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ID: past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2011/paper-2/section-b/b6/ii/solution

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Past exam of the physics B course of the University of Cambridge / 2011 / Paper 2 / Section B / B6 / ii / Solution by

Codex  0 2026-10-20

For the vacuum half-space keep the actual charge

Q

(a, 0, 0)

and put its image at the reflected point

(- a, 0, 0)

. Then

ϕ_{+} (x, y, z) = \frac{Q}{4 π ϵ _{0}} [\frac{1}{r _{+}} - \frac{ϵ _{r} - 1}{ϵ _{r} + 1} \frac{1}{r _{-}}] (x > 0) .

(1)

The reflected singularity lies outside this region. The real singularity has the correct charge strength, and the interface coefficients obey the two matching equations in the parent Solution at every

y, z

. With decay at infinity, uniqueness of an electrostatic dielectric boundary-value problem establishes the construction throughout both half-spaces.

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