- #1

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I know how to use Gauss' Law to find the electric field in- and outside a homogeneously charged sphere. But say I wanted to compute this directly via integration, how would I evaluate the integral

$$\vec{E}(\vec{r})=\frac{1}{4\pi\varepsilon_0}\int\rho(\vec{r}')\frac{\vec{r}-\vec{r}'}{|\vec{r}-\vec{r}'|^3} dx' dy' dz'$$

if ##\vec{r}## is inside the sphere? This doesn't seem to converge for ##\vec{r}'\rightarrow\vec{r}## and ##\rho(\vec{r}')\neq\ 0##.