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- A general electromagnetic wave pulse propagating in the
-direction at velocity
is written

where
,
,
,
,
, and
are arbitrary functions. In order
to exclude electrostatic and magnetostatic fields, these functions are subject to
the constraint that
, where

Suppose that the pulse propagates through a uniform dielectric medium of
dielectric constant
. Demonstrate from Maxwell's equation that
,
,
, and
. Incidentally, this result implies that a general
wave pulse is characterized by two arbitrary functions, corresponding
to the two possible independent polarizations of the pulse.

- Show that the mean energy flux due to an electromagnetic wave of angular frequency
propagating though a good
conductor of conductivity
can be written

where
is the peak electric field-strength, and
.

- Consider an electromagnetic wave propagating in the positive
-direction through a good
conductor of conductivity
. Suppose that the wave electric field is

where
is the skin-depth. Demonstrate that the mean electromagnetic energy flux across the plane
matches the
mean rate at which electromagnetic energy is dissipated per unit area due to Joule heating in the region
.

- A plane electromagnetic wave, linearly polarized in the
-direction, and propagating in the
-direction through an electrical
conducting medium of conductivity
and relative dielectric constant unity, is governed by

where
and
are the electric and magnetic components
of the wave.
Derive an energy conservation equation of the form

where
is the electromagnetic energy per unit volume, and
the electromagnetic energy flux.
Give expressions for
and
. What does the right-hand side of the
previous equation represent? Demonstrate that
obeys the wave-diffusion
equation

where
. Consider the high frequency, low conductivity, limit
. Show that a wave propagating into the medium varies as

where
,
, and
.
Demonstrate that
: that is, the wave penetrates many wavelengths into the medium.

- Consider a uniform plasma of plasma frequency
containing a uniform magnetic field
. Show that left-hand
circularly polarized electromagnetic waves can only propagate parallel to the
magnetic field provided that
, where
is the electron cyclotron frequency.
Demonstrate that right-hand circularly polarized electromagnetic waves can only propagate
parallel to the magnetic field provided that their frequencies do
not lie in the range
.

** Next:** Wave Propagation in Inhomogeneous
** Up:** Wave Propagation in Uniform
** Previous:** Signal Arrival
Richard Fitzpatrick
2014-06-27