Introduction to Electrodynamics · Problem 2.46

Electrostatics — David J. Griffiths

Physics · charge-density-from-electric-field

Problem

If the electric field in some region is given (in spherical coordinates) by the expression

\mathbf{E}(\mathbf{r}) = \frac{k}{r}\left[3\,\hat{r} + 2\sin\theta\cos\theta\sin\phi\,\hat{\theta} + \sin\theta\cos\phi\,\hat{\phi}\right],

for some constant

k

, what is the charge density?

[Answer: 3k\epsilon_0(1+\cos 2\theta \sin\phi)/r^2]