## 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: $ablacdot extbf{A} + frac{1}{c^2} frac{partial phi}{partial t} = 0.$

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: $abla^2phi - frac{1}{c^2}frac{partial^2 phi}{partial t^2} = - ho/ epsilon_0$ $abla^2 extbf{A} - frac{1}{c^2}frac{partial^2 extbf{A}}{partial t^2} = - mu_0 extbf{J}$

In other gauges, the equations are different. A different notation to write these same equations (using four-vectors) is shown below.

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