# Magnetostatics solving for constants after separation of variables spherical harmonic

1. Jan 30, 2013

### billybob5588

1. The problem statement, all variables and given/known data

I'm attempting to solve the constants that are attained after using separation of variables for a solid permeable prolate spheroidal core wrapped with N turns of wire using boundary conditions.

H$^{1}_{\delta}$ inside core

H$^{2}_{\delta}$ insulation

H$^{3}_{\delta}$ outside

2. Relevant equations

H$^{1}_{\delta}$ = K $\Sigma$$^{∞}_{n=1}$ B$_{n}$P$^{1}_{n}$($\delta$)P$_{n}$($\eta$)...(1)

H$^{2}_{\delta}$ = K $\Sigma$$^{∞}_{n=1}$ P$^{1}_{n}$($\delta$)C$_{n}$[P$_{n}$($\eta$)+D$_{n}$Q$_{n}$($\eta$)]...(2)

H$^{3}_{\delta}$ = K $\Sigma$$^{∞}_{n=1}$ E$_{n}$P$^{1}_{n}$($\delta$)Q$_{n}$($\eta$)...(3)

The conditions are:

H$^{2}_{\delta}$ = H$^{1}_{\delta}$...(4)

H$^{3}_{\delta}$ - H$^{2}_{\delta}$ = Surface current density...(5)

3. The attempt at a solution

H$^{2}_{\delta}$ = H$^{1}_{\delta}$

K $\Sigma$$^{∞}_{n=1}$ P$^{1}_{n}$($\delta$)C$_{n}$[P$_{n}$($\eta$)+D$_{n}$Q$_{n}$($\eta$)] = K $\Sigma$$^{∞}_{n=1}$ B$_{n}$P$^{1}_{n}$($\delta$)P$_{n}$($\eta$)...(6)

K $\Sigma$$^{∞}_{n=1}$ cancel out also P$^{1}_{n}$($\delta$) are orthogonal and cancel out. Therefore left with;

C$_{n}$[P$_{n}$($\eta$)+D$_{n}$Q$_{n}$($\eta$)] = B$_{n}$P$_{n}$($\eta$)

Solve for B$_{n}$ C$_{n}$ or D$_{n}$?

Where do I go from here? If i sub this back into (6) doesn't get me anywhere. Unless I am missing something obvious.

Thanks,