The bound-charge current
Wherever the polarization in matter changes with time, there is an electric current, a genuine motion of charge. Suppose there are N dipoles per cubic meter of dielectric, and that in the time interval dt each changes from p to p + dp. Then the macroscopic polarization density P changes from P = Np to P + dP = N(p + dp). Suppose the change dp was effected by moving a charge q through a distance ds, in each atom: q ds = dp. Then during the time dt there was actually a charge cloud of density ρ = Nq, moving with velocity v = ds/dt. This is a conduction current of a certain density J in coulombs per second per square meter:
J = ρv = Nq ds/dt = N dp/dt ⇒ J = dP/dt (10.74)
The connection between rate of change of polarization and current density, J = dP/dt, is independent of the details of the model. A changing polarization is a conduction current, not essentially different from any other. Note that if we take the divergence of both sides of Eq. (10.74) and use Eq. (10.61), we obtain div J = d(div P)/dt = −dρ_bound/dt, which is consistent with the continuity equation in Eq. (4.10).
Naturally, such a current is a source of magnetic field. If there are no other currents around, we should write Maxwell's equation, curl B = μ_0 ϵ_0 (∂E/∂t) + μ_0 J, as
curl B = μ_0 ϵ_0 ∂E/∂t + μ_0 ∂P/∂t. (10.75)
506 Electric fields in matter
The only difference between an "ordinary" conduction current density and the current density ∂P/∂t is that one involves free charge in motion, the other bound charge in motion. There is one rather obvious practical distinction—you can't have a steady bound-charge current, one that goes on forever unchanged. Usually we prefer to keep account separately of the bound-charge current and the free-charge current, retaining J as the symbol for the free-charge current density only. Then to include all the currents in Maxwell's equation we have to write it this way:
curl B = μ_0 ϵ_0 ∂E/∂t + μ_0 ∂P/∂t + μ_0 J. (10.76)
bound-charge free-charge
current density current density
In a linear dielectric medium, we can write ϵ_0 E + P = ϵ E, allowing a shorter version of Eq. (10.76):
curl B = μϵ∂E/∂t + μ_0 J. (10.77)
More generally, Eq. (10.76) can also be abbreviated by introducing the vector D, previously defined in any medium as ϵ_0 E + P (which reduces to ϵ E in a linear dielectric):
curl B = μ_0 ∂D/∂t + μ_0 J (10.78)
The term ∂D/∂t is usually referred to as the displacement current. Actually, the part of it that involves ∂P/∂t represents, as we have seen, an honest conduction current, real charges in motion. The only part of the total current density that is not simply charge in motion is the ∂E/∂t part, the true vacuum displacement current which we discussed in Chapter 9.
Incidentally, if we want to express all components of the full current density in units corresponding to those of J, we can factor out the μ_0 and write Eq. (10.76) as follows:
curl B = μ_0 (ϵ_0 ∂E/∂t + ∂P/∂t + J). (10.79)
vacuum bound- free-
displacement charge charge
current current current
density density density
Involved in the distinction between bound charge and free charge is a question we haven't squarely faced: can one always identify unambiguously the "molecular dipole moments" in matter, especially solid matter? The answer is no. Let us take a microscopic view of a thin wafer of sodium chloride crystal. The arrangement of the positive sodium ions