Jefimenko's equations

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Jefimenko's equations describe the behavior of the electric and magnetic fields in terms of the sources at retarded times. Combined with the continuity equation, Jefimenko's equations are equivalent to Maxwell's equations of electromagnetism.

[edit] Explanation

The electric field $\vec{E}$ and the magnetic field $\vec{B}$ are given in terms of the charge density $\rho\,$ and the current density $\vec{J}$ as

$\vec{E}(\vec{r},t) = \frac{1}{4\pi\epsilon_0}\int{\left(\frac{\rho(\vec{r'},t_r)\,\vec{R}}{R^3}+\frac{\vec{R}}{R^2c}\frac{\partial\rho(\vec{r'},t_r)}{\partial t} - \frac{1}{Rc^2}\frac{\partial \vec{J}(\vec{r'},t_r)}{\partial t}\right)\mathrm{d}^3\vec{r'}}$

$\vec{B}(\vec{r},t) = \frac{\mu_0}{4\pi}\int{\left(\frac{\vec{J}(\vec{r'},t_r)\times\vec{R}}{R^3}+\frac{1}{R^2c}\frac{\partial \vec{J}(\vec{r'},t_r)}{\partial t}\times\vec{R}\right)\mathrm{d}^3\vec{r'}}$