5B (Methods)

For an axisymmetric separated solution of the Laplace equation whose angular factor is the Legendre polynomial

P_{n} (cos θ)

r^{n}

and

r^{- n - 1}

. Thus

Φ (r, θ, ϕ) = (A r^{n} + B r^{- n - 1}) P_{n} (cos θ)

A, B

Write

x = cos θ

. Since

3 cos (2 θ) = 6 x^{2} - 3 = 4 P_{2} (x) - P_{0} (x)

, only the spherical harmonics of degrees zero and two occur. The degree-zero radial coefficient satisfying its two Dirichlet boundary conditions is

1 - 2/ r

. For degree two, write

C r^{2} + D r^{- 3}

; then

C + D = 4, 4 C + \frac{D}{8} = 0,

C = - 4/31

and

D = 128/31

Φ (r, θ) = 1 - \frac{2}{r} + \frac{4}{31} (\frac{32}{r ^{3}} - r^{2}) P_{2} (cos θ) .

Direct substitution at

r = 1, 2

verifies both boundary values.

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