Retarded Potentials in Lorenz Gauge

Derivation

Retarded Potentials in Lorenz Gauge

D-293 Home PU-305 Threads fields · waves Depends on Displacement Current and Maxwell's Equations, Laplacian of 1/r as a Delta Function

Statement

Starting from the microscopic Maxwell equations, the electromagnetic potentials \(V\) and \(\vec A\) defined by \(\vec B=\nabla\times\vec A\) and \(\vec E=-\nabla V-\partial_t\vec A\) are shown, once the Lorenz gauge condition \(\nabla\cdot\vec A+\tfrac{1}{c^2}\partial_t V=0\) is imposed, to satisfy two decoupled inhomogeneous wave equations \(\Box V=-\rho/\varepsilon_0\) and \(\Box\vec A=-\mu_0\vec J\). Their unique causally-propagating (retarded) solutions are the retarded potentials, in which each source contributes evaluated at the earlier time \(t_r=t-|\vec r-\vec r\,'|/c\).

Why it matters

The retarded potentials are the bridge between sources and radiation. Every antenna pattern, every Liénard–Wiechert field, every treatment of synchrotron light and of the self-force of an accelerating charge begins here. They are the explicit statement that electromagnetic influence is not instantaneous: it leaves the source, travels at \(c\), and arrives later.

They also demonstrate the power of gauge freedom. The raw field equations for the potentials are coupled and unwieldy; a single scalar condition on the potentials disentangles them into the same wave operator acting separately on \(V\) and \(\vec A\), with the sources appearing untouched on the right. This is the cleanest route from Maxwell to causality.

Assumptions

The fields derive from potentials in a simply-connected domain.If the domain has nontrivial topology (e.g. threading flux), \(\vec A\) may not be globally single-valued and the representation \(\vec B=\nabla\times\vec A\) can require patching; the local wave equations still hold but global solutions differ.

The medium is vacuum with sources \(\rho,\vec J\) prescribed.In linear media one replaces \(\varepsilon_0,\mu_0\) by \(\varepsilon,\mu\) and \(c\to c/n\); in dispersive or nonlinear media the simple wave operator with a single speed fails and Green-function methods must be reworked frequency by frequency.

Sources conserve charge: \(\partial_t\rho+\nabla\cdot\vec J=0\).Continuity is required for the two wave equations to be mutually consistent under the Lorenz condition; if it is violated no gauge can decouple them and the potentials are overdetermined.

The Lorenz gauge is attainable — a gauge function \(\lambda\) solving \(\Box\lambda=-(\nabla\cdot\vec A+\tfrac{1}{c^2}\partial_t V)\) exists.If no such \(\lambda\) exists (pathological boundary data), one cannot reach the Lorenz condition and the clean decoupling is unavailable; one must work in another gauge (e.g. Coulomb) with a nonlocal source term.

Causal (retarded) boundary conditions: no incoming radiation from infinity.The wave operator also admits the advanced Green function and homogeneous free-wave solutions; dropping the outgoing/no-incoming condition makes the solution non-unique — the retarded choice is a physical input, not a theorem.

Derivation

\[ \nabla\cdot\vec B=0 \;\Longrightarrow\; \vec B=\nabla\times\vec A \]

A divergence-free field on a simply-connected domain is the curl of some vector potential (Helmholtz/Poincaré). This solves one homogeneous Maxwell equation identically. A

\[ \nabla\times\vec E=-\partial_t\vec B=-\partial_t(\nabla\times\vec A)\;\Longrightarrow\;\nabla\times\!\left(\vec E+\partial_t\vec A\right)=0 \]

Substitute \(\vec B=\nabla\times\vec A\) into Faraday's law and commute \(\partial_t\) with \(\nabla\times\). A curl-free field is a gradient. A

\[ \vec E+\partial_t\vec A=-\nabla V \;\Longrightarrow\; \vec E=-\nabla V-\partial_t\vec A \]

Write the curl-free combination as \(-\nabla V\), defining the scalar potential \(V\). The two source-free Maxwell equations are now solved identically for any \(V,\vec A\). A

\[ \nabla\cdot\vec E=\frac{\rho}{\varepsilon_0}\;\Longrightarrow\;-\nabla^2 V-\partial_t(\nabla\cdot\vec A)=\frac{\rho}{\varepsilon_0} \]

