Exploring Excess Charge on a Capacitor with Gauss's Theorem

[maburne2]

Hey everyone,
I am working through all my graduate texts a couple problems each simultaneously to begin to see any overlap in physics. Anyhow, I started working on Jackson and the first problem has me using Gauss's theorem to prove any excess charge on a capacitor must exist on the surface. Informally I get it, due to the closed integral-then yes any excess charge is excluded from the charge density encompassed by the capacitor-but is there a rigorous mathematical way of showing this exclusivity? I thought about approaching it using work, i.e. applying a electric field to the capacitor to show that the work done on the charges in the capacitor is zero while the integral for the work done on the excess charge is nonzero.

Thanks for any and all help.

 

[Matterwave]

Isn't this just a consequence of the fact that there cannot be an electric field inside a conductor due to the requirement of electrostatics?

 

[maburne2]

I agree, but I am trying to find a quantitative method of proving this rather than qualitative.

 

[Matterwave]

Well, if you had charge within a surface, you can use Gauss's theorem to prove that there must be an electric field inside the conductor, thereby contradicting your original assumptions. Isn't that proof enough?