Important equations in Electrodynamics

First, we start with Maxwell’s equations

\[\nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _0}}\]

\[\nabla \cdot \mathbf {B} = 0 \]

\[\nabla \times \mathbf {E} =-\partial_t \mathbf{B} \]

\[\nabla \times \mathbf {B} =\mu _{0}\left(\mathbf {J} +\varepsilon _{0}\partial_t \mathbf {E} \right) \]

We then define the four-gradient as $ \partial ^{\nu }={\frac {\partial }{\partial x_{\nu }}}=\left({\frac {1}{c}}{\frac {\partial }{\partial t}},-\mathbf {\nabla } \right)$, and the d’Alembertian as $\partial ^{2}={\frac {1}{c^{2}}}{\partial^2 \over \partial^2 t}-\nabla ^{2}$.