Insert \(\vec E=-\nabla V-\partial_t\vec A\) into Gauss's law. This is the first source equation; \(V\) and \(\vec A\) are still coupled through \(\nabla\cdot\vec A\). A

\[ \nabla\times(\nabla\times\vec A)=\mu_0\vec J+\mu_0\varepsilon_0\,\partial_t\!\left(-\nabla V-\partial_t\vec A\right) \]

Insert \(\vec B=\nabla\times\vec A\) and \(\vec E=-\nabla V-\partial_t\vec A\) into the Ampère–Maxwell law (the displacement-current-completed form). A

\[ \nabla(\nabla\cdot\vec A)-\nabla^2\vec A=\mu_0\vec J-\mu_0\varepsilon_0\nabla(\partial_t V)-\mu_0\varepsilon_0\,\partial_t^2\vec A \]

Apply the identity \(\nabla\times(\nabla\times\vec A)=\nabla(\nabla\cdot\vec A)-\nabla^2\vec A\) to the left side. Purely algebraic vector calculus. A

\[ \left(\nabla^2\vec A-\mu_0\varepsilon_0\,\partial_t^2\vec A\right)-\nabla\!\left(\nabla\cdot\vec A+\mu_0\varepsilon_0\,\partial_t V\right)=-\mu_0\vec J \]

Group the pure-\(\vec A\) wave terms and collect everything else under one gradient. Use \(\mu_0\varepsilon_0=1/c^2\). This isolates the offending coupling as a single gradient. B

\[ \boxed{\;\nabla\cdot\vec A+\frac{1}{c^2}\partial_t V=0\;}\quad(\text{Lorenz gauge}) \]

Exploit gauge freedom \(V\to V-\partial_t\lambda,\ \vec A\to\vec A+\nabla\lambda\), which leaves \(\vec E,\vec B\) unchanged. Choosing \(\lambda\) to solve \(\Box\lambda=-(\nabla\cdot\vec A+\tfrac1{c^2}\partial_tV)\) enforces the condition. The gradient term in Step 7 vanishes. B

\[ \nabla^2\vec A-\frac{1}{c^2}\partial_t^2\vec A=-\mu_0\vec J,\qquad \nabla^2 V-\frac{1}{c^2}\partial_t^2 V=-\frac{\rho}{\varepsilon_0} \]

With the gauge term gone, Step 7 gives the \(\vec A\) wave equation. Using \(\nabla\cdot\vec A=-\tfrac1{c^2}\partial_t V\) in Step 4 turns \(-\nabla^2V-\partial_t(\nabla\cdot\vec A)=\rho/\varepsilon_0\) into the \(V\) wave equation. The potentials are decoupled and symmetric. B

\[ \Box\equiv\nabla^2-\frac{1}{c^2}\partial_t^2,\qquad \Box V=-\frac{\rho}{\varepsilon_0},\quad \Box\vec A=-\mu_0\vec J \]

Define the d'Alembertian. Both potentials obey the same scalar wave operator with their sources on the right; the four components \((V,\vec A)\) satisfy one universal equation. Consistency between them under the gauge condition requires exactly charge continuity \(\partial_t\rho+\nabla\cdot\vec J=0\). B

\[ \Box\,G(\vec r,t;\vec r\,',t')=-\delta^3(\vec r-\vec r\,')\,\delta(t-t') \]

Seek the Green function of \(\Box\). By translation invariance \(G=G(\vec R,\tau)\) with \(\vec R=\vec r-\vec r\,'\), \(\tau=t-t'\). Linearity then gives \(V=\int G\,(\rho/\varepsilon_0)\,d^3r'\,dt'\) and \(\vec A=\int G\,\mu_0\vec J\,d^3r'\,dt'\). C

\[ G_{\text{ret}}(\vec R,\tau)=\frac{1}{4\pi R}\,\delta\!\left(\tau-\frac{R}{c}\right),\qquad R=|\vec R| \]

