14.7 Alternative Treatment using Scalar and Vector Potentials

Our first step is to search for equations for E, B in terms of the scalar potential ? and the vector potential A. Again we assume that the interior of the wave guide is filled with some homogeneous material with electromagnetic constants ?, ?(e.g. those of the vacuum). Then inside the wave guide the following equations apply

But, we also have

and

and

Hence we can write

Now that we are using the vector potential A, we still have the freedom to choose some gauge fixing condition. We choose again (as in our treatment of the Lihard-Wiechert potentials) the Lorentz gauge, i.e. we set

Then

With the ansatz A ? e ^{? i ? t} (as above for E, H) the following equation results

Hence from Eq. (14.101):

with

Equations (14.103) and (14.104) show: E, B can be derived from A, which is a solution of Eq. (14.102).

We consider now the TM and TE cases separately.

1. TM: B _z = 0 everywhere.

   We set

   
   

   Then

   
   

   and from Eq. (14.103),

   
   

   i.e.

   
   

   However, we still have the equation ? E = 0, i.e. from Eq. (14.103)

   
   

   and this is satisfied in view of Eq. (14.102), for A = (0, 0, ?), i.e.

   
   

   All fields can now be derived from the one scalar function ?, which is a solution of this equation. The function ?