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 comparison with static potentials for time-independent fields
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.
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. However, provided we require that the potentials vanish at infinity, they can be expressed neatly in terms of fields:
In linearized gravity
The retarded potential in
linearized general relativity is closely analogous to the electromagnetic case. The trace-reversed tensor plays the role of the four-vector potential, the
harmonic gauge replaces the electromagnetic Lorenz gauge, the field equations are , and the retarded-wave solution is[6]
Using SI units, the expression must be divided by , as can be confirmed by dimensional analysis.
The potential of charge with uniform speed on a straight line has
inversion in a point that is in the recent position. The potential is not changed in the direction of movement.[7]
^Rohrlich, F (1993).
"Potentials". In Parker, S.P. (ed.). McGraw Hill Encyclopaedia of Physics (2nd ed.). New York. p. 1072.
ISBN0-07-051400-3.{{
cite encyclopedia}}: CS1 maint: location missing publisher (
link)
^Garg, A., Classical Electromagnetism in a Nutshell, 2012, p. 129
^Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008,
ISBN978-0-471-92712-9