PHGN462: Advanced E&M
Quiz 4 – October 5, 2009
NAME: KEY
1. Consider a hollow waveguide with a rectangular cross section of dimensions a × b. It is oriented along the z-axis with the
x-axis along the a side and y-axis along the b side. By z-translational symmetry, we may assume the electric and magnetic
fields take the form:
~ = E(x,
~
E
y)eı(kz z−ωt)
~ = B(x,
~
B
y)eı(kz z−ωt)
(Note the difference between bold-faced and normal-faced vectors.)
a. Take the x̂-component of Maxwell’s third equation and the ŷ-component of Maxwell’s fourth equation, and,using the
forms above, solve for Ey and Bx in terms of {Ez , Bz } and the plane wave factors, {kz , ω}. Show all work.
Solution:
~
~ = − ∂B
~ ×E
MIII: x̂ · ∇
∂t · x̂
~
~ = − 12 ∂ E · ŷ
~ ×B
MIV: ŷ · ∇
c ∂t
∂E
x
− ∂zy = − ∂B
∂t
Using the plane wave form for the z- and t-dependence gives
∂Ez
∂y
∂ ı(kz z−ωt)
∂z e
∂Bx
∂Bz
∂z − ∂x
ı(kz z−ωt)
= ıkz e
∂Ez
∂y
∂E
= c12 ∂ty
∂ ı(kz z−ωt)
and ∂t
e
= −ıωeı(kz z−ωt)
− ıkz Ey = ıωBx
ıkz Bx −
This is two equations for the two unknowns, {Ey , Bx }. Solve by substitution to find:
∂Bz
∂x
= − ıω
c2 Ey
ω ∂Ez
∂Bz
kz
− 2
Bx = 2 2
(ω /c − kz2 )
∂x
c ∂y
ı
∂Ez
∂Bz
Ey = 2 2
kz
−ω
(ω /c − kz2 )
∂y
∂x
ı
b. Consider the special case of transverse magnetic modes (Bz = 0). Assume the general solution of the z-component of
the electric field is:
Ez = Ez0 [α sin(kx x) + β cos(kx x)] [γ sin(ky y) + δ cos(ky y)]
(1)
where Ez0 is the amplitude of Ez (x, y). Apply the boundary condition that E|| is continuous across a boundary to find
{α, β, kx }. Show all work.
Solution:
Since E = 0 inside the conductor, for E|| to be continuous across the surface of the conductor, Ey (x = 0, y) = 0 and
Ey (x = a, y) = 0. Using the expression above, and setting Bz = 0 for transverse magnetic modes, we have
Ey ∝
∂Ez
∝ [α sin(kx x) + β cos(kx x)] .
∂y
(The y-derivative doesn’t change the x-dependence.) Thus, applying the boundary conditions gives β = 0 and kx =
Finally, since Ez0 is the stated amplitude of the wave, α = 1.
1
(2)
nπ
a .