Maxwell's equations for a region with no charge or current are, in differential form:
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: