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Magnetostatics solving for constants after separation of variables spherical harmonic

  1. Jan 30, 2013 #1
    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[itex]^{1}_{\delta}[/itex] inside core

    H[itex]^{2}_{\delta}[/itex] insulation

    H[itex]^{3}_{\delta}[/itex] outside

    2. Relevant equations

    H[itex]^{1}_{\delta}[/itex] = K [itex]\Sigma[/itex][itex]^{∞}_{n=1}[/itex] B[itex]_{n}[/itex]P[itex]^{1}_{n}[/itex]([itex]\delta[/itex])P[itex]_{n}[/itex]([itex]\eta[/itex])...(1)

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

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

    The conditions are:

    H[itex]^{2}_{\delta}[/itex] = H[itex]^{1}_{\delta}[/itex]...(4)

    H[itex]^{3}_{\delta}[/itex] - H[itex]^{2}_{\delta}[/itex] = Surface current density...(5)


    3. The attempt at a solution

    H[itex]^{2}_{\delta}[/itex] = H[itex]^{1}_{\delta}[/itex]

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

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

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

    Solve for B[itex]_{n}[/itex] C[itex]_{n}[/itex] or D[itex]_{n}[/itex]?

    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,
     
  2. jcsd
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