 Solution

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

For the dielectric half-space the two formal sources coincide at

(a, 0, 0)

, outside the region being represented. Their sum is

Q + Q^{'} = 2 ϵ_{r} Q / (1 + ϵ_{r})

in the dielectric Coulomb kernel, hence

ϕ_{-} (x, y, z) = \frac{2 Q}{4 π ϵ _{0} ( 1 + ϵ _{r} ) r _{+}} (x < 0) .

There is no charge singularity inside this half-space. The expression satisfies Laplace's equation there, decays at infinity, and matches the vacuum solution's tangential electric field and normal electric displacement. The coincident image is a mathematical representation, not an extra physical charge in the vacuum region.

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