Next: MHD Waves
Up: Waves
Previous: Sound Waves
In a vacuum where there are no currents and no charges, so that
Maxwell's equations become
 |
(6.12) |
 |
(6.13) |
 |
(6.14) |
 |
(6.15) |
Consider equations (6.13) and (6.15). If we take,
of (6.13) we obtain
Now we use the curl of (6.15) to rewrite the right hand side in
terms of
alone. Hence,
Finally, using (6.14) we obtain
 |
(6.16) |
where
is the speed of light squared. Thus, the magnetic field and the
electric field support electromagnetic waves that propagate at the
speed of light. Note, that this represents very rapid time variations
so that the usual MHD approximation does not apply.
In Cartesian coordinates, the wave equation has constant coefficients
and solutions can be represented in the form
where
and the dispersion relation is
or
 |
(6.17) |
Electromagnetic waves, such as gamma rays, x-rays, UV, visible
light, infra-red waves and radio waves, have wavelengths ranging
from
to
, all propagate at the
speed of light. (6.12) and (6.14) imply that
and so the electromagnetic waves are transverse since they
propagate in a direction that is at right angles to the direction of
the electric and magnetic field vectors. In addition, (6.13)
gives
so that
and the electric and magnetic fields associated with the waves are
perpendicular to each other. Finally, (6.15) implies
where
is a unit vector in the direction of the
magnetic field,
and so
Example 6. .2Suppose that
,
then (6.12) and (6.14) are satisfied identically
and (6.13) and (6.15) imply
 |
(6.18) |
 |
(6.19) |
and so
 |
(6.20) |
One possible solution is
and
Next: MHD Waves
Up: Waves
Previous: Sound Waves
Prof. Alan Hood
2000-11-06