Explicit Forms

The beauty of the definitions above is that they permit us to do algebra that initially skips the following fully expanded forms in terms of the vector spherical harmonics. However ultimately one has to do computations, of course - there are no free lunches. The following results come from actually working out the gradients, divergences, and curls in the definitions:

$\displaystyle \mbox{\boldmath$M$}_L$	$\textstyle =$	$\displaystyle f_\ell(kr) \mbox{\boldmath$Y$}_{\ell \ell}^{m}$	(13.11)
$\displaystyle \mbox{\boldmath$N$}_L$	$\textstyle =$	$\displaystyle \sqrt{\frac{\ell+1}{2 \ell +1}} f_{\ell - 1}(kr) \mbox{\boldmath$... ...\frac{\ell}{2 \ell + 1}} f_{\ell + 1}(kr) \mbox{\boldmath$Y$}_{\ell,\ell+1}^{m}$	(13.12)
$\displaystyle \mbox{\boldmath$L$}_L$	$\textstyle =$	$\displaystyle \sqrt{\frac{\ell}{2 \ell + 1}} f_{\ell - 1}(kr) \mbox{\boldmath$Y... ...frac{\ell+1}{2 \ell +1}} f_{\ell + 1}(kr) \mbox{\boldmath$Y$}_{\ell,\ell+1}^{m}$	(13.13)

$\displaystyle \mbox{\boldmath$M$}_L$	$\textstyle =$	$\displaystyle f_\ell(kr) \mbox{\boldmath$Y$}_{\ell \ell}^{m}$	(13.14)
$\displaystyle \mbox{\boldmath$N$}_L$	$\textstyle =$	$\displaystyle \frac{1}{kr} \bigg\{ \frac{d }{d(kr)} (kr f_\ell) \big(i\hat{\m... ...}^{m} \big) - \hat{\mbox{\boldmath$r$}}\sqrt{\ell(\ell + 1)} f_\ell Y_L \bigg\}$	(13.15)
$\displaystyle \mbox{\boldmath$L$}_L$	$\textstyle =$	$\displaystyle \sqrt{\ell(\ell + 1)} \frac{1}{kr} (i \hat{\mbox{\boldmath$r$}} \... ...ell}^{m}) - \hat{\mbox{\boldmath$r$}}\bigg[\frac{d }{d(kr)} f_\ell \bigg] Y_L$	(13.16)

As we will see, these relations allow us to construct the completely general solution to the EM field equations in a way that is intuitive, reasonable, and mathematically and numerically tractible. In other words, we're (mostly) done with the grunt work and can begin to reap the rewards.

What grunt work remains, you might ask? Well, there are a slew of identities and evaluations and relations developed from the definitions of the spherical harmonics themselves, the spherical bessel/neumann/hankel functions themselves, and the vector spherical harmonics and Hansen solutions that can be worked out and assembled in a table of sorts to simplify the actual process of doing algebra or computations using them.

Such a table is presented at the end of this chapter, and proving relations on that table constitute most of the homework related to the chapter, since once this work is done doing actual computations for specific charge/current densities is reduced to quadratures (another way of saying ``expressible as a bunch of definite integrals'' that can either be done analytically if they are relatively simple or numerically if not).

Those rewards are most readily apparent when we construct the vector Green's function for the vector IHE.