6.5 Retarded potentials

Retarded time in electromagnetism

Retarded potentials solve a fundamental problem: electromagnetic fields don't propagate instantaneously. When a charge wiggles at some source point, the effect isn't felt at a distant observation point until later, because the information travels at the speed of light. Retarded potentials build this time delay directly into the expressions for the scalar and vector potentials.

This matters most for dynamic situations: radiating antennas, relativistic particles, or any source whose charge and current distributions change in time. Without retarded potentials, you'd be stuck with the static Coulomb and Biot-Savart expressions, which silently assume infinite propagation speed.

The retarded time is defined as

$t_r = t - \frac{|\vec{r} - \vec{r}'|}{c}$

where $\vec{r}$ is the observation (field) point, $\vec{r}'$ is the source point, and $c$ is the speed of light. The quantity $|\vec{r} - \vec{r}'|/c$ is the light-travel time between source and observer. When you evaluate source densities at $t_r$ rather than at $t$, you're asking: what was the source doing at exactly the right earlier moment so that its influence arrives at $\vec{r}$ at time $t$?

Derivation of retarded potentials

Inhomogeneous wave equations

The retarded potentials emerge as solutions to the inhomogeneous wave equations for $\phi$ and $\vec{A}$. Working in the Lorenz gauge ($\nabla \cdot \vec{A} + \frac{1}{c^2}\frac{\partial \phi}{\partial t} = 0$), Maxwell's equations decouple into two wave equations with source terms:

• Scalar potential:

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

• Vector potential:

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

These have the same mathematical structure: a d'Alembertian ($\Box^2$) acting on the potential equals a source. Solving them amounts to finding the Green's function for the wave operator.

Green's function for the wave equation

A Green's function gives the field produced by an idealized point source that flashes on at a single instant. For the wave equation, the retarded Green's function is

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

The delta function enforces the constraint that only the retarded time contributes. Physically, this means the response at $(\vec{r}, t)$ comes from the source event at $(\vec{r}', t_r)$ that lies on the past light cone of the field point. There is also an advanced Green's function (using $t + |\vec{r}-\vec{r}'|/c$), but it violates causality and is discarded for physical radiation problems.

Mathematical form of retarded potentials

Convolving the Green's function with the source distributions gives the retarded potentials:

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

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

These look exactly like the static Coulomb and Biot-Savart potentials, with one critical difference: the source densities $\rho$ and $\vec{J}$ are evaluated at the retarded time $t_r = t - |\vec{r}-\vec{r}'|/c$, not at the present time $t$. Each volume element $d^3r'$ in the source contributes at its own retarded time, since each sits at a different distance from the field point.

Physical interpretation of retarded potentials

Causality and retarded potentials

Retarded potentials enforce causality: the field you measure at $(\vec{r}, t)$ depends only on source events inside the past light cone of that spacetime point. No information from the future or from spacelike-separated events can influence the field. This is built in automatically by the retarded time constraint, which restricts contributions to those that had enough time to propagate to the observer at speed $c$.

Retarded potentials vs. instantaneous potentials

Instantaneous (or "Coulomb-gauge") potentials assume the scalar potential responds to the source distribution at the current time, with no propagation delay. This isn't wrong in the Coulomb gauge, but it hides the causal structure: the retardation effects get shuffled entirely into the vector potential, making physical interpretation harder.

Retarded potentials in the Lorenz gauge treat $\phi$ and $\vec{A}$ on equal footing, with both exhibiting explicit retardation. This makes them the natural choice for radiation problems. The instantaneous picture is a reasonable approximation only in the quasi-static (near-field) regime, where the source dimensions and distances are much smaller than the characteristic wavelength, so the light-travel time is negligible compared to the timescale on which sources change.

Retarded electric and magnetic fields

Jefimenko's equations

You can bypass the potentials entirely and express $\vec{E}$ and $\vec{B}$ directly in terms of retarded source densities. The results are Jefimenko's equations:

$\vec{E}(\vec{r},t) = \frac{1}{4\pi\epsilon_0}\int\left[\frac{\rho(\vec{r}',t_r)}{R^2}\,\hat{R} + \frac{\dot{\rho}(\vec{r}',t_r)}{cR}\,\hat{R} - \frac{\dot{\vec{J}}(\vec{r}',t_r)}{c^2 R}\right]d^3r'$

$\vec{B}(\vec{r},t) = \frac{\mu_0}{4\pi}\int\left[\frac{\vec{J}(\vec{r}',t_r)}{R^2} + \frac{\dot{\vec{J}}(\vec{r}',t_r)}{cR}\right] \times \hat{R}\,d^3r'$

Here $\vec{R} = \vec{r}-\vec{r}'$, $R = |\vec{R}|$, $\hat{R} = \vec{R}/R$, and dots denote time derivatives evaluated at $t_r$.

