipulate scalars than vectors. Moreover, in terms of the potential V, Max-well's equation (26) has nothing to say. (It is just an identity.) Equation (24), on theother hand, becomes — ◊ H—V L = - 1ÅÅÅÅÅÅÅ

¶0 r, or

(29)“2 V = -1

ÅÅÅÅÅÅÅÅ¶0

r.

This is Poisson's equation. In the case of no charge densities present, r = 0, and

(30)“2 V = 0,

which is Laplace's equation. Both equations are extremely important in electrostatics(physics, in general) and we will study methods for solving them in this course. In fact,


solving either of them to find the potential is equivalent to solving both Maxwell'sequations for electrostatics, (24) and (26). [Having found V we simply use (28) todetermine E.] So, you see, we have traded two vector first-order partial differentialequations [(24) and (26)] for one second-order partial differential equation, either (29)or (30). We will have more to say about the electric potential later when start thediscussion of electrostatics in earnest.

In the case of the magnetic field, the curl is not necessarily zero, but the divergencealways is. Using the identity — ÿ H— μ f L = 0, valid for any vector function f , we canwrite the magnetic field as

(31)B = — μ A.

The vector field A is called the magnetic vector potential (or simply the vector poten-tial). Plugging (31) into (27),

— μ H— μ AL = m0 J,

and using the identity — μ H— μ f L = —H— ◊ f L - “2 f , valid for any vector function f ,

— H— ◊ AL - “2 A = m0 J.

Once again there is an ambiguity in the definition of A. We can add the gradient of anyscalar field y to A without affecting B. (Because — μ — y = 0.) Because of that we arefree to choose, for example, an A whose divergence is zero, — ◊ A = 0, turning theabove equation into

(32)“2 A = - m0 J.

Eq. (32) is three Poisson's equations, one for each component of A. Although not asuseful as the Poisson equation for the electric potential, eq. (32) together with (31)could be used as an alternative foundation for magnetostatics.

1.7 Potentials, Gauge Transformations, and Electrodynamics

What if the fields are not static? Can we still write them in terms of potentials? Well,obviously nothing changes for the magnetic field (— ÿ B = 0, no matter what). But theelectric field is no longer "curless". Nevertheless, since B = — μ A, writing

(33)E ª -—V -∑AÅÅÅÅÅÅÅÅÅÅÅ∑ t

,


we get

(34)— μ E = -— μ —V´ ¨¨¨ ¨ ¨≠ Æ¨¨¨¨ ¨=0

- — μ J ∑AÅÅÅÅÅÅÅÅÅÅÅ∑ t

N = -∑

ÅÅÅÅÅÅÅÅ∑ t

— μ J ∑AÅÅÅÅÅÅÅÅÅÅÅ∑ t

N = -∑BÅÅÅÅÅÅÅÅÅÅÅ∑ t

.

So, we see that — μ E = -∑B ê ∑ t (Faraday's law) and — ◊ B = 0 (no monopoles) areautomatically satisfied with the definitions (31) and (33). [Eq. (33) reduces to (28) fora static A.] Checking Gauss's law, — ◊ E = r ê e0,

(35)— ◊ E = -— ◊ —V - — ◊∑AÅÅÅÅÅÅÅÅÅÅÅ∑ t

= -“2 V -∑


— ◊ A,

which is satisfied if

(36)-“2 V -∑


— ◊ A =1

ÅÅÅÅÅÅÅe0

r.

The Ampère-Maxwell law, — μ B = m0 J + m0 ¶0 ∑E ê ∑ t, becomes

(37)— μ H— μ AL = m0 J + m0 ¶0 ∑


J-—V -∑AÅÅÅÅÅÅÅÅÅÅÅ∑ t

N.

Using again the identity — μ H— μ f L = —H— ◊ f L - “2 f ,

(38)—H— ◊ AL - “2 A = m0 J - m0 ¶0 —∑VÅÅÅÅÅÅÅÅÅÅÅ∑ t

- m0 ¶0 ∑2 AÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑ t2 .

