# Plugging $j= \sigma E$ into Maxwell's equations, but $q = 0$?

In order to derive the dispersion relation for a dielectric material (including the dielectric constant $$\epsilon$$ and the conductivity $$\sigma$$), one starts with the macroscopic maxwell equation, i.e. the maxwell's equations for a medium. One assumes the linear relation $$D = \epsilon_{0} \epsilon E$$, as well as the linear relation between current density and applied electric field $$j= \sigma E$$. The last equation is plugged into one of the macroscopic maxwell equations. $$j$$ is the free current inside the medium? There is also current from bound electrons, that is hidden in the Magnetization. However if we have a free current, should we then not also have free charges (because of the continuity equation)?

For all the derivations I am looking at, the free current is proportional to the electric field, however they assume no free charges?

Is it because of the positively charged nuclei in the solid giving a net zero charge?

• You can extend the discussion and include free charges, but you can have free currents without free charges, as long as $\nabla \cdot \vec j = 0$, so as long as all your currents flow in loops. Jan 28, 2022 at 20:06
• Considering an infinite crystal and the situation I described in the question. Divergence of $j$ being zero would mean a constant uniform electron cloud moving in the direction of applied electric field on a uniformly positively charged background, right? Jan 28, 2022 at 21:42
• @Newbie I never said it implies no free charges. My point was that divergence free, free currents are compatible with no (net) free charges. Jan 29, 2022 at 12:12
• @Hans-UlrichRudel No, divergence free just means that no free charge carriers are accumulated or depleted – that is the flow has now sources or sinks. (Like the velocity field of an incompressible fluid). So your electrons can arbitrarily move in the positively charged background as long as their density does not change. Your situation is a possible one for divergence free flow, but only if the conductor extends to infinity (otherwise charges would accumulate/deplete on the surfaces). Jan 29, 2022 at 14:14
• Why is it cleary wrong, that the electric field in a wave has zero divergence? The wave is transverse. Jan 30, 2022 at 13:52

Recall that Maxwell’s equations are linear. So in a metal you have the conduction electrons that have a negative charge density and a current density and in the lattice you have metal ions that have a positive charge density and no current density. The total charge and current density is the sum of those two, resulting in no charge density and a current density.

Now, you appear to be a bit confused about free vs bound charges. Remember that bound charges are associated with materials that become electrically polarized. So the metal lattice is a free charge, not a bound charge. The metal doesn’t become polarized in the sense of a bound charge. Some people erroneously think that the lattice should be a bound charge because it doesn’t move, but that isn’t the important part.

To further emphasize that point, recall that free charge is conserved. When you make a free negative charge on a piece of metal via electrostatic induction, it also produces a region of positive charge. That positive charge is necessarily a free charge since the negative charge is, and the positive charge consists of the metal lattice.

• In your first paragraph, you claim that there is no charge density, which is only true if one averages over one or several lattice sites. Thanks for the clarification about the free charges. Jan 30, 2022 at 12:37
• The goal of the derivation I'm interested, is to obtain a wave equation for the electric field inside the material. However with the linear relation between electric field and current density, the charge neutrality is no longer guaranteed at every point, because the divergence of the electric field is no longer zero at certain points. Thereforce the current density has non zero divergence, giving a non zero time derivative of the charge density (continuity eq.), which will make the charge density non zero at certain times. A contradiction to the inital assumptions. So it is not that easy. Jan 30, 2022 at 12:49
• @Hans-UlrichRudel certainly charge density is not guaranteed to be zero. But charge density can be zero even when current density is non zero. That was the question. As you say, for determining changes in charge density the important thing is the divergence of the current density, not the current density itself.
– Dale
Jan 30, 2022 at 13:00
• What you are asking here in the comments appears to be a different question than what you asked in the question. You should open a new question. Comments are not intended for further questions or discussion
– Dale
Jan 30, 2022 at 13:49
• I just wanted to correct my last comment: As @Sebastian Riese said in another comment, we can and maybe even must have zero divergence for a solution to the wave equation. The electric field of a plane wave has zero divergence. So there should be no need to define new bound charges and currents. Jan 30, 2022 at 14:43