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F16 PH333 3.3.2

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Richard Datwyler

on 23 October 2018

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Transcript of F16 PH333 3.3.2

Separation of Variables
Spherical Coordinates

3.3.2 Spherical coordinates
Spherical coordinates
We only do polar and radial components and leave azimuthal for graduate EM with Jackson.
Prove this
This one is harder.
But the solution is known
Legendre polynomials
Rodrigues formula
compute 3 or 4
That's it, now we have done separation of variables for spherical coordinates
l
But to deal with the A and the B , and to see which of the Legendre terms last.
This we still need to deal with, but some questions first
Potential on surface of hollow sphere R. Find V inside.
B goes away (problem at origin)
l
At surface of sphere
Fourier trick
Not super exciting, let us give Vo some substance.
only l = 0 and 1 survive and by inspection
(note we didn't need to 'do' the integral)

Find the potential inside and outside a sphere of radius R with the potential on surface as
inside (B is gone)
outside (A is gone)
match!!
"I don't even know what I should be asking"
"Can you explain why the polar angle is orthogonal? Can you elaborate on the idea of "inner products" pertaining to orthogonality? "
"all knowledge of the Fourier's trick that is used here and wasn't able to follow it in class either while you were showing it."
"Where did the l(l+1) come from? Why is the polar angle orthogonal? Can we go over Legendre polynomials. What words of hope do you have for me as I struggle through this class? "
"Im having trouble seeing the point of the Legendre polynomials and the Rodriguez formula"
"It asks which of the spherical solutions is orthogonal and the answer is the polar one. "
l
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