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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ib/paper-2/15d/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2008 / ib / Paper 2 / 15D / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

The axis-regular separation of variables expansion for an axisymmetric harmonic potential is

Φ (r, x) = \sum_{n = 0}^{\infty} (A_{n} r^{n} + B_{n} r^{- n - 1}) P_{n} (x) .

(1)

The radial powers solve

r^{2} R^{''} + 2 r R^{'} - n (n + 1) R = 0

, while the regular angular solutions are Legendre polynomials. For an exterior solution tending to zero at infinity, retain only the negative radial powers. The boundary polynomial decomposes as

1 + x^{2} = \frac{4}{3} P_{0} (x) + \frac{2}{3} P_{2} (x)

. Therefore

Φ (r, x) = \frac{4 a}{3 r} + \frac{2 a ^{3}}{3 r ^{3}} P_{2} (x) = \frac{4 a}{3 r} + \frac{a ^{3}}{3 r ^{3}} (3 x^{2} - 1), r \geq a .

(2)

It has the required boundary value and decay. Uniqueness follows from the maximum principle for harmonic functions for a harmonic difference, using its zero boundary values and vanishing limit at infinity.

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