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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/2023/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 / 2023 / Paper 2 / 2 / b / Solution by

Codex  0 2026-10-18

Treat each angular Fourier mode independently. Boundedness at the axis excludes the negative radial powers and the logarithm inside; decay at infinity excludes the positive powers, constant and logarithm outside. For the cosine mode write

Φ_{n, in} = A_{n} r^{n} cos n θ, Φ_{n, out} = B_{n} r^{- n} cos n θ .

(1)

Continuity at

R

gives

B_{n} = A_{n} R^{2 n}

. The jump of the normal derivative is then

- n B_{n} R^{- n - 1} - n A_{n} R^{n - 1} = - 2 n A_{n} R^{n - 1} = - a_{n} .

(2)

Thus

A_{n} = a_{n} R^{1 - n} / (2 n)

and

B_{n} = a_{n} R^{n + 1} / (2 n)

. The sine mode has the same relation with

b_{n}

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

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

(3)

There is no angular mean term: the given surface charge density has zero average. The interior constant is zero because its boundary value must match the vanishing exterior mean. A nonzero mean charge per unit length would instead produce a logarithmic electrostatic potential, incompatible with the specified decay at infinity.

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