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Wikipedia states the macroscopic Gauss' law as $\nabla \cdot \overrightarrow{D} = \rho_f$, where $\rho_f$ is the charge density of free charge carriers.

I understand that conducting electrons in a metal are considered free charge carriers, while the dipoles in a dielectric medium, for example induced by an external electric field, are considered bound charge carriers. This way, the fluctuations of those microscopic dipoles which average out when a volume of a couple atoms is considered and the macroscopic charge density goes to $0$.

But what about bound charge carriers where the charge density doesn't average out to $0$? A medium containing predominantly negative ions, which are not mobile but bound, wouldn't necessarily be counted as having free charges. The charges are physically there tho, and should certainly produce an electric field. Are those ions somehow considered "free charges", even tho they couldn't travel the medium? Or am I fundamentally missing something here?

Thanks much in advance!

Frikandel
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2 Answers2

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A few things to consider usually you are essentially averaging the properties of the material when you consider the Macroscopic point of few. So if you had a few charges trapped in random in a material you could choose to treat most of the material with the bulk property and the use superposition if you knew the location of the charges.

But you are correct you could have an amount of bound charge so the the whole material has a net positive or negative charge. This is sometimes called an electret. For example could take a insulating material and hit it with an electron beam. Then you could have a net electric field from the bound charge.

The entry for Electric Displacement Field has a short proof showing why. Essentially you are burying the physics in the Polarization. If you changed the distribution or amount of bound charge then you would have to change the polarization P.

UVphoton
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In a metal conductor both the negatively charged valence electrons and the positively charged “lattice” are considered free charge. The question is not one of charge mobility as the electrons are mobile and the lattice is not. The question is whether it polarizes or not. Materials that have a polarization have bound charges, and materials that do not polarize have free charges even if the charges are not mobile.

Dale
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