Notice the structure: terms with $1/R^2$ are near-field (Coulomb/Biot-Savart-like), while terms with $1/R$ involve time derivatives of the sources and dominate in the far field. Those $1/R$ terms are the radiation fields.

Retarded fields of moving point charges

For a single point charge $q$ with trajectory $\vec{r}_q(t)$ and velocity $\vec{v}_q(t)$, the retarded electric field is the Liénard-Wiechert electric field:

$\vec{E}(\vec{r},t) = \frac{q}{4\pi\epsilon_0}\left[\frac{(\hat{n}-\vec{\beta})}{\gamma^2(1-\hat{n}\cdot\vec{\beta})^3 R^2}\right]_{t_r} + \text{acceleration terms}$

where:

• $\hat{n} = (\vec{r}-\vec{r}_q(t_r))/|\vec{r}-\vec{r}_q(t_r)|$ points from the retarded position to the field point
• $\vec{\beta} = \vec{v}_q(t_r)/c$ is the normalized velocity at the retarded time
• $\gamma = 1/\sqrt{1-\beta^2}$ is the Lorentz factor
• $R = |\vec{r}-\vec{r}_q(t_r)|$ is the retarded distance

The retarded magnetic field is always related to the electric field by

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

The factor $(1 - \hat{n}\cdot\vec{\beta})^3$ in the denominator is characteristic of retarded-time physics. It arises because a charge moving toward the observer "compresses" the signals it emits (a relativistic beaming effect), amplifying the field in the forward direction.

Liénard-Wiechert potentials

Derivation from retarded potentials

To get the Liénard-Wiechert potentials, substitute the point-charge densities $\rho(\vec{r}',t) = q\,\delta^3(\vec{r}'-\vec{r}_q(t))$ and $\vec{J} = q\vec{v}_q\,\delta^3(\vec{r}'-\vec{r}_q(t))$ into the retarded potential integrals. Performing the spatial integration collapses the delta function, but you must be careful: the retarded-time condition $t_r = t - |\vec{r}-\vec{r}'|/c$ implicitly depends on $\vec{r}'$, so the delta function picks up a Jacobian factor of $1/(1-\hat{n}\cdot\vec{\beta})$. The results are:

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

$\vec{A}(\vec{r},t) = \frac{\mu_0 q}{4\pi}\left[\frac{\vec{v}_q}{(1-\hat{n}\cdot\vec{\beta})\,R}\right]_{t_r}$

Everything inside the brackets is evaluated at the retarded time. The $(1-\hat{n}\cdot\vec{\beta})$ factor in the denominator is the same Doppler-like compression factor that appears in the fields.

Electric and magnetic fields of moving charges

The fields follow from the standard relations $\vec{E} = -\nabla\phi - \partial\vec{A}/\partial t$ and $\vec{B} = \nabla \times \vec{A}$, though the derivatives are nontrivial because $t_r$ itself depends on $\vec{r}$ and $t$. The full result splits into two physically distinct pieces:

• Velocity (Coulomb) field: Falls off as $1/R^2$, depends only on the charge's velocity (not acceleration), and carries no net energy to infinity. This is the generalized Coulomb field of a uniformly moving charge.
• Acceleration (radiation) field: Falls off as $1/R$, depends on the charge's acceleration at the retarded time, and carries energy outward. This is the radiation field.

Only the acceleration term contributes to the radiated power at large distances.

Applicability and limitations

The Liénard-Wiechert results are exact for a classical point charge at arbitrary velocity (including relativistic). They're the workhorse for:

• Radiation from single particles (synchrotrons, antennas modeled as current elements)
• Relativistic electrodynamics problems

For extended charge distributions, you go back to the full retarded-potential integrals. For quantum-scale problems or self-force issues (radiation reaction), the classical point-charge picture breaks down and requires additional treatment.

