Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2011 / Paper 2 / 3 / ii / Solution

For the insulating wall, the image instead has the same strength

q

. With the distances defined above,

V_{N} (x, y) = \frac{q}{2 π} (lo g r + lo g r_{*}) + C, x < b .

It has the required source in the physical domain. Its normal derivative is

\partial_{x} V_{N} = \frac{q}{2 π} [\frac{x}{x ^{2} + y ^{2}} + \frac{x - 2 b}{( x - 2 b ) ^{2} + y ^{2}}] .

At

x = b

the terms are

b / (b^{2} + y^{2})

and

- b / (b^{2} + y^{2})

, so they cancel. The Neumann boundary condition leaves the additive constant

C

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