14. Ch. 6, part 1

Lecture 14 - Maxwell’s equations

Just as moving charges generate magnetic fields, so moving magnets generate electric fields. Faraday was the first to understand this. He showed more generally that there is an electric field associated with a time-varying magnetic field. His results are summarized in the law that bears his name, so that after his work one had the four equations

Coulomb′s law :  ∇ · E = 4πρ
Ampère′s law  ∇ × B = \frac{4π}{c} J

Faraday′s law ∇ × E + \frac{1}{c} \frac{∂B}{∂t} = 0
∇ · B = 0

Maxwell realized that Ampère’s law cannot be correct in general, because it implies that ∇ · J = 0, which is true for steady currents, but not always true for time-dependent currents. He fixed up Ampère’s law by adding to it a term that makes it look more like Faraday’s law. The amended equation :

∇ × B − \frac{1}{c} \frac{∂E}{∂t} = \frac{4π}{c} J

could be valid in general because (using also Coulomb’s law) it implies

−\frac{∂ρ}{∂t} = ∇ · J

which is the equation of continuity for the electrical current and expresses the principle of charge conservation (net charge is never created or destroyed). Today we have a very strong belief in charge conservation and further we know that charge is quantized. It is current practice to construct field equations by requiring consistency with conservation laws. One can in fact claim, as we have seen, that the full set of Maxwell equations is dictated by charge conservation.

From a modern perspective, one can also claim that Faraday’s law is a consequence of Ampère’s law and of the basic principle of relativity that goes back to Galileo. With apologies to Faraday and his beautiful experiments, here is the argument.

14.1. Faraday’s law of induction

We can start from the Lorentz force law, which is Ampère’s force law for a current consisting of a point charge q moving with velocity v:

F = \frac{q}{c} v × B

Consider then a conducting rod parallel to the x axis and moving with velocity v in the y direction, in the presence of a field B in the z direction. A charge q in the rod is subject to a the force F = qvB/c in the x direction. A force acting on a particle with a strength proportional to its charge is caused by an electric field, by definition. Thus there is in the rod an electric field E′ = vB/c in the x direction. This field will cause accumulation of opposite charges on the ends of a moving, conducting rod; the charge will accumulate until it causes an electrostatic field (in the frame of the moving rod) that cancels E′. The direction of E′ is to the right when the rod is moving upwards in the figure and B is out of the sheet.

If the rod is part of a loop, its upward motion during time dt increases the magnetic flux through the loop by dΦ = Bl vdt, where l is the length of the rod. At the same time there is an “electromotive force” in the circuit that will cause a current to flow clockwise. In this example the electromotive force is a potential difference E′l = Blv/c between the ends of the rod, with the left end at the higher potential. In general one defines

Electromotive force E = ∮_{C} E′ · dl

with the integral going around the loop counterclockwise with respect to a normal vector direction n, and one also defines with the same orientation of n

Magnetic flux Φ = ∫_{S} B · n da

In our example E = −Blv/c and the rate of change of the flux is dΦ/dt = Bl v, so we obtain Faraday’s law

E = −\frac{1}{c} \frac{dΦ}{dt}

The negative sign means physically that E induces a current (clockwise in our example, i.e. negative) that opposes the change of flux through the circuit. This sign rule is known as Lenz’s law.

If a circuit moves rigidly through a uniform magnetic field, there is no net electromotive force, and no net current, but some charge accumulates on one side. If a circuit moves rigidly trough a non-uniform field, there is a net E in accordance with Faraday’s law, it is easy to see. By the principle of relativity, the same E must appear if the circuit is fixed and the (non-uniform) magnetic field is moving. Finally, it cannot make any difference if B varies because it is generated by moving sources or because it is generated by varying currents. Faraday’s law holds generally, whatever is the cause of dΦ/dt.

Back at the start of the argument, we have denoted the field in the moving rod by E′, rather than E, to stress that it is the field as seen by an observer moving with the rod. For this observer the rod is fixed and the field is moving and thus E′ also is the field in the lab frame when the rod is fixed and the field is moving.

