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Feb 09, 20251 min read

Describes electric and magnetic fields for retarded potentials

$E (r, t) = k \int [\frac{ρ ( r ^{'} , t _{r} )}{r ^{2}} \overset{r}{^} + \frac{ρ ˙ ( r ^{'} , t _{r} )}{cr} \overset{r}{^} - \frac{J ˙ ( r ^{'} , t _{r} )}{c ^{2} r}] d τ^{'}$

$\displaystyle \vec{E}(\vec{r},t)=k\int \left[ \frac{\rho(\vec{r}',t_{r})}{\mathscr{r}^{2}} \mathscr{\hat{r}}+ \frac{\dot{\rho}(\vec{r}',t_{r})}{c\mathscr{r}}\mathscr{\hat{r}}- \frac{\dot{\vec{J}}(\vec{r}',t_{r})}{c^{2}\mathscr{r}}\right] \, \mathrm{d}\tau'$
$\displaystyle \vec{B}(\vec{r},t)=\frac{{\mu}_{0}}{4\pi}\int \left[\frac{\vec{J}(\vec{r}',t_{r})}{\mathscr{r}^{2}}+ \frac{\dot{\vec{J}}(\vec{r}',t_{r})}{c\mathscr{r}}\right] \, \mathrm{d}\tau'$