5.6 Liénard-Wiechert potentials

Fields of moving point charges

When a charge moves, the electromagnetic fields it produces don't update instantaneously everywhere in space. Information about the charge's position travels outward at the speed of light, so the fields at any observation point depend on where the charge was at an earlier moment. This is the core idea behind the Liénard-Wiechert potentials, and it connects electrodynamics directly to special relativity.

Charge in arbitrary motion

Consider a point charge $q$ moving along an arbitrary trajectory described by the position vector $\vec{w}(t)$. Its velocity $\vec{v}(t) = d\vec{w}/dt$ and acceleration $\vec{a}(t) = d\vec{v}/dt$ follow from differentiating the trajectory. The electromagnetic fields at a field point $\vec{r}$ depend not on the charge's current position, but on its position, velocity, and acceleration at the retarded time.

Retarded time

The retarded time $t_r$ is the moment when the signal that arrives at the observation point $\vec{r}$ at time $t$ was actually emitted by the charge. Because that signal travels at speed $c$, the retarded time satisfies:

$t_r = t - \frac{|\vec{r} - \vec{w}(t_r)|}{c}$

This is an implicit equation for $t_r$: the charge's position $\vec{w}(t_r)$ itself depends on $t_r$, so you generally can't solve it in closed form. For a given field point and time, there is exactly one retarded time (assuming the charge moves slower than light).

Retarded position

The retarded position $\vec{w}_r \equiv \vec{w}(t_r)$ is simply where the charge was at the retarded time. All quantities in the Liénard-Wiechert formulas are evaluated at this position. Define the separation vector and distance:

• $\vec{R} = \vec{r} - \vec{w}_r$ (points from retarded position to field point)
• $R = |\vec{R}|$
• $\hat{n} = \vec{R}/R$ (unit vector from retarded position to field point)

Liénard-Wiechert potentials

The Liénard-Wiechert potentials give the scalar and vector potentials produced by a point charge in arbitrary motion. They were derived independently by Alfred-Marie Liénard (1898) and Emil Wiechert (1900). From these potentials you can obtain the full electromagnetic fields without solving Maxwell's equations from scratch.

Scalar potential

$\phi(\vec{r},t) = \frac{q}{4\pi\epsilon_0} \frac{1}{R(1 - \hat{n} \cdot \vec{\beta})}\bigg|_{t_r}$

Here $\vec{\beta} = \vec{v}_r/c$ is the charge's velocity at the retarded time normalized by $c$. The factor $(1 - \hat{n} \cdot \vec{\beta})$ in the denominator is critical. It accounts for the fact that a charge moving toward the observer "compresses" the signal, boosting the potential, while a charge moving away "stretches" it. For a stationary charge ($\vec{\beta} = 0$), this reduces to the familiar Coulomb potential.

Vector potential

$\vec{A}(\vec{r},t) = \frac{\mu_0 q}{4\pi} \frac{\vec{v}_r}{R(1 - \hat{n} \cdot \vec{\beta})}\bigg|_{t_r}$

Notice that $\vec{A} = \vec{v}_r \phi / c^2$, so the vector potential is simply the scalar potential scaled by the retarded velocity (divided by $c^2$). Both potentials share the same denominator $R(1 - \hat{n}\cdot\vec{\beta})$, sometimes written as $\kappa R$ where $\kappa = 1 - \hat{n}\cdot\vec{\beta}$.

Fields in terms of potentials

The electric and magnetic fields follow from the standard relations:

$\vec{E}(\vec{r},t) = -\nabla\phi - \frac{\partial\vec{A}}{\partial t}$

$\vec{B}(\vec{r},t) = \nabla \times \vec{A}$

Taking these derivatives is nontrivial because $t_r$ itself depends on $\vec{r}$ and $t$. The chain rule introduces extra factors of $(1 - \hat{n}\cdot\vec{\beta})$ throughout the calculation, which is where the complexity of the Liénard-Wiechert fields comes from.

Derivation of Liénard-Wiechert potentials

The derivation starts from the retarded solutions to the inhomogeneous wave equations for $\phi$ and $\vec{A}$, then specializes to a point-charge source. The key subtlety is correctly handling the delta-function integration over a moving source.

Retarded Green's function

The retarded Green's function for the wave equation in free space is:

$G(\vec{r},t;\vec{r}',t') = \frac{\delta\!\left(t' - [t - |\vec{r} - \vec{r}'|/c]\right)}{4\pi|\vec{r} - \vec{r}'|}$

This enforces causality: the field at $(\vec{r}, t)$ depends only on the source at the earlier time $t' = t - |\vec{r} - \vec{r}'|/c$. The delta function picks out exactly the retarded time.

