Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 2 / 15 / b

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- Solution b

Solution

For a radial electric potential,

E = - V^{'} (r) r / r

. Comparing radial coefficients gives

V_{in}^{'} (r) = \frac{k r}{a ^{3}}, V_{out}^{'} (r) = \frac{k}{r ^{2}} .

Integrating these one-variable derivatives gives the general local expressions

V_{in} (r) = \frac{k r ^{2}}{2 a ^{3}} + C_{in}, V_{out} (r) = - \frac{k}{r} + C_{out} .

The two constants are initially independent because these integrations were performed in separate radial regions.

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