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Their Significant Properties

The virtue of the Hansen solutions is that they ``automatically'' work to decompose field components into parts that are mutual curls (as required by Faraday/Ampere's laws for the fields) or divergences (as required by Gauss's laws for the fields):

$\displaystyle \mbox{\boldmath$\nabla$}\cdot {\mbox{\boldmath$M$}_L}$ $\textstyle =$ $\displaystyle 0$ (13.5)
$\displaystyle \mbox{\boldmath$\nabla$}\cdot {\mbox{\boldmath$N$}_L}$ $\textstyle =$ $\displaystyle 0$ (13.6)
$\displaystyle \mbox{\boldmath$\nabla$}\cdot {\mbox{\boldmath$L$}_L}$ $\textstyle =$ $\displaystyle i k f_\ell(kr) Y_L(\hat{\mbox{\boldmath$r$}})$ (13.7)

Hence $\mbox{\boldmath$M$}_L$ and $\mbox{\boldmath$N$}_L$ are divergenceless, while the divergence of $\mbox{\boldmath$L$}_L$ is a scalar solution to the HHE! $\mbox{\boldmath$L$}_L$ is related to the scalar field and the gauge invariance of the theory in an interesting way we will develop. Also:
$\displaystyle \mbox{\boldmath$\nabla$}\times {\mbox{\boldmath$M$}_L}$ $\textstyle =$ $\displaystyle -ik \mbox{\boldmath$N$}_L$ (13.8)
$\displaystyle \mbox{\boldmath$\nabla$}\times {\mbox{\boldmath$N$}_L}$ $\textstyle =$ $\displaystyle ik \mbox{\boldmath$M$}_L$ (13.9)
$\displaystyle \mbox{\boldmath$\nabla$}\times {\mbox{\boldmath$L$}_L}$ $\textstyle =$ $\displaystyle 0$ (13.10)

which shows how $\mbox{\boldmath$M$}_L$ and $\mbox{\boldmath$N$}_L$ are now ideally suited to form the components of electric and magnetic multipole fields mutually linked by Ampere's and Faraday's law.


next up previous contents
Next: Explicit Forms Up: The Hansen Multipoles Previous: The Basic Solutions   Contents
Robert G. Brown 2013-01-04