Solution of wave equation

In the Lorenz gauge, the potentials satisfy:

$\nabla^2\phi - \frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2} = -\frac{\rho}{\epsilon_0}$

$\nabla^2\vec{A} - \frac{1}{c^2}\frac{\partial^2\vec{A}}{\partial t^2} = -\mu_0\vec{J}$

Using the retarded Green's function, the general solutions are:

$\phi(\vec{r},t) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\vec{r}',t_r)}{|\vec{r} - \vec{r}'|} \, d^3r'$

$\vec{A}(\vec{r},t) = \frac{\mu_0}{4\pi} \int \frac{\vec{J}(\vec{r}',t_r)}{|\vec{r} - \vec{r}'|} \, d^3r'$

For a point charge, $\rho(\vec{r}',t') = q\,\delta^3(\vec{r}' - \vec{w}(t'))$ and $\vec{J} = q\vec{v}\,\delta^3(\vec{r}' - \vec{w}(t'))$. When you perform the spatial integration, the delta function over the moving trajectory introduces a Jacobian factor of $1/(1 - \hat{n}\cdot\vec{\beta})$. This is the origin of the $(1 - \hat{n}\cdot\vec{\beta})$ denominator in the Liénard-Wiechert potentials.

Lorenz gauge condition

The Lorenz gauge condition:

$\nabla \cdot \vec{A} + \frac{1}{c^2}\frac{\partial\phi}{\partial t} = 0$

is what decouples the wave equations for $\phi$ and $\vec{A}$. The Liénard-Wiechert potentials automatically satisfy this condition, which you can verify by direct (if tedious) substitution.

Properties of Liénard-Wiechert fields

Once you differentiate the potentials, the resulting fields split naturally into two physically distinct pieces: a velocity field and an acceleration (radiation) field.

Electric field of moving charge

The full electric field is:

$\vec{E}(\vec{r},t) = \frac{q}{4\pi\epsilon_0} \left[ \frac{(\hat{n} - \vec{\beta})(1 - \beta^2)}{\kappa^3 R^2} + \frac{\hat{n} \times [(\hat{n} - \vec{\beta}) \times \dot{\vec{\beta}}]}{c\,\kappa^3 R} \right]_{t_r}$

where $\kappa = 1 - \hat{n}\cdot\vec{\beta}$ and $\dot{\vec{\beta}} = \vec{a}/c$.

• First term (velocity field): Falls off as $1/R^2$. Present even for unaccelerated charges. Does not radiate.
• Second term (acceleration/radiation field): Falls off as $1/R$. Only present when the charge accelerates. This is the term responsible for radiation.

The $1/R$ vs $1/R^2$ distinction matters because radiated power goes as $|\vec{E}|^2 R^2$. The velocity field contribution vanishes at large $R$, while the radiation field gives a finite power flow through any distant sphere.

Magnetic field of moving charge

The magnetic field has a simple relationship to the electric field:

$\vec{B}(\vec{r},t) = \frac{1}{c}\,\hat{n} \times \vec{E}(\vec{r},t)$

This means $\vec{B}$ is always perpendicular to both $\hat{n}$ and $\vec{E}$, and its magnitude is $|\vec{E}|/c$ (for the radiation part, at least). This is the same relationship that holds for plane electromagnetic waves.

Field of charge in uniform motion

For a charge moving at constant velocity ($\vec{a} = 0$), the acceleration field vanishes. Only the velocity field survives:

$\vec{E} = \frac{q}{4\pi\epsilon_0} \frac{(\hat{n} - \vec{\beta})(1 - \beta^2)}{\kappa^3 R^2}\bigg|_{t_r}$

The field still points radially outward from the present position of the charge (not the retarded position), which you can show by rewriting the expression in terms of the current position. The field is compressed in the transverse direction by a factor of $\gamma^2$ relative to the Coulomb field, so a fast-moving charge has stronger fields perpendicular to its motion and weaker fields along its direction of travel.

Field of accelerating charge

When $\vec{a} \neq 0$, both terms contribute. The radiation field dominates at large distances, and the emitted radiation carries energy irreversibly away from the charge. The angular pattern and polarization of the radiation depend on the relationship between $\vec{v}$ and $\vec{a}$, as discussed below.

Relativistic transformations

The four-potential $A^\mu = (\phi/c, \vec{A})$ transforms as a four-vector under Lorentz transformations. This provides an elegant alternative route to the Liénard-Wiechert potentials: start from the Coulomb potential in the charge's instantaneous rest frame, then boost.