For \(R\neq0\) this is a spherical impulse travelling outward at \(c\), and one checks \(\Box G_{\text{ret}}=0\) there. Integrating \(\Box G\) over an infinitesimal ball about \(R=0\) reduces (the time-derivative term integrates to zero over the shrinking volume) to the static identity \(\nabla^2\tfrac1R=-4\pi\delta^3(\vec R)\), fixing the coefficient \(1/4\pi\) and the sign. The retarded (not advanced) root \(\tau=+R/c\) enforces no incoming radiation. C

\[ V(\vec r,t)=\frac{1}{4\pi\varepsilon_0}\int\frac{\rho(\vec r\,',t')}{R}\,\delta\!\left(t-t'-\tfrac{R}{c}\right)d^3r'\,dt' \]

Fold \(G_{\text{ret}}\) against \(\rho/\varepsilon_0\). The \(\delta\) collapses the \(t'\) integral, pinning \(t'=t-R/c\). B

\[ V(\vec r,t)=\frac{1}{4\pi\varepsilon_0}\int\frac{\rho\!\left(\vec r\,',\,t-\tfrac{|\vec r-\vec r\,'|}{c}\right)}{|\vec r-\vec r\,'|}\,d^3r',\quad \vec A(\vec r,t)=\frac{\mu_0}{4\pi}\int\frac{\vec J\!\left(\vec r\,',\,t-\tfrac{|\vec r-\vec r\,'|}{c}\right)}{|\vec r-\vec r\,'|}\,d^3r' \]

Perform the \(t'\)-integral for both potentials. Each source element is evaluated at its own retarded time \(t_r=t-|\vec r-\vec r\,'|/c\). These are the retarded potentials. A

Result

\[ V(\vec r,t)=\frac{1}{4\pi\varepsilon_0}\int\frac{[\rho]}{|\vec r-\vec r\,'|}\,d^3r',\qquad \vec A(\vec r,t)=\frac{\mu_0}{4\pi}\int\frac{[\vec J]}{|\vec r-\vec r\,'|}\,d^3r',\qquad [\,\cdot\,]\equiv \text{evaluated at }t_r=t-\frac{|\vec r-\vec r\,'|}{c} \]

Reading. The potential here-and-now is the Coulomb/Biot–Savart-like sum over all source elements, but each element speaks to us from the moment its light-cone reaches us. Information about a source point a distance \(R\) away is always old by \(R/c\). The functional form is identical to the static case except that "now" is replaced by "then." When the sources change slowly (or not at all), \(t_r\to t\) and the static expressions are recovered.

Units check. Scalar: \([\rho]\,\text{C·m}^{-3}\times d^3r'\,\text{m}^3\div R\,\text{m}=\text{C·m}^{-1}\); times \((4\pi\varepsilon_0)^{-1}\) which carries \(\text{V·m·C}^{-1}\), giving volts. Vector: \(\mu_0\,[\text{T·m·A}^{-1}]\times [\vec J]\,[\text{A·m}^{-2}]\times d^3r'\div R=\text{T·m}\), the correct units for \(\vec A\) since \(\vec B=\nabla\times\vec A\) has \(\text{T·m}/\text{m}=\text{T}\).

Limiting cases

Static sources (\(\partial_t\rho=\partial_t\vec J=0\)): \(t_r\) dependence disappears, giving the Coulomb potential \(V=\frac{1}{4\pi\varepsilon_0}\int\rho/R\,d^3r'\) and the magnetostatic \(\vec A=\frac{\mu_0}{4\pi}\int\vec J/R\,d^3r'\).
Quasi-static / near zone (\(R\ll\lambda\), source size \(\ll\lambda\)): expand \([\rho]\approx\rho(t)-\tfrac{R}{c}\dot\rho+\dots\); the leading correction to the instantaneous potential is \(O(R/c)\), the retardation delay.
Point charge: the volume integrals collapse (with the Jacobian of \(t_r\)) to the Liénard–Wiechert potentials \(V=\frac{q}{4\pi\varepsilon_0}\frac{1}{(R-\vec R\cdot\vec\beta)}\), \(\vec A=\vec\beta V/c\).
Free field (\(\rho=\vec J=0\)): the particular solution vanishes and \(\Box V=\Box\vec A=0\) leave source-free radiation, i.e. light propagating with no charges present.
Far zone / radiation (\(r\gg\lambda\gg\) source): \(R\approx r-\hat r\cdot\vec r\,'\), the \(1/R\) prefactor \(\to1/r\), and \(\vec A\) is dominated by the retarded-time variation, producing the radiation fields.

