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
= ε
0
E
+
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
J
e
or as surface currents on boundaries.
is often the best option when it is possible to specify all source currents as external currents Je or as surface currents on boundaries.
vanishes from the equations, only the second one is needed,