Classical Electrodynamics — PHY5347

HOMEWORK 17

(January 16, 2013)

Due on Thursday, February 21, 2013

PROBLEM 49

We have shown in class that the electromagnetic field carries energy and momentum.In this problem you will show that the electromagnetic field also carries angularmomentum. The total angular momentum carried by the electromagnetic field isgiven by

L =1

4πc

∫r× (E×B) d3r ,

where r is the radius vector from some chosen origin. Recall that we have shownin class that the (linear) momentum density carried by the electromagnetic field (aquantity we denoted by Pγ and we denote here by g) is given by

g =E×B

4πc.

(a) Show that if the magnetic field B is eliminated in favor of the vector potentialA, the total angular momentum may be written as L≡ l + s, where the orbitalangular momentum (which depends on the choice of origin) is given by

l =i

4πc

∫Ej

[r× (−i∇)

]Aj d

3r .

and the intrinsic—or “spin”—angular momentum by

s =1

4πc

∫E×A d3r .

Hint: Use the fact that the ∇ · E=0 and that the fields vanish at infinity.

(b) Assuming a plane-wave form for the vector potential, namely,

A(r, t) = ae± ei(kz−ωt) ,

where e± are polarization vectors of positive and negative helicity, show thatthe intrinsic angular momentum s is related to the energy U by the followingexpression:

s = ±Uωz .

(c) Assuming that the energy of the electromagnetic field is “quantized”, namely,U = ~ω, conclude that the spin is also quantized and given by s = ±~z. Thatis, the “photon” is a spin-1 boson—but with only two (not three!) allowed spinprojections.

PROBLEM 50

Consider a source-free region of space in which the electric field is given by:

E(r, t) = xE0 cos(kz − ωt) ,

where E0 is a constant and k=ω/c.

(a) Compute the magnetic field B(r, t).

(b) Compute all elements of the energy-momentum tensor T µν for this configurationof electromagnetic [E(r, t) and B(r, t)] fields.

(c) Verify explicitly that all conservation laws (or continuity equations) involving theenergy-momentum tensor are satisfied for this configuration of electromagneticfields.

PROBLEM 51

A material is called “Ohmic” (i.e., it obeys Ohm’s law) if the response of its freecarriers (e.g., electrons) to an electric field is linear. That is, the current density Jinduced in the material in response to the applied electric field E satisfies J = σE,where σ is the conductivity of the material. Assume for this problem that the chargedensity in the material vanishes.

(a) Starting from Maxwell’s equations, derive the wave equation satisfied by theelectric field E(r, t).

(b) Assuming a plane-wave solution of the form

E(r, t) = xE0 ei(kz−ωt) ,

obtain the dispersion relation for the propagating wave, i.e., obtain an expres-sion for k in terms of ω and σ (it may be complex!).

(c) Derive an expression for the magnetic field B(r, t).

(d) Obtain the phase difference between E(r, t) and B(r, t) in the limit of a verygood conductor, i.e., in the limit of ω/σ→ 0.