Returning to the lab frame, we rewrite Faraday’s law as

∮_{C} E · dl + \frac{1}{c} \frac{∂}{∂t} ∫_{S} B · n da = 0

and apply Stokes’ theorem to obtain

∫_{S} (∇ × E + \frac{1}{c} \frac{∂B}{∂t}) ·n da = 0

and

∇ × E+\frac{1}{c} \frac{∂B}{∂t} = 0

14.2. Maxwell’s equations in matter

Maxwell also derived the equations for the suitably averaged fields in the presence of material media. They are

Coulomb′s law :  ∇ · D = 4πρ
Ampère′s law ∇ × H = \frac{4π}{c} J+\frac{1}{c} \frac{∂D}{∂t}

Faraday′s law  ∇ × E = −\frac{1}{c} \frac{∂B}{∂t}
∇ · B = 0

with

D = E + 4πP
H = B − 4πM

Maxwell’s correction to Ampère’s law corresponds to a displacement current (1/4π)∂D/∂t. One part of it is the polarization current J_{P} = ∂P/∂t. Similarly, one part of ∇ × H corresponds to the magnetization current J_{M} = c∇ × M. This is consistent with our earlier definitions of P and M. We note that ∇ · J_{M} = 0, or, in words, J_{M} is a transverse current. All this suggests a simple derivation of the equation in matter as follows.

Start from the microscopic Ampère-Maxwell equation

∇ × B = \frac{4π}{c} (J + J_{ind})

where J_{ind} is the current induced in matter by the application of external fields. Separate from J_{ind} a transverse part that you are going to call J_{M} and call the rest J_{P} . Define M and P by J_{M} = c∇ × M and J_{P} = ∂P/∂t. It is a bit arbitrary what should go into J_{M} and what should stay in J_{P} . In a linear, homogeneous medium P = χ^{(e)}E and

∇ × J_{P} = χ^{(e)} \frac{∂}{∂t} ∇ × E = 0

if B is time-independent. Then J_{P} must be the longitudinal part of J_{ind} and there is no ambiguity. Anyhow, the Ampère-Maxwell equation becomes

∇ × B − \frac{1}{c} \frac{∂E}{∂t} = \frac{4π}{c} J+4π∇ × M + \frac{4π}{c} \frac{∂P}{∂t}

and it is just a matter of rearranging to get

∇ × H = \frac{4π}{c} J+\frac{1}{c} \frac{∂D}{∂t}

Poisson’s equation follows by taking the divergence. Using the continuity equation, one gets

0 = 4π∇ · J+\frac{∂}{∂t}∇ · D = −4π \frac{∂ρ}{∂t}+\frac{∂}{∂t}∇ · D

At this point these are still exact microscopic equations. The next step is to average spatially to get the approximate macroscopic equations. In this process one gets the constitutive relations between J_{P} and E and between J_{M} and H. Some aspects of this subject are postponed by Jackson until Sections 7.5 – 7.7. We discuss here briefly the averaging process for J_{P} in a linear, homogeneous medium.

14.2.1. Microscopic response and averaging.

In general the linear response is not instantaneous and not local, meaning that the polarizability P at point x and time t depends on the electric field E at earlier times t′, according to

P(x, t) = ∫ d^{3}x′ ∫_{−∞}^{t} χ^{(e)}(x, x′, t − t′)E(x′, t′) dt′

The integral over x′ is formally over all space, but the function χ^{(e)}(x, x′, t − t′) goes quickly to zero when |x − x′| is greater than some characteristic length l (to be discussed later). It also goes to zero when t − t′ is larger than some characteristic time, but that does not concern us here.

To obtain the macroscopic equations, the main approximation is to replace the actual medium by an average medium that is homogeneous over a region of size l. Then χ^{(e)} depends only on x − x′. Fourier transforming according to

χ^{(e)}(x − x′, t) = ∫ χ^{(e)}(k, ω) exp [i (k· (x − x′) − ωt)] \frac{d^{3}k dω}{(2π)^{4}}

one finds by the convolution theorem

P(k, ω) = χ^{(e)}(k, ω)E(k, ω)

Note for future reference that a pair of direct and inverse Fourier transforms is

P(x, t) = ∫ P(k, ω) exp [i (k · x − ωt)] \frac{d^{3}k dω}{(2π)^{4}}

P(k, ω) = ∫ P(x, t) exp [−i (k · x − ωt)] d^{3}x dt

The next step is to say that we are interested in small values of k and to replace χ^{(e)}(k, ω) by its limit for k → 0. This approximation is valid when P and E vary slowly over distances of order l. Then the constitutive relation is reduced to P(k, ω) = χ^{(e)}(k = 0, ω)E(k, ω), which is in effect a local relation: P(x, ω) = χ^{(e)}( ω)E(x, ω). We have dropped the indication that k = 0 in χ^{(e)}. Finally, one allows the properties of the medium to vary, even though at each point the medium is treated as homogeneous. Then

P(x, ω) = χ^{(e)}(x, ω)E(x, ω)

This relation is also used at an interface, where χ^{(e)}(x, ω) varies abruptly. As a result of the approximations, the macroscopic E_{n} and D_{∥} are discontinuous and one should be careful when using fields that are to be computed “on the surface”, although generally E_{∥} and D_{n} can be used with confidence.

