Calculation of Magnetic and Electric Fields from Displacement Currents

We calculate the magnetic induction from the integral form of the Biot-Savart law, $B = μ0/4π∫(J+Ḋ) × rdV/r3$⁠. In the quasistatic case, using only the scalar electric potential, we transform the volume integral of $Ḋ$ into an inner and an outer surface integral, both of which vanish except for a contribution from the polarization current in dielectric materials. Hence the induction is calculable from the sum of the conduction- and polarization-current densities alone. This result has no bearing on the calculation of the induction from Ampere's law, where we must use the entire displacement-current density. In the converse problem of an induced electric field there is no magnetic conduction current, and the quasistatic magnetic displacement current arises from a changing electric current in a closed circuit. We express H through the scalar magnetic potential, which is discontinuous on a surface bounded by the electric circuit. We obtain the induced electric field from a volume integral like the Biot-Savart integral above, and as before we transform it into surface integrals. There are two nonvanishing contributions. One is over both sides of the surface which is bounded by the electric circuit; it gives an expression for the electric field which we commonly compute from the vector potential. The second contribution reappears as a volume integral of magnetization currents in magnetic material.