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Concept
Retarded potential
Summary
In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light c, so the delay of the fields connecting cause and effect at earlier and later times is an important factor: the signal takes a finite time to propagate from a point in the charge or current distribution (the point of cause) to another point in space (where the effect is measured), see figure below.
Maxwell's equations and Mathematical descriptions of the electromagnetic field
The starting point is Maxwell's equations in the potential formulation using the Lorenz gauge:
where φ(r, t) is the electric potential and A(r, t) is the magnetic vector potential, for an arbitrary source of charge density ρ(r, t) and current density J(r, t), and is the D'Alembert operator. Solving these gives the retarded potentials below (all in SI units).
For time-dependent fields, the retarded potentials are:
where r is a point in space, t is time,
is the retarded time, and d3r' is the integration measure using r'.
From φ(r, t) and A(r, t), the fields E(r, t) and B(r, t) can be calculated using the definitions of the potentials:
and this leads to Jefimenko's equations. The corresponding advanced potentials have an identical form, except the advanced time
replaces the retarded time.
In the case the fields are time-independent (electrostatic and magnetostatic fields), the time derivatives in the operators of the fields are zero, and Maxwell's equations reduce to
where ∇2 is the Laplacian, which take the form of Poisson's equation in four components (one for φ and three for A), and the solutions are:
These also follow directly from the retarded potentials.
In the Coulomb gauge, Maxwell's equations are
although the solutions contrast the above, since A is a retarded potential yet φ changes instantly, given by:
This presents an advantage and a disadvantage of the Coulomb gauge - φ is easily calculable from the charge distribution ρ but A is not so easily calculable from the current distribution j.
Official source
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