14.2 Transverse Fields Derived from Longitudinal Fields

We now show that it suffices to know the z-components E _z, B _z; the transverse fields E _?, B _? can then be obtained from these. We consider the two curl-equations of Eq. (14.4):

Case (1). We have

or with (14.9) and

the following equation

In the direction of z

and in the ( x, y)-plane

We multiply Eq. (14.12) from the left by ? _z and obtain (see comments at the end)

In the case of the first two expressions on the right of Eq. (14.14) we used the relation " curl curl = grad div ? div grad", in the case of the remaining terms the relation

taking care of the ordering. The first and the last contributions on the right of Eq. (14.14) are scalar products of orthogonal vectors and therefore vanish. Hence we are left with

But

so that

Case (2). In a corresponding way we deal with

i.e.

where the left hand side is the following sum

so that in the direction of z:

and transversally

i.e.

We insert this in Eq. (14.16) and obtain

i.e. arranging the contributions in a different way,

Hence

Owing to the symmetry of the Maxwell equations (14.4) in E and B we have also

We see therefore: The transverse components of the fields can be obtained from the longitudinal components, in fact in each case from both...