Three Dimensional Charge Density in Capacitor Plates

[Conservation]

Hello everyone,

I was wondering how one would calculate the three-dimensional charge density (per volume) in a capacitor plate with a given thickness. I know how to calculate the charge density on the surface based on the capacitance and voltage, but how would one calculate the comprehensive charge density that also incorporates the charge distribution in the direction of the thickness?

Thanks.

 

[Born2bwire]

Well, the charge will reside only on the surface of the plate, so the thickness is more or less immaterial. You can model the plate as a two dimensional plate then. You can then use various computational techniques to calculate the charge distribution due to a known voltage. One method is the Method of Moments and the case of the charge on a plate at uniform voltage is solved in Harrington's text, "Method of Moments." Balanis' "Advanced Engineering Electromagnetics" discusses integral methods. While he does not do the plate problem, one can learn enough to knock out a code in Matlab for example.

 

[Conservation]

I see. Thanks for a detailed answer for the calculation process.

However, does the charge truly reside only on the surface? I mean, there are still electrons in regions that are not the surface directly in contact with the dielectric which would be attracted to the opposite plate, causing an uneven electron distribution, right?

 

[Born2bwire]

Electrostatics is concerned with the steady state solution. So if there was any net electric field on the interior of the conducting plate, then it would simply whisk away the charges as a result. The only way that we reach a steady state is having no net electric field on the interior of a conductor. Or in other words, the voltage difference between any two interior points is zero. For this to be true, then there cannot be any net charge in the interior.

So regardless of what configuration of charge we can think of, when we let them move about in response to each other they will always end up with a net charge only on the surface of conductors.

Now electrodynamics can allow for a net charge density to appear in the interior of imperfect conductors, like real world copper. In this case, we would have to model a capacitor that has a time-varying voltage across it and the charges would permeate into the interior based upon the conductivity of the metal and the frequency of the time-varying fields. But to solve that case we have to use a more complicated set of equations, but we can still use the Method of Moments as the solver.