As we have now seen repeatedly from Chapter J6 on, in a source free
region of space, harmonic electromagnetic fields are divergenceless and
have curls given by:

(14.19) | |||

(14.20) |

By massaging these a little bit (recall and for we can easily show that both and must be divergenceless solutions to the HHE:

(14.21) |

If we know a solution to this equation for we can obtain from its curl from the equation above:

(14.22) |

and vice versa. However, this is annoying to treat directly, because of the vector charactor of and which complicate the description (as we have seen - transverse electric fields are related to magnetic multipoles and vice versa). Let's eliminate it.

By considering the action of the Laplacian on the scalar product of with a well-behaved vector field ,

(14.23) |

and using the divergenceless of and , we see that the scalars and

(14.24) | |||

(14.25) |

We already know how to write a general solution to either of these equations in terms of the spherical bessel, neumann, and hankel functions times spherical harmonics.

Recall, that when we played around with multipole fields, I kept emphasizing
that electric n-pole fields were transverse magnetic and vice versa? Well,
transverse electric fields have
by definition,
right? So now we *define* a **magnetic multipole field of order L** by

(14.26) | |||

(14.27) |

Similarly, a

(14.28) | |||

(14.29) |

In these two definitions, and are arbitrary linear combinations of spherical bessel functions

Now, a little trickery. Using the curl equation for we get:

(14.30) |

so that is a scalar solution to the HHE for magnetic multipolar fields. Ditto for in the case of electric multipolar fields. Thus,

(14.31) |

etc. for .

Now we get *really* clever. Remember that
.
Also,
. We have arranged things just so that if
we write:

(14.32) | |||

(14.33) |

we exactly reconstruct the solutions above. Neato! This gives us a completely general TE, MM EMF. A TM, EM EMF follows similarly with and (and a minus sign in the second equation).

This is good news and bad news. The good news is that this is a hell of a lot
simpler than screwing around with symmetric and antisymmetric vector
decompositions and integrations by parts *ad nauseam*. The radial part of
the solutions is straightforward, and the angular part is written in a concise
notation. The bad news is we have never *seen* that notation, good or
bad, ever before. We have two choices. Either we can laboriously crank out
the operator products and curls for each problem as we need to (which is
really just as bad as what we have been doing) or we have to work out the
algebra of these new objects once and for all so we can plug and chug out the
most difficult of answers with comparative ease.

Guess which one we're about to do.