Introduction to Electrodynamics · Problem 10.13

Potentials and Fields — David J. Griffiths

Physics · jefimenko-constant-current

Problem

Suppose

\mathbf{J}(\mathbf{r})

is constant in time, so (Prob. 7.60)

\rho(\mathbf{r},t) = \rho(\mathbf{r},0) + \dot{\rho}(\mathbf{r},0)t

. Show that

\mathbf{E}(\mathbf{r},t) = \frac{1}{4\pi\epsilon_0}\int \frac{\rho(\mathbf{r}',t)}{i^2}\hat{\boldsymbol{i}}\,d\tau';

that is, Coulomb's law holds, with the charge density evaluated at the non-retarded time.