for 
? A full answer is needed for the following.
1.
(a) What is the general behavior of for ? A full answer is needed for the following.
(b) Show that the sum rule
follows from the dispersion relation
and from the behavior of 
for 
Supply the meaning of N and m.
(c) The intensity of a plane wave propagating in the z direction
with frequency 
is observed to be proportional to 
Relate the attenuation constant 
to the imaginary part of the index
of diffraction in the medium.
(d) What is the relation between 
and the
complex index of diffraction 
? Can one write dispersion
relations for 
(e) By an argument similar to that in (b), or otherwise, evaluate
for a non-magnetic medium (
2.
Consider a cylindrical cavity (not necessarily a circular cylinder) with the axis in the z direction. Assume that the walls are perfectly conducting.
(a) What are the boundary conditions for the E and H fields on the side walls?
(b) What are the boundary conditions for E and H on the end walls?
(c) Describe qualitatively the lowest TM mode. You may use a cavity with a rectangular or circular cross section as an example.
(d) Describe qualitatively the lowest TE mode. You may use an example as above.
(e) Which mode has lowest frequency for a long, thin cavity? For a short cavity?
3.
(a) Write down the expression for the skin depth 
in a
medium of d.c. conductivity 
and
magnetic permeability 
at
frequency 
Describe conditions under which this expression is valid.
(b) Under the above conditions, how is the surface current K related to the E field at the surface? How is it related to the B field at the surface?
(c) What is the power dissipated per unit surface area?
SOLUTIONS
1.
(a) 
where N is the electron density, m the electron mass. This
formula also implies that 
faster than 
.
(b) Compare 
with 
.
(c) 
(d) 
One can,
because 
and 
have no zeros for 
.
(e) Compare 
with
.
Get 
or
----------------------
2.
(a) and (b) If we introduce components parallel and
perpendicular to the wall, such as 
and 
the
conditions on any wall are 
and 
In particular on the
end walls 
and 
(c) In the lowest TM mode 
does not depend on z, hence the
frequency does not depend on the length of the cavity 
and is equal to 
the lowest TM cutoff for a waveguide with the same cross section.
(d) In the lowest TE mode 
and the frequency 
depends on 
according to
where 
is the lowest TE cutoff for a waveguide with the
same cross section.
(e) Since 
(see the Examples), the
lowest mode is TM for a short cavity, TE for a long cavity.
Example 1. For a box of sides 
with 
the lowest mode is TM
for 
(short cavity) and TE
for 
(long cavity). Explicitly, with 
and 
:
mode has fields
where 
is the frequency. We see that 
does not depend on 
and is equal to the TM threshold 
(in the usual notation).
mode has fields
where 
is the frequency. We see that 
is of the form (1)
with 
(in the usual notation).
Example 2. For a circular cylinder of radius R and
length 
the lowest mode is TM
for 
(short cavity)
and TE
for 
(long cavity). Explicitly, with 
and 
:
mode has fields
The frequency is 
. We see that 
does not
depend on 
and is equal to the TM threshold 
(in the usual notation).
mode is doubly degenerate (
). For comparison
with the fields in a box it is good to use the linear combination
where 
is the frequency. We see that 
is
of the form (1) with 
(in the usual notation).
Comment: Jackson uses k for 
For the box he uses 
and for the circular cylinder he uses 
for 
-------------------
3.
(a) 
is valid
when 
. One can always write 
and the condition is
then 
In many good conductors 
and 
depend weakly on 
up to
frequencies in the infrared. The surface must also be nearly flat on the
scale of 
(b) 
and 
(c) 