We can make everything more symmetrical and easier to handle by adding a termm0 ¶0 ∑2 V ê ∑ t2 to both sides of (36) and rearranging,

(39)m0 ¶0 ∑2 VÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑ t2 - “2 V =

1ÅÅÅÅÅÅÅe0

r +∑


J— ◊ A + m0 ¶0 ∑VÅÅÅÅÅÅÅÅÅÅÅ∑ t

N,

and rewriting (38) as

(40)m0 ¶0 ∑2 AÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑ t2 - “2 A = m0 J - — J— ◊ A + m0 ¶0

∑VÅÅÅÅÅÅÅÅÅÅÅ∑ t

N.

Eqs. (39) and (40) both contain the piece — ◊ A + m0 ¶0 ∑V ê ∑ t which, as we will see,can be made equal to zero without altering Maxwell's equations. In fact, definitions(31) and (33) give us the freedom to make the changes to the potentials

(41)A Ø A£ = A + — y,

and

(42)V Ø V £ = V -∑yÅÅÅÅÅÅÅÅÅÅ∑ t

,

without altering E and B. Checking,


(43)B = — μ A Ø B£ = — μ HA + — yL = — μ A + — μ — y´ ¨¨¨ ¨ ¨≠ Æ¨¨¨¨ ¨=0

= B,

and

(44)E = -—V -

∑AÅÅÅÅÅÅÅÅÅÅÅ∑ t

Ø E£ =

-—JV -∑ yÅÅÅÅÅÅÅÅÅÅ∑ t

N -∑


HA + — yL = -—V + —∑yÅÅÅÅÅÅÅÅÅÅ∑ t

-∑AÅÅÅÅÅÅÅÅÅÅÅ∑ t

-∑


— y.

But —H∑y ê ∑ tL = H∑ ê ∑ tL — y. So, E£ = E.

The transformations (41) and (42) are known as gauge transformations. These arethe general statements for the "ambiguities" in the definition of V and A that we men-tioned in the previous section. The invariance of electromagnetic theory under gaugetransformations is an example of a gauge symmetry. This "gauge freedom" allows usto choose any scalar function y and transform the potentials according to (41) and (42)without affecting any physical quantities (the one we measure with our instruments).Thus, in classical electromagnetic theory (what we are studying in this course), thepotentials A and V are assumed to have no absolute physical significance, since we canalways redefine them by a gauge transformation without altering the underlying phys-ics. This is not the case in quantum mechanics, though. As you will see when youstudy the topic, potentials do have a more physical (as opposed to just mathematicallyconvenient) role to play. (For example, in the Bohm-Aharonov effect.)

With the gauge freedom afforded by (42) and (43), we can choose a specific scalarfunction y that will make — ◊ A + m0 ¶0 ∑V ê ∑ t equal to zero. (Choosing a particular yis known as fixing the gauge or choosing a gauge condition.) The gauge condition

(45)— ◊ A + m0 ¶0 ∑VÅÅÅÅÅÅÅÅÅÅÅ∑ t

= 0

is known as the Lorentz gauge. In general, A and V don not satisfy (45), but we canalways make a gauge transformation that will ensure that the new fields A£ and V £ dosatisfy it. Using the Lorentz gauge, eqs. (39) and (40) become

(46)m0 ¶0 ∑2 VÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑ t2 - “2 V =

1ÅÅÅÅÅÅÅe0

r,

and

(47)m0 ¶0 ∑2 AÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑ t2 - “2 A = m0 J,

respectively. Equation (46) is the inhomogeneous wave equation for V, and (47) theinhomogeneous wave equation for A. They are direct consequences of the Gauss and


Ampère-Maxwell laws and are very important.

Equations (46) and (47), together with (31) and (33) which define the electric andmagnetic fields in terms of A and V, are equivalent to the full Maxwell equations andform the foundation for the study of electrodynamics (next semester!). This is analo-gous to the way that eqs. (29) and (32), together with (28) and (31), are equivalent toMaxwell's equations for static fields, and constitute the foundation of electrostatics andmagnetostatics (this semester!).


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