Charge density definition

CrusaderSean

silly question i guess, but is charge density defined only for stationary charges?

i'm asking this because I did a couple homework problems on finding electric field in linear dielectric material. Dielectric is between two surfaces held at constant potential difference (ie. parallel plate, concentric sphere or cylinder.. well neglect fringe effects). From what I understand, there is no charge (free or induced) density in the dielectric material because they only occur on boundary between conductor and dielectric. So I solved the problems using Laplace's equation (instead of Poisson). I found there is constant current going through the dielectric material. Since current is flux of charges, there must be charges moving through dielectric. Those charges aren't in the charge density definition because they're acting as current or some other reason? Perhaps i'm confusing some issues here...

dextercioby

[itex]\rho_{el}\left(\vec{r},t\right) [/itex] describes the volumic density of electric charge at the point [itex] \vec{r} [/itex] at the moment "t",no matter if the charge is moving or not...

Daniel.

CrusaderSean

my class is only on electrostatics.. so the [tex]\rho[/tex] in the equations below are static charges only?
[tex]\nabla{^2} \cdot V = \frac{-\rho}{\epsilon}[/tex]
[tex]\nabla \cdot E = \frac{\rho}{\epsilon}[/tex]

dextercioby

Yeah.Electrostatics means static fields created,obviously by time independent electric charge densities.

Daniel.

SpaceTiger

Those equations apply to moving charges as well. Both of them are equivalent to one of Maxwell's Equations. In the presence of moving charge, however, that won't be enough to determine the electric field at a given point because it will have a non-zero curl.

dextercioby

Nope,for moving charges (in vacuum) we have

[tex] \square V\left(\vec{r},t\right)=-\frac{\rho\left(\vec{r},t\right)}{\epsilon_{0}} [/tex]

Daniel.

CrusaderSean

i thought those equations were general (for static and dynamic) as well... guess i was wrong.

dextercioby

Gauss's law is the same (in mathematical form,not as functional dependence of the quantities involved) both for static & dynamic description.

The potential's equation is diff,however...Poisson vs.d'Alembert...

Daniel.

SpaceTiger

You only need worry yourself about such things if your charges are moving relativistically, but dexter's equation is more general.

dextercioby

That potential equation i posted is valid in the Coulomb gauge.Typically relativistic...

Daniel.

Meir Achuz

An electric current can occcur with a zero charge density.
The current usually consists of negative electrons moving through the material,
while the positive charges remain fixed.
The density of negative charges equals that of the positive charges in the material.
Think of a hall with 100 men and 100 women in it. If the men start walking, but the women remain standing. The density of people will remain the same, but men will pile u at one end of the hall. There could be a charge buildup if the men walk at different rates. That would be described by the continutity eqation.