Download Physics PHYS 354 Electricity and Magnetism II Problem Set #9

Physics PHYS 354
Electricity and Magnetism II
Problem Set #9
1.
Normal Modes in a Cavity
Construct solutions of Maxwell's equations in the empty interior of a box with perfectly
conducting plane walls at x 0 and a, y 0 and b, and z 0 and c, as follows:
Each of the field components, Ex , E y , Ez , Bx , By , and Bz is of the form:
Ei eit Ki fi x gi  y hi z 
where: K i is a constant,
f x  is sin x  or cos x 
g  y  is sin y  or cos y 
and
h z  is sin z  or cos z  .
(  , ,  , and  are the same for all components, but K , f , g , and h are different.)
a)

Choose forms for E and values of  ,  , and  to satisfy the conditions on
 
the faces of the box, and  E 0 .
b)
Each field component must satisfy
 2 1 2 
   2 2 F 0
c t 

(the wave equation in free space).
Express w in terms of  ,  , and  .
c)

Now determine the form of B .
d)

Does this solution for B satisfy the boundary condition that the normal

component of B equal zero at the walls?
e)
You have found the frequency of the normal modes of oscillation in this
cavity. How many modes are there?
2.
3.
Plane Waves in Conductors
a)
Show that the expressions for the electric and magnetic fields in good
conductors satisfy all of Maxwell's equations.
b)
Compare the ratios E H for an electromagnetic wave in air and in sea
water at 20 kHz and at 20 MHz.
c)
Calculate the attenuation of an electromagnetic wave in sea water and in
copper at 20 kHz and at 20 MHz. State your result in dB/m.
Circular Polarization
A circularly polarized wave results from the superposition of two waves that are
a)
b)
c)
of the same frequency and amplitude,
plane-polarized in perpendicular directions, and
90° out of phase.
Show that the average value of the Poynting vector for such a wave is the sum of
the average values of the Poynting vectors for the two plane polarized waves.
2