A similar discussion holds for the magnetic susceptibility χ^{(m)} and for the conductivity σ. Actually, for time-dependent phenomena there is no real distinction between Ohm’s law J_{cond}(ω) = σ(ω)E(ω) and the relation

J_{P} (ω) = −iωχ^{(e)}(ω)E(ω)

which follows from J_{P} (ω) = −iωP(ω) and P(ω) = χ^{(e)}(ω)E(ω). A conductor is characterized by a finite value of σ(ω) for ω = 0, and an insulator by a finite value of χ^{(e)}(ω) for ω = 0. In general (except for superconductors) the total current induced by E at small frequencies (up to 10^{12} Hz, typically) is

J_{cond}(ω) + J_{P} (ω) = [σ(0) − iωχ^{(e)}(0)] E(ω)

and we can equivalently say that

σ(ω) = σ(0) − iωχ^{(e)}(0)

or that

χ^{(e)}(ω) = χ^{(e)}(0) − \frac{σ(0)}{iω}

One can say that σ(0) is due to “free” charges, while χ^{(e)}(0) is due to “bound” charges. The characteristic length l for “bound” charges is an atom size, while for “free” charges l is the mean free path between collisions, which can be thousands of Ångstroms. Thus non-local effects are more likely to be important in the response of “free” charges.

What response function one uses is a matter of convenience and convention. At higher frequencies (infrared and optical, 10^{13} – 10^{16} Hz) it is customary to use ϵ(ω) = 1+4πχ^{(e)}(ω), for metals as well as for insulators.

14.2.2. Longitudinal and transverse fields

This is an appropriate time to show how a general vector field can be broken up into its longitudinal and transverse parts. It is convenient to work with fourier transforms of the space coordinates. Taking J as an example, we could work with J(k, t) or with J(k, ω) equally well, so we suppress the dependence on t or ω.

• The longitudinal current J_{l}(k) is the component of J(k) in the direction of k

J_{l}(k) = \frac{k}{k} (\frac{k}{k} · J(k)) = \frac{1}{4π} k (\frac{4π}{k^{2}} k · J(k))

Recalling that 4π/k^{2} is the Fourier transform of 1/|x|, we find from this

J_{l}(x) = −\frac{1}{4π} ∇ ∫ \frac{1}{|x − x′|} ∇′·J(x′) d^{3}x′

• The transverse current J_{t}(k) is the component of J(k) in the direction perpendicular to k. One can write

J_{t}(k) = J(k) − J_{l}(k) = J(k) − \frac{k}{k} ( \frac{k}{k} · J(k))

but it is often better to use the equivalent form

J_{t}(k) = −\frac{k}{k} × (\frac{k}{k} × J(k)) = −\frac{1}{4π} k× ( k×\frac{4π}{k^{2}} J(k))

which in direct space gives

J_{t}(x) = \frac{1}{4π} ∇× ∫ \frac{1}{|x − x′|} ∇′ × J(x′) d^{3}x′

An immediate application of this decomposition is the solution of quasistatic problems of electricity, which are governed by

Coulomb′s law :  ∇ · E = 4πρ
Faraday′s law ∇ × E + \frac{1}{c} \frac{∂B}{∂t} = 0

under the assumption that ∂B/∂t is assigned (and is not influenced appreciably by the displacement current in the Ampère-Maxwell equation). We have simply E = E_{l} +E_{t} with

E_{l}(x) = − ∇ ∫ \frac{ρ(x′)}{|x − x′|} d^{3}x′
E_{t}(x) = −\frac{1}{4π} ∇× ∫ \frac{1}{|x − x′|} \frac{∂B(x′)}{∂t} d^{3}x′

In other words, we can solve separately the two problems defined by

∇ · E_{l} = 4πρ ∇ × E_{l} = 0

and

∇ · E_{t} = 0 ∇ × E_{t} + \frac{1}{c} \frac{∂B}{∂t} = 0

and add the two solutions. It is a theorem that a vector field is completely determined if its divergence and curl are given everywhere.

