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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-2/2/b/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 2 / 2 / b / Solution by

Codex  0 2026-10-18

Boundedness at the origin allows only

r^{n}

for nonconstant interior modes; decay at infinity allows only

r^{- n}

outside. Write the common surface potential in its Fourier series as

\sum_{n \geq 1} v_{n} (θ)

, where

v_{n} = A_{n} cos n θ + B_{n} sin n θ

. Continuity then makes the corresponding interior and exterior factors

(r / R)^{n}

and

(R / r)^{n}

.

Their outward radial derivatives at the surface are

n v_{n} / R

inside and

- n v_{n} / R

outside. Thus the jump is

- 2 n v_{n} / R

. Equating it to minus the prescribed surface-charge mode gives

v_{n} (θ) = \frac{R}{2 n} (a_{n} cos n θ + b_{n} sin n θ) .

Therefore the potential of a charged cylindrical shell with angular Fourier data is

Φ (r, θ) = n = 1 \sum \infty \frac{R}{2 n} {(r / R)^{n}, (R / r)^{n}, 0 \leq r \leq R, r \geq R (a_{n} cos n θ + b_{n} sin n θ) .

Either branch gives the same value at

R

. The mean angular mode is zero: outside, neither a constant nor a logarithm can survive the condition at infinity; continuity then forces the interior constant to zero. Consequently

Φ (0) = 0

.

The absence of a mean term in the surface charge density is essential. A nonzero mean would require a logarithmic exterior potential and would be incompatible with decay at infinity in this two-dimensional reduction. The normalization follows the stated derivative jump, with no additional permittivity factor introduced.

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