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

(1) | |||

(2) | |||

(3) | |||

(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:

When taking , one can use to eliminate the first term on the right hand side, leaving only the second curl-of-curl term. The interior curl can be found by again using Maxwell's equations, leaving

Upon exchanging the order of differentiation on the right hand side, and using Maxwell's equations one last time for , I verify the the wave equation (5) is correct.