Solving Constants for Prolate Spheroidal Core Wrapped with N Turns of Wire

[billybob5588]

Homework Statement

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

Homework 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)

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,

 

[mark726]

your first step would be to carefully review the equations and boundary conditions given and make sure that they are correct. You can also do some background research on prolate spheroidal coordinates and the use of separation of variables in solving this type of problem.

Once you have a solid understanding of the problem and its mathematical formulation, you can start by substituting the boundary conditions (4) and (5) into the equations (1), (2), and (3). This will help you eliminate some of the unknown constants and simplify the equations.

Next, you can use the orthogonality of the prolate spheroidal functions to solve for the remaining unknown constants. This may involve some algebraic manipulation and possibly using some known identities for the prolate spheroidal functions.

Once you have solved for all the constants, you can then substitute them back into the original equations (1), (2), and (3) to obtain the final solution for H^{1}_{\delta}, H^{2}_{\delta}, and H^{3}_{\delta}.

It's important to keep in mind that solving this type of problem can be complex and may require multiple iterations and careful checking of the solution. It's also helpful to consult with colleagues or a mentor for guidance and to double-check your work. Good luck!