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I'm trying to derive the vector field $\vec{E} = \frac{1}{4\pi\epsilon_0}\frac{q\vec{r}}{r^3}$ surrounding a point charge, starting with $\oint_S \vec{E} \cdot \mathrm{d}\vec{A}$. My uneducated guess would be to get the magnitude of the electric field from gauss' law, then integrate to get the scalar potential, before taking the gradient to get the vector field. Is there a more elegant way to achieve this?

Thanks.

James