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Maxwell’s partial differential equations show how the electric $\vec{E}$ and magnetic $\vec{B}$ fields vary with position in space $\vec{r}$ and time $t$ depending on charge density $\rho$ and currents $\vec{j}$:
\[\vec{\nabla} \cdot \vec{E}(\vec{r},t) = \frac{\rho(\vec{r},t)}{\varepsilon_0}\] \[\vec{\nabla} \cdot \vec{B}(\vec{r},t) = 0\] \[\vec{\nabla} \times \vec{E}(\vec{r},t) = - \frac{\partial \vec{B}(\vec{r},t)}{\partial t}\] \[\vec{\nabla} \times \vec{B}(\vec{r},t) = \mu_0 \vec{j}(\vec{r},t) + \frac{1}{c^2}\frac{\partial \vec{E}(\vec{r},t)}{\partial t}\]Additionally, we often define the potentials $\phi$ and $\vec{A}$ such that
\[\vec{B} = \vec{\nabla} \times \vec{A}\] \[\vec{E} = - \vec{\nabla}\phi - \frac{1}{c}\frac{\partial \vec{A}}{\partial t}\]In vacuum, when the source terms $\rho$ and $\vec{j}$ vanish, the equations result in 4 wave equations
\[\nabla^2 \begin{bmatrix} \phi \\ \vec{A }\end{bmatrix} = \frac{1}{c^2} \frac{\partial }{\partial t} \begin{bmatrix} \phi \\ \vec{A }\end{bmatrix}\]This has the general solution $f(\vec{k} \cdot \vec{x} \pm \omega t) $ with $\omega = |\vec{k}| c$. Therefore, the magnetic and electric fields are
\[\vec{B} = \vec{\nabla} \times \vec{A} = \vec{k} \times \vec{A}( \vec{k} \cdot \vec{x} \pm \omega t )\] \[\vec{E} = \mp \hat{k} \times \vec{B}\]