Time-Harmonic Magnetic Fields

In the time-harmonic case, there is no computational cost for including the displacement current in Ampère’s law (then called Maxwell-Ampère’s law):

In the transient case, the inclusion of this term leads to a second-order equation in time, but in the harmonic case there are no such complications. Using the definition of the electric and magnetic potentials, the system of equations becomes:

The constitutive relation D = ε0E + P has been used for the electric field.

To obtain a particular gauge that reduces the system of equation, choose Ψ = −jV/ω in the gauge transformation. This gives:

When vanishes from the equations, only the second one is needed,

Working with is often the best option when it is possible to specify all source currents as external currents Je or as surface currents on boundaries.