Liénard–Wiechert potential

The potentials of a point charge $q$ arewith every source quantity evaluated at the retarded time.

The Liénard-Wiechert field is the electric field and magnetic field obtained from the retarded potentials of a moving point charge. In the nonrelativistic radiation zone, the electric radiation field is ${\mathbf{E}}_{\mathrm{rad}}=q\hat{\mathbf{n}}\times (\hat{\mathbf{n}}\times \mathbf{a})/(4\pi {ϵ}_0{c}^{2}R)$ at retarded time. Its outgoing Poynting vector integrates to the Larmor formula. Coulomb and induction fields must also be retained when a near-field calculation is required.

Let $\mathbf{R}$ join the instantaneous position of charge q to the field point, with constant $\mathbf{\beta }=\mathbf{v}/c$. Transforming its rest-frame Coulomb field givesThe electric field is radial from the current position, reduced along the velocity and enhanced transversely. This present-position form is compatible with the causal retarded solution specifically because motion is uniform. There is no acceleration-dependent radiation field, although the moving charge has a magnetic field and transports electromagnetic energy.