Fields in different reference frames

If $\vec{E}$ and $\vec{B}$ are the fields in one inertial frame, and a second frame moves with velocity $\vec{v}$ relative to the first, the fields in the primed frame are:

$E'_\parallel = E_\parallel \qquad B'_\parallel = B_\parallel$

$\vec{E}'_\perp = \gamma(\vec{E}_\perp + \vec{v} \times \vec{B}_\perp)$

$\vec{B}'_\perp = \gamma\!\left(\vec{B}_\perp - \frac{\vec{v} \times \vec{E}_\perp}{c^2}\right)$

Here $\parallel$ and $\perp$ denote components parallel and perpendicular to $\vec{v}$, and $\gamma = 1/\sqrt{1 - v^2/c^2}$. The parallel components are unchanged; the perpendicular components mix $\vec{E}$ and $\vec{B}$.

Lorentz transformation of potentials

The four-potential transforms as:

$\phi' = \gamma(\phi - \vec{v}\cdot\vec{A})$

$\vec{A}' = \vec{A} + (\gamma - 1)\frac{(\vec{v}\cdot\vec{A})}{v^2}\vec{v} - \frac{\gamma\phi}{c^2}\vec{v}$

You can derive the Liénard-Wiechert potentials for a uniformly moving charge by starting with $\phi = q/(4\pi\epsilon_0 r)$ and $\vec{A} = 0$ in the rest frame, then applying this transformation.

Invariance of Maxwell's equations

Maxwell's equations take the same form in every inertial frame. This Lorentz covariance was historically one of the key motivations for special relativity. The field tensor $F^{\mu\nu}$, which packages $\vec{E}$ and $\vec{B}$ into a single antisymmetric tensor, transforms as a rank-2 tensor, guaranteeing this invariance.

Radiative and velocity fields

The physical distinction between velocity fields and radiative fields is one of the most important results from the Liénard-Wiechert analysis.

Field of slowly moving charge

When $v \ll c$ (so $\beta \approx 0$ and $\gamma \approx 1$), the velocity field reduces to the Coulomb field:

$\vec{E} \approx \frac{q}{4\pi\epsilon_0}\frac{\hat{n}}{R^2}$

The magnetic field in this limit is given by the Biot-Savart expression, proportional to $\vec{v}/c^2$. The radiation field from acceleration still exists but is typically very weak at low speeds.

Velocity fields vs radiative fields

The velocity field is sometimes called the "generalized Coulomb field." It's the field the charge would have if it continued at constant velocity forever. The radiation field exists only when the charge accelerates and represents genuinely new electromagnetic energy being created at the expense of the charge's kinetic energy.

Angular distribution of radiation

The radiation pattern depends on the geometry of $\vec{v}$ and $\vec{a}$:

• Linear acceleration ($\vec{a} \parallel \vec{v}$): In the non-relativistic limit, radiation peaks perpendicular to the motion with a $\sin^2\theta$ pattern (where $\theta$ is measured from the acceleration direction). At relativistic speeds, the pattern is swept forward into a narrow cone of half-angle $\sim 1/\gamma$.
• Circular motion ($\vec{a} \perp \vec{v}$): The radiation is concentrated in the plane of the orbit. At relativistic speeds, it again beams sharply forward into a cone of angular width $\sim 1/\gamma$, producing the characteristic synchrotron radiation pattern.

The relativistic beaming effect (concentration into a $1/\gamma$ cone) is a direct consequence of the $\kappa^3 = (1 - \hat{n}\cdot\vec{\beta})^3$ factor in the denominator of the radiation field.

Radiated power and Larmor's formula

For a non-relativistic accelerating charge, the total radiated power is given by the Larmor formula:

$P = \frac{q^2 a^2}{6\pi\epsilon_0 c^3}$

The relativistic generalization (sometimes called the Liénard formula) is:

$P = \frac{q^2 \gamma^6}{6\pi\epsilon_0 c^3}\left(a^2 - \frac{|\vec{v} \times \vec{a}|^2}{c^2}\right)$

The $\gamma^6$ factor means that radiated power increases dramatically at relativistic speeds. For circular motion ($\vec{v} \perp \vec{a}$), this gives $P \propto \gamma^4 a^2$, which is why synchrotron radiation losses are a major concern in circular particle accelerators.

Applications and examples

Synchrotron radiation

Synchrotron radiation is emitted by relativistic charged particles (typically electrons) following curved trajectories in magnetic fields. Because $\vec{a} \perp \vec{v}$, the radiated power scales as $\gamma^4$, making it extremely intense at high energies. The radiation spans a broad spectrum from infrared through hard X-rays and is highly collimated in the forward direction. Synchrotron light sources are used extensively in materials science, structural biology (protein crystallography), and medical imaging.

Bremsstrahlung

Bremsstrahlung ("braking radiation") occurs when a charged particle, usually an electron, is deflected and decelerated by the Coulomb field of a nucleus. The sudden change in velocity produces a broad-spectrum radiation pulse. Bremsstrahlung is the dominant mechanism for X-ray production in medical X-ray tubes and is also the primary energy-loss mechanism for high-energy electrons passing through matter.