14.3 Boundary Conditions

We now come to the important aspect of boundary conditions. We recall for this reason the continuity conditions at a boundary surface. We had in particular

where " n" stands for "normal component" and " t" for "tangential component". The two conditions in Eq. (14.22) are exact. The other two conditions which we obtained and used previously are not needed in the present case since surface charge and surface current are here zero (for ? = ?).The two conditions above can also be written ( e _n e _z = 0)

with B ⁽¹⁾ = B, E ⁽¹⁾ = E in the interior of the wave guide and B ⁽²⁾ = 0 = E ⁽²⁾ outside (again for ? = ?, i.e. ideal conductors). We assume here that the wave guides are infinitely long in order to avoid finite end effects.

We can write the two boundary conditions:

i.e. (since n ? B _z):

and

These conditions hold at the boundary surface (i.e. not anywhere else). Since we saw that the fields E _?, B _? follow from E _z, B _z, we are interested in the boundary conditions and the differential equations of E _z, B _z. Thus we have to find the boundary condition of B _z. This is our next step.

Vector multiplication of the Maxwell equation (14.19) by e _z