Solution to Laplace's equation in spherical co-ordinates

[jdmo]

I have a question about the general solution to Laplace's equation in spherical co-ordinates, which takes the form of a linear combination of the spherical harmonics. In my problem, I am solving for the potential within two concentric spherical shells, each with its own conductivity. Now, since the outer volume does not contain the origin, am I right to assume that the coefficients B_{lm} do not vanish? For the inner volume, which does contain the origin, am I right in assuming that the coefficients B_{lm} do vanish?

 

[Meir Achuz]

Yes. In between the spheres, neither A nor B vanishes.

 

[charng]

I can confirm that your assumptions are correct. In the general solution to Laplace's equation in spherical co-ordinates, the coefficients B_{lm} represent the contribution of each spherical harmonic to the overall potential. Since the outer volume does not contain the origin, the coefficients B_{lm} do not vanish as there is a non-zero potential at those points. However, for the inner volume where the origin is included, the coefficients B_{lm} do vanish as the potential is assumed to be zero at the origin. This is a fundamental property of the spherical harmonics and is essential in solving problems involving concentric spherical shells with different conductivities. I hope this clarifies your question.