Jefimenko's equations

Jefimenko's equations are generalized formulas for the electric and magnetic field generated by a time-dependent electric charge and electric current distribution

\mathbf{E}(\mathbf{r},t)=\frac{1}{4\pi \varepsilon_{0}}\int_{\mathcal{V}}\frac{\rho(\mathbf{r}',t)}{\mathfrak{r}^{2}}\hat{\boldsymbol{\mathfrak{r}}}+ \frac{\dot{\rho}(\mathbf{r}',t)}{c\mathfrak{r}}\hat{\boldsymbol{\mathfrak{r}}}- \frac{\dot{\mathbf{J}}(\mathbf{r}',t)}{c^{2}\mathfrak{r}}d\tau'

\mathbf{B}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\int_{\mathcal{V}}\left[ \frac{\mathbf{J}(\mathbf{r}',t_{r})}{\mathfrak{r}^{2}}+ \frac{\dot{\mathbf{J}}(\mathbf{r}',t_{r})}{c\mathfrak{r}} \right]\times \hat{\boldsymbol{\mathfrak{r}}}\ d\tau'

where $t_{r}$ is the retarded time and $c$ is the speed of light. They are the solutions to Maxwell's equations in time-dependent form.

Derivation#

Given the retarded potentials

V(\mathbf{r},t)=\frac{1}{4\pi \varepsilon_{0}}\int \frac{\rho(\mathbf{r}',t_{r})}{\mathfrak{r}} \ d\tau',\qquad \mathbf{A}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\int \frac{\mathbf{J}(\mathbf{r}',t_{r})}{\mathfrak{r}} \ d\tau'

for retarded time $t_{r}$ , the electric and magnetic fields can be obtained from

\mathbf{E}=-\nabla V-\frac{ \partial \mathbf{A} }{ \partial t } ,\qquad \mathbf{B}=\nabla\times \mathbf{A}

The calculation itself is complicated by the fact that the retarded time is itself dependent on position and complicates the derivatives. The Gradient of $V$ can be found in the retarded potentials article and is

\nabla V=\frac{1}{4\pi \varepsilon_{0}}\int_{\mathcal{V}}\left[ - \frac{\dot{\rho}}{c} \frac{\hat{\boldsymbol{\mathfrak{r}}}}{\mathfrak{r}}-\rho \frac{\hat{\boldsymbol{\mathfrak{r}}}}{\mathfrak{r}^{2}} \right]d\tau'

The time derivative of $\mathbf{A}$ is

\frac{ \partial \mathbf{A} }{ \partial t } =\frac{\mu_{0}}{4\pi}\int_{\mathcal{V}}\frac{\dot{\mathbf{J}}}{\mathfrak{r}}d\tau'

Using $c^{2}=1/\mu_{0}\varepsilon_{0}$ gives us

\boxed{\mathbf{E}(\mathbf{r},t)=\frac{1}{4\pi \varepsilon_{0}}\int_{\mathcal{V}}\frac{\rho(\mathbf{r}',t)}{\mathfrak{r}^{2}}\hat{\boldsymbol{\mathfrak{r}}}+ \frac{\dot{\rho}(\mathbf{r}',t)}{c\mathfrak{r}}\hat{\boldsymbol{\mathfrak{r}}}- \frac{\dot{\mathbf{J}}(\mathbf{r}',t)}{c^{2}\mathfrak{r}}d\tau'}

This is the time-dependent form of the electric field generated by a volume charge distribution.

The magnetic field can be calculated much in the same way by finding the Curl of $\mathbf{A}$

\nabla\times \mathbf{A}=\frac{\mu_{0}}{4\pi}\int_{\mathcal{V}}\left[ \frac{1}{\mathfrak{r}}(\nabla\times \mathbf{J})-\mathbf{J}\times \nabla\left( \frac{1}{\mathfrak{r}} \right) \right]d\tau'

We need the curl of $\mathbf{J}$ . The individual component of the curl is

(\nabla \times \mathbf{J})_{x}=\frac{ \partial J_{z} }{ \partial y } -\frac{ \partial J_{y} }{ \partial z }

where we used $x$ as an example. Since $J\equiv J(\mathbf{r},t)$ we have

\frac{ \partial J_{z} }{ \partial y } =\dot{J}_{z}\frac{ \partial t_{r} }{ \partial y } =- \frac{1}{c}\dot{J}_{z}\frac{ \partial \mathfrak{r} }{ \partial y }

Similarly for the derivative of $J_{y}$ . So

(\nabla\times \mathbf{J})_{x}=\frac{1}{c}\left[ \dot{J}_{z}\frac{ \partial \mathfrak{r} }{ \partial y } -\dot{J}_{y}\frac{ \partial \mathfrak{r} }{ \partial z } \right]=\frac{1}{c}[\dot{\mathbf{J}}\times(\nabla \mathfrak{r})]_{x}=\frac{1}{c}[\dot{\mathbf{J}}\times \hat{\boldsymbol{\mathfrak{r}}}]_{x}

\nabla\times \mathbf{J}=\frac{1}{c}\dot{\mathbf{J}}\times \hat{\boldsymbol{\mathfrak{r}}}

Combine this with $\nabla(1/\mathfrak{r})=-\hat{\boldsymbol{\mathfrak{r}}}/\mathfrak{r}^{2}$ to get

\boxed{\mathbf{B}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\int_{\mathcal{V}}\left[ \frac{\mathbf{J}(\mathbf{r}',t_{r})}{\mathfrak{r}^{2}}+ \frac{\dot{\mathbf{J}}(\mathbf{r}',t_{r})}{c\mathfrak{r}} \right]\times \hat{\boldsymbol{\mathfrak{r}}}\ d\tau'}

This is the time-dependent form of the magnetic field generated by a volume current distribution. Combined with the electric field formula, these are known as Jefimenko's equations.