 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ib/paper-2/16d/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Successive reflections in the two grounded planes require four charges:

position (a, b, 0) (- a, b, 0) (a, - b, 0) (- a, - b, 0) charge q - q - q q .

Their total charge and electric dipole moment vanish. Applying the second-order expansion from part (a), the terms proportional to

a^{2} x^{2}

b^{2} y^{2}

, and

a^{2} + b^{2}

also cancel, while the mixed terms add. Thus

Φ (r) \sim \frac{q}{4 π ϵ _{0}} \frac{12 ab x y}{r ^{5}} = \frac{3 q ab x y}{π ϵ _{0} r ^{5}}, r ≫ a, b .

This is an electric quadrupole potential.

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