Maxwell's equation
Using the above definition of the potentials and applying it to the other two Maxwell's equations (the ones that are not automatically satisfied) results in a complicated differential equation that can be simplified using the Lorenz gauge where A is chosen to satisfy:
Using the Lorenz gauge, Maxwell's equations can be written compactly in terms of the magnetic vector potential A and the electric scalar potential ϕ:
In other gauges, the equations are different. A different notation to write these same equations (using four-vectors) is shown below.