displacement current - electrodynamcis
I just wonder if someone of you has already developed a transient solver that can calculate the ohmic and displacement current, e.g. in a parallel-plate capacitor with a dielectric material that has an electric conductivity > 0.
In other words, I would like to solve an equation looking like this:
div(J) = 0,
where J is the electric current density [A/m2] which is given by
J = J_ohmic + J_displacement
J = sigma*E + d(D)/dt,
where sigma is the electric conductivity [S/m], E the electric field strength [V/m], and D the electric flux density [As/m2] which is differentiated by the time t [s]. The electric flux density D is given by
D = epsilon*E,
where epsilon is the permittivity in [F/m].
As the electric field strength E can be determined using the electric potential V (E= -grad(V)) the entire equation to be solved can be expressed as:
div( sigma*grad(V) + d(epsilon*grad(V))/dt ) = 0
I was spending quite some time trying to develop a solver that can do that. Unfortunately, without success. The time derivative inside the divergence operation is giving me a hard time. Any help would be very appreciated.
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