Mathematical Methods in the Physical Sciences · Problem 7.3

This is a boundary-value problem in spherical coordinates where you're solving Laplace's equation in the radial direction. The key insight is that the steady-state temperature must be finite at the center ($r=0$), which rules out negative powers of $r$—so your solution is a sum of terms $r^n P_n(\cos\theta)$ where the $P_n$ are Legendre polynomials. Expand the boundary condition in Legendre polynomials, match coefficients at $r=1$, and you'll read off each radial coefficient directly.

This is a boundary-value problem in spherical coordinates where you're solving Laplace's equation in the radial direction. The key insight is that the steady-state temperature must be finite at the center ($r=0$), which rules out negative powers of $r$—so your solution is a sum of terms $r^n P_n(\cos\theta)$ where the $P_n$ are Legendre polynomials. Expand the boundary condition in Legendre polynomials, match coefficients at $r=1$, and you'll read off each radial coefficient directly.