Breaks when

Charge is not conserved. If \(\partial_t\rho+\nabla\cdot\vec J\neq0\), the two wave equations are incompatible with the Lorenz condition; no gauge decouples them and the retarded-potential pair is not a solution of Maxwell's equations.
Dispersive or nonlinear media. A single propagation speed \(c\) no longer exists; the operator \(\Box\) with one \(c\) is wrong, and the causal Green function must be built frequency-by-frequency (each Fourier component sees \(c/n(\omega)\)).
Strong gravity / curved spacetime. The flat-space \(\Box\) and the \(1/R\) Green function are replaced by curved-space wave operators with tails; signals no longer travel strictly on the light cone.
Incoming radiation is present. If the physical boundary condition includes waves arriving from infinity, the retarded solution alone is incomplete — one must add homogeneous free-wave or advanced pieces.

Failure modes

Retarding the wrong slot. Evaluating \(\rho\) at \(t_r\) but forgetting that \(t_r\) itself depends on \(\vec r\,'\); the retarded time varies across the source, so it cannot be pulled outside the integral except in the far-zone approximation.
Differentiating through the retarded time. Computing fields by \(\vec E=-\nabla V-\partial_t\vec A\) while treating \([\rho]\) as if \(t_r\) were constant — the spatial gradient must act on \(t_r=t-R/c\) too, generating the crucial \(1/r\) radiation terms.
Coulomb-gauge slip. Writing \(V\) as the instantaneous Coulomb potential (true in Coulomb gauge) but then using the Lorenz-gauge \(\vec A\); mixing gauges gives wrong fields.
Sign/root of the delta. Selecting the advanced root \(t_r=t+R/c\) by algebra without imposing outgoing boundary conditions, yielding acausal potentials.
Dropping displacement current. Using \(\nabla\times\vec B=\mu_0\vec J\) (no \(\mu_0\varepsilon_0\partial_t\vec E\)) destroys the \(\partial_t^2\vec A\) term and gives a Poisson equation, not a wave equation — no retardation at all.

Discussion

The heart of the derivation is that gauge freedom is not a nuisance but a tool: the same transformation \((V,\vec A)\to(V-\partial_t\lambda,\vec A+\nabla\lambda)\) that leaves the observable fields untouched is exactly what lets us kill the coupling term in Step 7. In the Lorenz gauge the four potential components become four copies of one scalar wave equation, each fed by one component of the four-current \((\rho c,\vec J)\). This is the first hint of the relativistic structure: \(\Box\) is Lorentz-invariant, the Lorenz condition \(\partial_\mu A^\mu=0\) is a four-scalar, and \(\Box A^\mu=\mu_0 J^\mu\) packages both equations into one covariant statement. The gauge is named for Ludvig Lorenz, and its invariance under boosts is why it survives special relativity while the Coulomb gauge does not transform simply.

Physically, the retarded potentials are the mathematical form of the statement "electromagnetic news travels at \(c\)." The Green function \(G_{\text{ret}}\) is a single expanding spherical shell of influence launched by an impulsive point source; any real source is a superposition of such impulses, and the field at \((\vec r,t)\) is the sum over all past impulses whose shells have just reached the observer. The whole causal content of electrodynamics is compressed into the argument \(t_r=t-R/c\).

The choice of retarded over advanced Green function is a boundary condition, not a consequence of Maxwell's equations, which are time-symmetric. Both roots solve \(\Box G=-\delta^4\); we select the retarded one to match the observed arrow of radiation. Wheeler and Feynman's absorber theory showed that a fully time-symmetric \(\tfrac12(G_{\text{ret}}+G_{\text{adv}})\) can reproduce ordinary radiation once one demands complete future absorption, dramatizing that causality here is imposed by the thermodynamic/cosmological environment rather than derived. The retarded solution is also what makes the self-energy and radiation-reaction of a point charge subtle: differentiating the retarded field of a charge with respect to its own worldline yields the Abraham–Lorentz force and its notorious pre-acceleration pathologies.

Common misconceptions. (1) "Retarded potentials mean the fields lag the sources by a fixed time" — the delay \(R/c\) is different for every source element, so a spread-out source contributes a smear of retarded times. (2) "The retarded potential is just Coulomb with a time delay, so its gradient is the delayed Coulomb field" — false; because \(t_r\) depends on position, differentiating produces additional radiation (\(1/r\)) terms absent from the static field. (3) "Lorenz gauge fixes the potentials uniquely" — it does not; residual gauge freedom remains for any \(\lambda\) with \(\Box\lambda=0\).

Worked examples

\[ \textbf{Retardation of a static point charge.}\ q=1.0\ \text{nC at origin},\ \text{observer at }r=30\ \text{m}. \]

Symbolic: for a static source \(\rho(\vec r\,',t_r)=\rho(\vec r\,')\), the retarded integral collapses to \(V=\dfrac{q}{4\pi\varepsilon_0 r}\); the retarded delay is \(\Delta t=r/c\). Retardation shifts when the value applies, not the value itself. A

\[ \Delta t=\frac{r}{c}=\frac{30}{3.00\times10^{8}}=1.0\times10^{-7}\ \text{s}=100\ \text{ns} \]

\[ V=\frac{q}{4\pi\varepsilon_0 r}=\frac{(8.99\times10^{9})(1.0\times10^{-9})}{30}=0.30\ \text{V} \]

\[ V=0.30\ \text{V},\qquad \Delta t=100\ \text{ns} \]

Reading. The potential equals the ordinary Coulomb value; retardation is invisible because the source never changed. The 100 ns is the age of the information — relevant only if \(q\) were switched.

Units check. \((\text{V·m·C}^{-1})(\text{C})/\text{m}=\text{V}\); \(\text{m}/(\text{m·s}^{-1})=\text{s}\).

\[ \textbf{Vector potential of an oscillating current element.}\ I(t)=I_0\cos\omega t,\ I_0=1.0\ \text{A},\ \text{length }\ell=1.0\ \text{cm}. \]

A short current element at the origin gives \(\vec A(\vec r,t)=\dfrac{\mu_0\,\ell}{4\pi r}\,I\!\left(t-\dfrac{r}{c}\right)\hat z\), the retarded potential of \(\vec J\,d^3r'\to I\ell\,\hat z\). Take \(f=30\ \text{MHz}\), observe on axis at \(r=10\ \text{m}\), at \(t=0\). B

\[ \omega=2\pi f=2\pi(3.0\times10^{7})=1.885\times10^{8}\ \text{rad·s}^{-1},\qquad \frac{r}{c}=\frac{10}{3.0\times10^{8}}=33.3\ \text{ns} \]

\[ A_z(0)=\frac{\mu_0\ell I_0}{4\pi r}\cos\!\left(-\omega\,\tfrac{r}{c}\right) =\frac{(10^{-7})(1.0\times10^{-2})(1.0)}{10}\cos\!\big(-1.885\times10^{8}\times3.33\times10^{-8}\big) \]

\[ =1.0\times10^{-10}\times\cos(-6.28)=1.0\times10^{-10}\ \text{T·m} \]

\[ A_z\approx1.0\times10^{-10}\ \text{T·m} \]

Reading. The retarded phase \(\omega r/c=2\pi(r/\lambda)\approx2\pi\) means \(r\approx\lambda=10\) m, so the element is exactly one wavelength away and the observer sees the current as it was 33 ns ago — one full period earlier, hence \(\cos\approx1\).

Units check. \((\text{T·m·A}^{-1})(\text{m})(\text{A})/\text{m}=\text{T·m}\).