Radiative vs. non-radiative fields

Near-field and far-field regions

The boundary between near-field and far-field is set by the wavelength $\lambda$ of the radiation (or equivalently, the timescale of source variation):

• Near field ($R \ll \lambda$): The $1/R^2$ terms dominate. Fields look quasi-static and store energy locally without radiating it away.
• Far field ($R \gg \lambda$): The $1/R$ terms dominate. Fields are transverse, $\vec{E} \perp \vec{B} \perp \hat{n}$, and carry energy outward as electromagnetic waves. The Poynting flux falls as $1/R^2$, so the total power through a sphere at large $R$ is constant, confirming genuine radiation.

Retarded potentials in the radiation zone

In the far-field limit, several simplifications apply. The retarded distance $R$ varies slowly across the source, so you can pull $1/R$ out of the integral and approximate the retarded time using $|\vec{r}-\vec{r}'| \approx r - \hat{n}\cdot\vec{r}'$. The potentials then reduce to forms that depend on the Fourier components of the source distribution, which is the starting point for multipole radiation theory.

For a point charge, the far-field potentials simplify to:

$\phi(\vec{r},t) \approx \frac{q}{4\pi\epsilon_0\, r}\frac{1}{1-\hat{n}\cdot\vec{\beta}(t_r)}$

$\vec{A}(\vec{r},t) \approx \frac{\mu_0 q}{4\pi\, r}\frac{\vec{v}_q(t_r)}{1-\hat{n}\cdot\vec{\beta}(t_r)}$

where $r = |\vec{r}|$ and $\hat{n} \approx \hat{r}$ is now essentially constant across the source.

Non-radiative fields and retarded potentials

In the near-field regime ($R \ll \lambda$), the light-travel time across the region of interest is much shorter than the period of source variation. The retarded time $t_r \approx t$, and the retarded potentials reduce to the familiar instantaneous Coulomb and Biot-Savart forms. The fields in this region are responsible for:

• The electrostatic Coulomb interaction between charges
• Inductive and capacitive coupling in circuits
• Reactive (stored) energy near antennas

These fields don't carry energy to infinity. They fall off too fast ($\sim 1/R^2$ or steeper) for the integrated Poynting flux to survive at large distances.

Applications of retarded potentials

Electromagnetic radiation from accelerated charges

Any accelerated charge radiates. The instantaneous radiated power (non-relativistic case) is given by the Larmor formula:

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

where $\vec{a}$ is the acceleration. For relativistic charges, the generalized (relativistic) Larmor formula replaces $|\vec{a}|^2$ with a Lorentz-invariant expression involving $\gamma^6$ factors.

The angular distribution and polarization of the radiation follow from evaluating the Liénard-Wiechert fields in the far zone. The radiation is concentrated in the direction of motion for relativistic particles (beaming), with the opening half-angle of the radiation cone scaling as $\sim 1/\gamma$.

Synchrotron and bremsstrahlung radiation

Synchrotron radiation occurs when relativistic charged particles (typically electrons) are deflected by a magnetic field. Because the acceleration is perpendicular to the velocity, the relativistic beaming effect produces a narrow, swept beam. Key features:

• Broad continuous spectrum, from radio to hard X-rays depending on $\gamma$
• Highly polarized (linearly in the orbital plane)
• Used in dedicated synchrotron light sources for materials science, biology, and chemistry

Bremsstrahlung ("braking radiation") is emitted when a charged particle decelerates in the Coulomb field of a nucleus. Here the acceleration has a component along the velocity. Key features:

• Continuous spectrum up to the kinetic energy of the incident particle
• Dominant X-ray production mechanism in medical X-ray tubes
• Important energy-loss mechanism for electrons in matter

Both types of radiation are fully described by the retarded potentials and the Liénard-Wiechert fields, applied to the specific trajectory of the charge.

Čerenkov radiation and retarded potentials

Čerenkov radiation is emitted when a charged particle moves through a dielectric medium at a speed $v$ exceeding the phase velocity of light in that medium, $c/n$. The retarded-potential framework still applies, but with the replacement $c \to c/n$ in the wave equation for the medium.

The radiation is emitted in a cone at an angle $\theta$ to the particle's trajectory, given by:

$\cos\theta = \frac{c}{nv} = \frac{1}{n\beta}$

This is the electromagnetic analog of a sonic boom. The condition $v > c/n$ (equivalently $n\beta > 1$) is the threshold for emission. Čerenkov radiation is used in particle physics detectors (RICH detectors, water Čerenkov detectors) to identify particles by measuring $\theta$ and inferring $\beta$, and hence the particle mass when combined with momentum measurements.