Mathematical Methods in the Physical Sciences · Problem 7.6

The key is recognizing that steady-state heat flow inside the sphere obeys Laplace's equation, and the boundary condition $T(1, \theta) = \pi/2 - \theta$ tells you exactly what temperature profile to match. Expand the boundary condition as a Legendre polynomial series, then use the fact that each $P_n(\cos\theta)$ mode decays radially as $r^n$ inside the sphere. Your job is to find which Legendre polynomials appear in $\pi/2 - \theta$ and read off the coefficients that determine the full 3D solution.

The key is recognizing that steady-state heat flow inside the sphere obeys Laplace's equation, and the boundary condition $T(1, \theta) = \pi/2 - \theta$ tells you exactly what temperature profile to match. Expand the boundary condition as a Legendre polynomial series, then use the fact that each $P_n(\cos\theta)$ mode decays radially as $r^n$ inside the sphere. Your job is to find which Legendre polynomials appear in $\pi/2 - \theta$ and read off the coefficients that determine the full 3D solution.