Maxwell's equations for a region with no charge or current are, in differential form:
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(1) |
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(2) |
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(3) |
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(4) |
Here I have assumed that the the charge density and current
density
are zero, and that the electric displacement vector
can be expressed as
and the magnetic flux
can be expressed as
, which are common
assumptions. Thus, the speed
in the equations above refer to the
speed of light in the particular medium
,
and in the case of a vacuum,
is the standard speed of light in
vacuum.
In this problem I will focus exclusively on the magnetic field
components. Since the above equations are coupled, it suffices to
solve either for the electric field or magnetic field
,
and then the other vector can be inferred. This leads to a set of
vector wave equation: