In classical electrodynamics, the finite speed of light imposes a fundamental causal structure on electromagnetic interactions: changes in charge and current distributions cannot instantaneously affect distant points, but instead propagate outward at the speed of light. To capture this, the concepts of retarded potentials are essential.
The retarded potentials generalize the familiar electrostatic and magnetostatic potentials to time-dependent sources by incorporating the appropriate time delay — the retarded time — at which the sources influence the fields at a given point. These potentials are solutions to the inhomogeneous wave equations for the scalar and vector potentials in the Lorenz gauge and form the foundation for describing dynamic electromagnetic phenomena, including radiation.
Specializing these results to the case of a point charge moving arbitrarily yields the celebrated Liénard–Wiechert potentials, which provide the exact electromagnetic potentials generated by such a charge. These potentials encapsulate both the near-field behavior and the radiation emitted due to the charge’s acceleration and motion, including relativistic effects through the dependence on the charge’s velocity and retarded position.
In electrostatics, the scalar potentialV(r) satisfies Poisson’s equation
∇2V=−ϵ0ρ,
and is given by the well-known integral solution
V(r)=4πϵ01∫∣r−r′∣ρ(r′)d3r′.
This solution assumes the charge distribution ρ is static, or at least varies so slowly that the finite speed of light can be ignored. However, this instantaneous action-at-a-distance is inconsistent with special relativity and electromagnetic causality.
When charges vary with time, electromagnetic effects propagate at the finite speed c. Therefore, the potentials at time t must depend on the earlier state of the sources, evaluated at a retarded timetr:
tr=t−c∣r−r′∣.
Here, ∣r−r′∣/c is the time taken for the electromagnetic influence to travel from the source point r′ to the field point r.
A Green’s function is a powerful mathematical tool used to solve linear differential equations with specified source terms and boundary conditions.
Intuitively, the Green’s function acts like the “impulse response” of the differential operator: it tells us how the system responds to a point source or delta function input.
Consider the inhomogeneous wave equation for a scalar function ψ(r,t) with source f(r,t):
(∇2−c21∂t2∂2)ψ(r,t)=−f(r,t).
The Green’s function, G(r,t;r′,t′), is defined as the solution to
(∇2−c21∂t2∂2)G(r,t;r′,t′)=δ(3)(r−r′)δ(t−t′).
This equation means that G describes the response of the wave equation at point (r,t) due to a unit impulsive source applied at (r′,t′).
Because the wave equation is linear, the solution for an arbitrary source f(r,t) can be constructed by superposing the responses to many point impulses weighted by f.
Formally, the solution ψ(r,t) is given by the convolution of the Green’s function with the source:
ψ(r,t)=∫d3r′∫dt′G(r,t;r′,t′)f(r′,t′).
Physically, this integral sums the effect of all point sources f(r′,t′) at earlier times and positions, weighted by how strongly and how quickly their influence propagates to (r,t).
The choice of Green’s function depends on boundary and initial conditions. For electrodynamics, causality requires that disturbances propagate forward in time at speed c, so only past sources can affect the field at (r,t).
The retarded Green’s function satisfies this causal condition:
The potentials at time t depend on the charge and current distributions at the retarded timetr, accounting for the finite propagation speed of electromagnetic interactions.
The factor 1/∣r−r′∣ retains the spatial decay familiar from electrostatics.
The retarded potentials are the unique causal solutions to the Maxwell wave equations in the Lorenz gauge and form the foundation for the theory of electromagnetic radiation.
In the static limit where sources vary slowly, the retarded potentials reduce to the instantaneous Coulomb and magnetostatic potentials.
The Liénard–Wiechert potentials describe the exact scalar and vector potentials produced by a point charge moving arbitrarily in space and time. They are derived by substituting the point charge’s charge and current densities into the retarded potentials and carefully handling the implicit dependence on the retarded time.
This means the potentials depend on the position and velocity of the charge at the earlier time tr, when the charge emitted the signal that arrives at r at time t.
¶ 4. Deriving the Explicit Form: Using the Delta Function Identity
To find a useful closed form, we express the charge density in spacetime as
ρ(r′,t′)=qδ(3)(r′−r0(t′)).
The retarded scalar potential can also be written as
The retarded potentials are the unique, causal solutions to Maxwell’s equations in the Lorenz gauge that correctly account for the finite speed at which electromagnetic effects propagate. Unlike their static counterparts, these potentials incorporate a time delay — the retarded time — ensuring that the fields at a given point and time depend only on the source configuration in the past, in accordance with special relativity.
Using the machinery of Green’s functions, the retarded potentials are derived as integrals over the charge and current distributions, evaluated at the retarded time:
The motion of the source through the velocity dependence in the denominator;
The causality via evaluation at the retarded time tr;
The directionality of emission via the unit vector R^=(r−r0)/∣r−r0∣.
Together, they describe both near-field (Coulomb-like) and far-field (radiative) effects of the moving charge. This formulation underlies all classical descriptions of electromagnetic radiation from accelerating charges and is foundational for applications ranging from antenna theory to high-energy astrophysical processes such as synchrotron and bremsstrahlung radiation.