1

EE311: Assignment 4

1. Suppose

E(r, θ, φ, t) = E0sin θ

r

[cos(kr − ωt) − 1

krsin(kr − ωt)

]φ , with

ω

k= c .

(This is, incidentally, the simplest possible spherical wave.)

(a) Show that E obeys all four of Maxwell’s equations, in vacuum, and find the associated magnetic field.

(b) Calculate the Poynting vector. Average S over a full cycle to get the intensity vector I. Does it point inthe expected direction ? Does it fall off like r−2, as it should ?

(c) Integrate I · da over a spherical surface to determine the total power radiated.

[Answer:4πE2

0

3µ0c]

2.

(a) Suppose you imbedded some free charge in a piece of glass (n = 1.5, σ = 10−12 mho/m). About how longwould it take for the charge to flow to the surface ?

(b) Silver is an excellent conductor (ρ = 1.59 × 10−8 Ω-m), but it’s expensive. Suppose you were designing amicrowave experiment to operate at a frequency of 1010 Hz. How thick would you make the silver coatings?

(c) Find the wavelength and propagation speed in copper (σ = 6 × 107 mho/m) for radio waves at 1 MHz.Compare the corresponding values in air (or vacuum).

3.

(a) Show that the skin depth in a poor conductor (σ << ωε) is (2/σ)√ε/µ, (independent of frequency). Find

the skin depth (in meters) for pure water (ε = 80.1ε0, µ = µ0 and ρ = 2.5 × 105 Ω-m).

(b) Show that the skin depth in a good conductor (σ >> ωε) is λ/2π (where λ is the wavelength in theconductor). Find the skin depth (in nanometers) for a typical metal (σ = 107 mho/m) in the visible range(ω = 1015/s), assuming ε = ε0 and µ = µ0. Why are metals opaque?

(c) Show that in a good conductor the magnetic field lags the electric field by 45, and find the ratio of theiramplitudes. For a numerical example, use the ”typical metal” in part (b).

4.

(a) Calculate the (time averaged) energy density of an electromagnetic plane wave

E(z, t) = E0e−κz cos(kz − ωt+ δE) x

B(z, t) = B0e−κz cos(kz − ωt+ δE + φ) y .

in a conducting medium. Show that the magnetic contribution always dominates.

[Answer: k2

2µω2E20 e

−2κz]

(b) Show that the intensity is k2µωE

20 e

−2κz.

5. Find the width of the anomalous dispersion region for the case of a single resonance at frequency ω0. Assumeγ << ω0. Show that the index of refraction assumes its maximum and minimum values at points where theabsorption coefficient is at half-maximum.