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.

Explanation

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

$\backslash mathbf\{E\}(\backslash mathbf\{r\},t)\; =\; \backslash frac\{1\}\{4\backslash pi\backslash epsilon\_0\}\backslash int\{d^3r\text{'}\backslash left(\backslash frac\{\backslash rho(\backslash mathbf\{r\text{'}\},t\_r)\backslash ,\backslash mathbf\{R\}\}\{R^3\}+\backslash frac\{\backslash mathbf\{R\}\}\{R^2c\}\backslash frac\{\backslash partial\backslash rho(\backslash mathbf\{r\text{'}\},t\_r)\}\{\backslash partial\; t\}\; -\; \backslash frac\{1\}\{Rc^2\}\backslash frac\{\backslash partial\; \backslash mathbf\{J\}(\backslash mathbf\{r\text{'}\},t\_r)\}\{\backslash partial\; t\}\backslash right)\}$

$\backslash mathbf\{B\}(\backslash mathbf\{r\},t)\; =\; \backslash frac\{\backslash mu\_0\}\{4\backslash pi\}\backslash int\{d^3r\text{'}\backslash left(\backslash frac\{\backslash mathbf\{J\}(\backslash mathbf\{r\text{'}\},t\_r)\backslash times\backslash mathbf\{R\}\}\{R^3\}+\backslash frac\{1\}\{R^2c\}\backslash frac\{\backslash partial\; \backslash mathbf\{J\}(\backslash mathbf\{r\text{'}\},t\_r)\}\{\backslash partial\; t\}\backslash times\backslash mathbf\{R\}\backslash right)\}$

where $\backslash mathbf\{R\; =\; r\; -\; r\text{'}\}$, and $t\_r\; =\; t\; -\; R/c\; \backslash ,$ (the retarded time).

Electrodynamics | Electromagnetism

Ecuaciones de Jefimenko