Problems

Show that in the Lorenz gauge charge continuity \(\partial_t\rho+\nabla\cdot\vec J=0\) is exactly the consistency condition between \(\Box V=-\rho/\varepsilon_0\) and \(\Box\vec A=-\mu_0\vec J\).
Solution
Take \(\nabla\cdot\) of the \(\vec A\) equation and \(\tfrac1{c^2}\partial_t\) of the \(V\) equation: \(\Box(\nabla\cdot\vec A)=-\mu_0\nabla\cdot\vec J\) and \(\Box(\tfrac1{c^2}\partial_t V)=-\tfrac{1}{c^2\varepsilon_0}\partial_t\rho=-\mu_0\partial_t\rho\) (using \(1/c^2=\mu_0\varepsilon_0\)). Add them: \(\Box(\nabla\cdot\vec A+\tfrac1{c^2}\partial_t V)=-\mu_0(\nabla\cdot\vec J+\partial_t\rho)\). The left side is \(\Box(0)=0\) by the Lorenz condition, so \(\partial_t\rho+\nabla\cdot\vec J=0\). Consistency and continuity are equivalent.
A source is switched on at \(t=0\). At an observation point \(r=1.5\times10^{11}\ \text{m}\) (1 AU), when does the potential first respond, and what is that delay in minutes?
Solution
The retarded potential responds only when the light cone arrives: \(t=r/c=(1.5\times10^{11})/(3.0\times10^{8})=500\ \text{s}=8.3\ \text{min}\). Before \(t=500\) s, \([\rho]=0\) everywhere within reach and \(V=0\). This is the familiar 8-minute Sun–Earth light delay.
For a static current loop of radius \(a\) carrying \(I\), argue that the retarded \(\vec A\) reduces to the magnetostatic dipole result at large \(r\), and give the far-field magnitude scaling.
Solution
Static \(\vec J\) removes all \(t_r\) dependence, so \(\vec A=\frac{\mu_0}{4\pi}\int\vec J/R\,d^3r'\), the magnetostatic integral. Expanding \(1/R=1/r+\hat r\cdot\vec r\,'/r^2+\dots\), the monopole term vanishes (\(\oint d\vec l=0\)); the next term gives \(\vec A=\frac{\mu_0}{4\pi}\frac{\vec m\times\hat r}{r^2}\) with \(\vec m=I\pi a^2\hat n\). Magnitude \(|\vec A|\sim\frac{\mu_0 m}{4\pi r^2}\sin\theta\), falling as \(1/r^2\).
Verify by direct differentiation that \(V=\dfrac{f(t-r/c)}{r}\) satisfies \(\Box V=0\) for \(r\neq0\), where \(f\) is any twice-differentiable function.
Solution
For a spherically symmetric \(V(r,t)\), \(\nabla^2 V=\frac1r\partial_r^2(rV)\). Here \(rV=f(t-r/c)\), so \(\partial_r^2(rV)=\frac1{c^2}f''\), giving \(\nabla^2V=\frac{f''}{c^2 r}\). Also \(\partial_t^2 V=\frac{f''}{r}\), so \(\frac1{c^2}\partial_t^2V=\frac{f''}{c^2 r}\). Hence \(\Box V=\nabla^2V-\frac1{c^2}\partial_t^2V=\frac{f''}{c^2r}-\frac{f''}{c^2r}=0\). The outgoing spherical wave solves the homogeneous equation, as required of \(G_{\text{ret}}\) away from the origin.
A point charge \(q=2.0\ \text{nC}\) sits at the origin; its magnitude begins to grow at \(t=0\) as \(q(t)=q_0(1+t/\tau)\) with \(\tau=1.0\ \mu\text{s}\) (imagine current fed in radially). At \(r=300\ \text{m}\) and lab time \(t=2.0\ \mu\text{s}\), evaluate the retarded scalar potential. (Take the charge as effectively point-like.)
Solution
Retarded time: \(t_r=t-r/c=2.0\times10^{-6}-\frac{300}{3.0\times10^{8}}=2.0\times10^{-6}-1.0\times10^{-6}=1.0\ \mu\text{s}\). Charge then: \(q(t_r)=q_0(1+t_r/\tau)=2.0\ \text{nC}\times(1+1.0/1.0)=4.0\ \text{nC}\). Potential: \(V=\frac{q(t_r)}{4\pi\varepsilon_0 r}=\frac{(8.99\times10^{9})(4.0\times10^{-9})}{300}=0.12\ \text{V}\). The observer sees the charge as it was 1 µs ago (\(=4.0\) nC), not its present value \(q(2\mu\text{s})=6.0\) nC. Note: strictly a growing isolated point charge violates continuity unless fed by an inward radial current \(\vec J\) supplying \(dq/dt\); that current also contributes to \(\vec A\).