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

For a radial electric potential, $\mathbf{E}=-V^{\prime }(r)\mathbf{r}/r$. Comparing radial coefficients givesIntegrating these one-variable derivatives gives the general local expressionsThe two constants are initially independent because these integrations were performed in separate radial regions.