Uniform electromagnetic fields

Uniform electric fields

For a time-independent uniform electric field, i.e. an electric field whose components are constant throughout space and time: \[\begin{equation} \require{physics} \vb{E} = (E_x, E_y, E_z) \thinspace , \end{equation}\] the scalar potential can be written as \[\begin{equation} \phi(\vb{r}) = - \vb{E} \vdot \vb{r} \end{equation}\] and the vector potential is zero.

Uniform magnetic fields

For a time-independent uniform magnetic field, the scalar potential is zero and the vector potential can be written in the following, form: \[\begin{equation} \vb{A}(\vb{r}) = \frac{1}{2} \vb{B} \cross \vb{r} \thinspace . \end{equation}\] This expression automatically satisfies Coulomb gauge.