Note well that we have written the mutilated Maxwell Equations so that
the components are *all on the right hand side*. If they are
known functions, and if the only dependence is the complex
exponential (so we can do all the -derivatives and just bring down a
) then *the transverse components
and
are determined!*

In fact (for propagation in the direction,
):

(10.52) | |||

(10.53) |

and

or

(10.54) | |||

(10.55) |

(where we started with the second equation and eliminated to get the second equation just like the first).

Now comes the relatively tricky part. Recall the boundary conditions for
a perfect conductor:

They tell us basically that ( ) is strictly perpendicular to the surface and that ( ) is strictly parallel to the surface of the conductor

This means that it is *not* necessary for or *both*
to vanish everywhere inside the dielectric (although both can, of
course, and result in a TEM wave or no wave at all). All that is
strictly required by the boundary conditions is for

(10.56) |

(10.57) |

We therefore have two possibilities for *non-*zero or
that can act as source term in the mutilated Maxwell Equations.