Classical Electrodynamics — PHY5347

HOMEWORK 20

(February 28, 2013)

Due on Tuesday, March 21, 2013

PROBLEM 58

Consider a waveguide made of an ideal conductor with cross sectional area in the shapeof a quarter circle of radius R, as shown in the figure. In what follows assume thatyou are interested in computing only transverse magnetic (TM) waves, i.e., Bz ≡ 0everywhere.

(a) We have shown in class that knowledge of the longitudinal component of theelectric field (Ez) is sufficient to completely specify E and B. Assuming atraveling-wave solution of the form:

Ez(x, t) = Ez(ρ, φ)ei(kz−ωt) ,

compute the longitudinal component of the electric field Ez(ρ, φ) by solving theappropriate eigenvalue equation.

(b) From the results of part (a) obtain Ez(ρ, φ) for the lowest-frequency mode ofpropagation and give the value of the cutoff frequency.

You might find the following table useful:

Function First rootJ0(J

′0) 2.405(3.832)

J1(J′1) 3.832(1.841)

J2(J′2) 5.136(3.054)

Here Jm are Bessel functions satisfying the differential equation:

d2

dr2Jm(kr) +

1

r

d

drJm(kr) +

(k2 − m2

r2

)Jm(kr) = 0 .

R

z

PROBLEM 59

A resonant cavity in the form of a cube of side a (as shown in the figure) is made ofperfectly conducting walls. For a certain cavity mode the electric field is given by thefollowing expression:

E(r, t) = x̂E0 sin(πya

)sin

(πza

)e−iωt .

In what follows assume that the material inside the cavity is vacuum.

(a) Compute the magnetic field B.

(b) Show explicitly that all of Maxwell’s equations and its associated boundaryconditions are satisfied.

(c) What is the resonant frequency of this mode?

(d) Find the currents that are flowing through the walls.

PROBLEM 60

In Section 8.4 Jackson solves the problem of the propagation of TE waves in a rect-angular wave guide with inner dimensions a and b (see Jackson’s Figure 8.5). In thisproblem you will consider the same geometry but now for TM waves. Throughoutthis problem you may assume that ε=µ=1.

(a) Using Maxwell’s equation and its associated boundary conditions, compute thelongitudinal component of the electric field Ez(r, t) by solving a suitable eigen-value problem.

(b) For the lowest propagating mode, obtain all the components of both the electricand magnetic fields.

(c) For the lowest propagating mode, obtain the surface charge density σ induced inthe walls of the waveguide.

(d) For the lowest propagating mode, obtain the surface current density K inducedin the walls of the waveguide.