 Solution

ID: 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/solution

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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