14.3. Gauge invariance and gauge choice

The two homogeneous Maxwell equations, ∇ · B = 0 and Faraday’s law ∇ × E = −\frac{1}{c} \frac{∂B}{∂t} , are automatically satisfied if we put

B = ∇ × A
E  = −∇Φ−\frac{1}{c} \frac{∂A}{∂t}

for arbitrary A and Φ. Coulomb’s law ∇ · E = 4πρ and Ampère’s law ∇ × B − \frac{1}{c} \frac{∂E}{∂t} =

\frac{4π}{c} J then become

∇^{2}Φ + \frac{1}{c} \frac{∂}{∂t} ∇ · A = − 4πρ
∇^{2}A − \frac{1}{c^{2}} \frac{∂^{2}A}{∂t^{2}} − ∇ (∇ · A+\frac{1}{c} \frac{∂Φ}{∂t}) = −\frac{4π}{c} J

We have reduced the six first-order equations for the components of E and B to four second-order equations for the scalar potential Φ and the three components of the vector potential A. Not much progress, it seems; but we have not yet used gauge invariance.

If we make the gauge transformation

A′ = A + ∇Λ
Φ′ = Φ − \frac{1}{c} \frac{∂Λ}{∂t}

we get the same B and E from A′ and Φ′ as we do from A and Φ, for any gauge function Λ. A particular choice of Λ is equivalent to imposing an additional condition on A and Φ that leads to an equation for Λ, provided of course that this equation has at least one solution. Here are the two most popular gauges.

14.3.1. Lorentz Gauge

If we impose the condition

∇ · A_{L} + \frac{1}{c} \frac{∂Φ_{L}}{∂t} = 0

the equations for A_{L} and Φ_{L} are standard wave equations, with sources:

∇^{2}Φ_{L} − \frac{1}{c^{2}} \frac{∂^{2}Φ_{L}}{∂t^{2}} = − 4πρ
∇^{2}A_{L} − \frac{1}{c^{2}} \frac{∂^{2}A_{L}}{∂t^{2}} = −\frac{4π}{c} J

One advantage of the Lorentz gauge is obvious: the equations for A and Φ are decoupled. Another advantage is that they are manifestly covariant under Lorentz transformations, as we will see. The third bonus is that the equations have an explicit solution in infinite space. We discuss this more fully in the next Section; we note here only that the Fouriertransformed equation

−k^{2}Φ_{L} + \frac{ω^{2}}{c^{2}} Φ_{L} = − 4πρ

has the solution

Φ_{L}(k, ω) = \frac{4πρ(k, ω)}{k^{2} − \frac{ω^{2}}{c^{2}}}

If we start with A and Φ that do not satisfy the Lorentz condition and make a gauge transformation, we see that the gauge function Λ_{L} must satisfy

∇^{2}Λ_{L} − \frac{1}{c^{2}} \frac{∂^{2}Λ_{L}}{∂t^{2}} = − (∇ · A + \frac{1}{c} \frac{∂Φ}{∂t})

which is again a wave equation with sources and has the solution shown above. There are no boundary conditions on Λ_{L} and one is still free to add to it any solution of the homogeneous wave equation. One says it is still possible to carry out a restricted gauge transformation.

14.3.2. Coulomb gauge

If we impose the condition

∇ · A_{C} = 0

the equation for Φ_{C} is Poisson’s equation

∇^{2}Φ_{C} = − 4πρ

and the equation for A_{C} is again an inhomogeneous wave equation

∇^{2}A_{C} − \frac{1}{c^{2}} \frac{∂^{2}A_{C}}{∂t^{2}} = −\frac{4π}{c} J+\frac{1}{c}∇\frac{∂Φ_{C}}{∂t}

These equations are not manifestly Lorentz-invariant, but are effectively decoupled and are even easier to solve than those in the Lorentz gauge. If we start with ∇ · A ≠ 0 the equation for the gauge function Λ_{C} is of the Poisson type:

∇^{2}Λ_{C} = −∇ · A

and has the familiar solutions.

The Coulomb gauge is also called the transverse gauge, because the source of A_{C} is the transverse current. As usual, this is more easily seen in k space, where Φ_{C} = 4πρ/k^{2}

and \frac{∂ρ}{∂t} = −ik · J, so that

4π J−ik\frac{∂Φ_{C}}{∂t} = 4π J−ik\frac{4π}{k^{2}} \frac{∂ρ}{∂t} = 4π [J−\frac{k}{k} (\frac{k}{k} · J)] = 4πJ_{t}

We are left then with the equation

∇^{2}A_{C} − \frac{1}{c^{2}} \frac{∂^{2}A_{C}}{∂t^{2}} = −\frac{4π}{c} J_{t}

In particular, if there are no longitudinal currents or static charges we can take Φ_{C} = 0. As we will see, radiation fields are generated by transverse currents and are conveniently described in the radiation gauge, which is the gauge with Φ = 0 and ∇ · A = 0.