Electric Field of an Accelerating Electron

For a point charge (electron) in motion, the electric field at an observation event can be decomposed into two parts:

a velocity (near/Coulomb) field that scales like $1/R^2$ , and
an acceleration (radiation) field that scales like $1/R$ .

Below, we first present the low-velocity (nonrelativistic) result for intuition, and then the fully relativistic (Liénard–Wiechert) fields.

Geometry and Retarded Kinematics

Let the source (electron) be at retarded position $\mathbf{r}_s(t_r)$ with velocity $\mathbf{v}(t_r)$ and acceleration $\mathbf{a}(t_r)$ . Define:

$\mathbf{R}=\mathbf{r}-\mathbf{r}_s(t_r),\qquad R=\|\mathbf{R}\|,\qquad \mathbf{n}=\frac{\mathbf{R}}{R},\qquad \boldsymbol{\beta}=\frac{\mathbf{v}}{c},\qquad \dot{\boldsymbol{\beta}}=\frac{\mathbf{a}}{c}.$

All source quantities are to be evaluated at the retarded time $t_r$ , defined implicitly by $t_r = t - R(t_r)/c$ .

Nonrelativistic Limit ( $\|\boldsymbol{\beta}\|\ll 1$ )

Keeping terms to leading order in $\beta$ , the electric field separates cleanly into a velocity piece $\mathbf{E}_{\text{vel}}\propto 1/R^2$ and an acceleration piece $\mathbf{E}_{\text{acc}}\propto 1/R$ :

$\boxed{\; \mathbf{E}(\mathbf{r},t) = \underbrace{\frac{q}{4\pi\varepsilon_0}\,\frac{\mathbf{n}}{R^{2}}}_{\displaystyle \mathbf{E}_{\text{vel}}} \;+\; \underbrace{\frac{q}{4\pi\varepsilon_0 c^{2}}\,\frac{\mathbf{n}\times\left(\mathbf{n}\times \mathbf{a}\right)}{R}}_{\displaystyle \mathbf{E}_{\text{acc}}} \Bigg|_{t_r}\;}$

Key features:

$\mathbf{E}_{\text{vel}}$ is essentially the instantaneous Coulomb field (evaluated at the retarded time).
$\mathbf{E}_{\text{acc}}$ is transverse ( $\mathbf{n}\cdot \mathbf{E}_{\text{acc}}=0$ ) and falls off as $1/R$ : it is the radiation field.

The magnetic field follows from $\mathbf{B}=\mathbf{n}\times \mathbf{E}/c$ in this limit.

Fully Relativistic Fields (Liénard–Wiechert)

For arbitrary velocities and accelerations, the exact fields are the Liénard–Wiechert fields. They naturally split into a velocity part (near field) and an acceleration part (radiation field):

$\boxed{\; \mathbf{E}(\mathbf{r},t) = \frac{q}{4\pi\varepsilon_0} \left[ \frac{\mathbf{n}-\boldsymbol{\beta}}{\gamma^{2}\left(1-\mathbf{n}\cdot\boldsymbol{\beta}\right)^{3}R^{2}} \;+\; \frac{\mathbf{n}\times\left[(\mathbf{n}-\boldsymbol{\beta})\times \dot{\boldsymbol{\beta}}\right]}{c\left(1-\mathbf{n}\cdot\boldsymbol{\beta}\right)^{3}R} \right]_{t_r} \;}$

$\boxed{\; \mathbf{B}(\mathbf{r},t)=\left[\mathbf{n}\times \mathbf{E}(\mathbf{r},t)\right]_{t_r} \;}$

Here $\gamma=(1-\beta^{2})^{-1/2}$ . The first bracketed term is the velocity field ( $\propto R^{-2}$ ); the second is the acceleration (radiation) field ( $\propto R^{-1}$ ). Both include the characteristic beaming and retardation factor $\bigl(1-\mathbf{n}\cdot\boldsymbol{\beta}\bigr)^{-3}$ .

Checks and Limits

Low-velocity check: For $\beta\to 0$ , $\gamma\to 1$ , and keeping leading order in $\beta$ , the expressions reduce to the nonrelativistic formulas above.
Transversality of radiation: The acceleration term is transverse to $\mathbf{n}$ , as seen from the double cross product.
Beaming: At relativistic speeds, fields concentrate in a narrow cone around the instantaneous velocity direction due to the $\bigl(1-\mathbf{n}\cdot\boldsymbol{\beta}\bigr)^{-3}$ factor.

Sketch of Derivation (from Retarded Potentials)

Start with the retarded Liénard–Wiechert potentials
$\phi(\mathbf{r},t)=\frac{q}{4\pi\varepsilon_0}\,\frac{1}{\bigl(1-\mathbf{n}\cdot\boldsymbol{\beta}\bigr)R}\Big|_{t_r},\qquad \mathbf{A}(\mathbf{r},t)=\frac{\boldsymbol{\beta}}{c}\,\phi(\mathbf{r},t).$
Compute fields via
$\mathbf{E}=-\nabla \phi-\frac{\partial \mathbf{A}}{\partial t},\qquad \mathbf{B}=\nabla\times \mathbf{A},$

carefully accounting for the implicit dependence of $t_r$ in $\mathbf{n},R,\boldsymbol{\beta}$ .
Group terms by their distance scaling: $R^{-2}$ (velocity field) and $R^{-1}$ (acceleration field). The algebra yields the boxed expressions above.

Power and Radiation (for context)

The far (acceleration) field governs radiation. The instantaneous radiated power is given by the relativistic Liénard formula,
\[
P=\frac

Fields of an accelerating charge