Electromagnetic Four-potential - in The Lorenz Gauge

In The Lorenz Gauge

Often, the Lorenz gauge condition in an inertial frame of reference is employed to simplify Maxwell's equations as:

SI units	cgs units

where Jα are the components of the four-current, and

is the d'Alembertian operator. In terms of the scalar and vector potentials, this last equation becomes:

SI units	cgs units

For a given charge and current distribution, ρ(r, t) and j(r, t), the solutions to these equations in SI units are:

where

is the retarded time. This is sometimes also expressed with

where the square brackets are meant to indicate that the time should be evaluated at the retarded time. Of course, since the above equations are simply the solution to an inhomogeneous differential equation, any solution to the homogeneous equation can be added to these to satisfy the boundary conditions. These homogeneous solutions in general represent waves propagating from sources outside the boundary.

When the integrals above are evaluated for typical cases, e.g. of an oscillating current (or charge), they are found to give both a magnetic field component varying according to r −2 (the induction field) and a component decreasing as r −1 (the radiation field).