Fields of a Moving Point Charge: Lienard-Wiechert

The Physical Picture

Imagine a charge moving through space. At any instant, the field you measure at a distant point depends on where the charge was when the light signal left it, not where it is now. This delay is the retarded time. The field has two distinct personalities. The first is the "near field," which points roughly toward the retarded position of the charge. It is strong nearby but weakens rapidly with distance, behaving like a static Coulomb field but distorted by the motion. The second is the "radiation field," which arises only if the charge accelerates. This component points perpendicular to the line of sight and falls off much more slowly with distance. It is this component that constitutes the electromagnetic wave carrying energy away from the source. The total field is the vector sum of these two parts.

Retarded Potentials

The starting point is the scalar potential \(V\) and vector potential \(\mathbf{A}\). For a point charge \(q\) at position \(\mathbf{w}(t_r)\) at the retarded time \(t_r\), the potentials are given by the Liénard–Wiechert expressions. The denominator \(Rc - \mathbf{R}\cdot\mathbf{v}\) accounts for the change in distance during the light travel time. If the charge moves towards the observer, the effective distance is shorter, increasing the potential.

Here \(\mathbf{R} = \mathbf{r} - \mathbf{w}(t_r)\) is the vector from the retarded position to the field point. The relation \(\mathbf{A} = \frac{\mathbf{v}}{c^2}V\) follows because the current density of a point charge is its charge density times its velocity, \(\mathbf{J} = \rho\mathbf{v}\), and both potentials are built from the same retarded Green's function. Note that all quantities on the right-hand side, including \(\mathbf{v}\) and \(\mathbf{R}\), are evaluated at the retarded time \(t_r\).

Deriving the Electric Field

To find the electric field, we differentiate the potentials: \(\mathbf{E} = -\nabla V - \frac{\partial \mathbf{A}}{\partial t}\). This calculation is notoriously tedious because the gradient \(\nabla\) and time derivative \(\frac{\partial}{\partial t}\) act on the retarded time \(t_r\), which is a function of \(\mathbf{r}\) and \(t\). The key identity is that \(\nabla t_r = -\frac{\hat{\mathbf{n}}}{c(1 - \hat{\mathbf{n}}\cdot\boldsymbol{\beta})}\) and \(\frac{\partial t_r}{\partial t} = \frac{1}{1 - \hat{\mathbf{n}}\cdot\boldsymbol{\beta}}\), where \(\boldsymbol{\beta} = \mathbf{v}/c\). After careful algebraic manipulation, the field separates into two terms. The first term involves the velocity \(\mathbf{v}\) and is proportional to \(1/R^2\). The second term involves the acceleration \(\mathbf{a}\) and is proportional to \(1/R\).

where \(\mathbf{u} \equiv c\hat{\mathbf{n}} - \mathbf{v}\). The term \((c^2 - v^2)\mathbf{u}\) is the velocity field, and \(\mathbf{R}\times(\mathbf{u}\times\mathbf{a})\) is the acceleration field. The denominator \((\mathbf{R}\cdot\mathbf{u})^3\) ensures the correct angular dependence and singularity structure. The magnetic field is simply \(\mathbf{B} = \frac{1}{c}\hat{\mathbf{n}}\times\mathbf{E}\), which guarantees that \(\mathbf{B}\) is perpendicular to both \(\mathbf{E}\) and the direction of propagation \(\hat{\mathbf{n}}\) in the radiation zone.

Velocity vs. Acceleration Fields

The distinction between the two parts of \(\mathbf{E}\) is crucial for understanding radiation. The velocity field, often called the "Coulomb" or "near" field, dominates at small distances. It falls off as \(1/R^2\), similar to the static electric field of a charge. However, it is not spherically symmetric; it is compressed in the direction of motion due to relativistic effects. This field does not carry net energy to infinity; it is bound to the charge.

The acceleration field, or "radiation" field, dominates at large distances. It falls off as \(1/R\). Because the Poynting vector \(\mathbf{S} = \frac{1}{\mu_0}\mathbf{E}\times\mathbf{B}\) is proportional to \(E^2\), the energy flux from the radiation field falls off as \(1/R^2\). This means that the total power radiated through a sphere of radius \(R\) is independent of \(R\), indicating that energy is being carried away to infinity. The velocity field's energy flux falls off as \(1/R^4\), so its contribution to the total radiated power vanishes as \(R \to \infty\).