Is the Electric Field Unique with Given Charge Density and Boundary Conditions?

[ehrenfest]

[SOLVED] Griffiths Problem 3.4

Homework Statement

Prove that the field is uniquelly determined when the charge density rho is given and either V or the normal derivative $\frac{\partial{V}}{\partial{n}}$ is specified on each boundary surface. Do not assume the boundaries are conductors, or that V is constant over any given surface.

Homework Equations

The Attempt at a Solution

If you know V on the surface, this is the Corollary to the First Uniqueness Theorem. I don't see how knowing the normal derivative, $\frac{\partial{V}}{\partial{n}}= \nabla{V}\cdot \hat{n}$, helps at all.

 

[vincebs]

My guess: probably one of those Stoke's theorems (curl, divergence, gradient) will simplify things.

 

[ehrenfest]

The divergence theorem and Poisson's equation tell us that $$\int_{S} \nabla V \cdot \hat{n} da = \int_{V}\nabla ^2V d\tau = \int_{V}\rho/\epsilon_0 d\tau$$. We know rho, so we didn't even need the normal derivative to get that integral. Thus, I don't see how the curl, divergence, gradient theorems will help.

 

[ehrenfest]

anyone?

 

[ehrenfest]

Never mind. I got it.