- #1

ELB27

- 117

- 15

## Homework Statement

Prove that the field is uniquely determined when the charge density ##\rho## is given and

*either*##V##

*or*the normal derivative ##\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

Poisson's equation which the potential ##V## must satisfy:

[tex]\nabla^2V = -\frac{\rho}{\epsilon_0}[/tex]

Gauss' law in differential form:

[tex]\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}[/tex]

The definition of the potential:

[tex]\vec{E} = -\nabla V[/tex]

Perhaps various product rules of vector calculus will also be useful.

## The Attempt at a Solution

I must admit that I'm not sure about the strategy I should use in this problem so I just played around with the equations trying to get some insight and using the proofs given for the two classical uniqueness theorems as models.

First, suppose there are

*two*fields that satisfy the conditions of the problem:

[tex] \vec{E_1} = \frac{\rho}{\epsilon_0} ; \vec{E_2} = \frac{\rho}{\epsilon_0}[/tex]

Looking at their difference ##\vec{E_3} = \vec{E_1} - \vec{E_2}##, it must satisfy ##\nabla \cdot \vec{E_3} = 0## or in integral form: ##\oint \vec{E_3}\cdot d\vec{a} = 0## over each boundary surface. Looking at the potential difference ##V_3 = V_1 - V_2##, it must satisfy ##V_3 = 0## on the boundary

*or*##\partial V_3/\partial n = 0## on the boundary.

In addition, ##\nabla^2V = 0## thus ##V_3## satisfies Laplace's equation and must have it's minima or maxima on the boundary.

Now I don't have any idea on how to proceed from here. In particular, I feel that the normal derivative can be exploited somehow but since I can't assume that the surfaces are conductors, I don't know how. I would highly appreciate any insight on this problem as well as general tips for going about such a problem in the future.

Thanks very much in advance.

**EDIT:**After thinking a bit more, if ##V## is given then it follows from the above that ##V_3 = 0## in the region (since it is zero on the boundary and has both maxima and minima zero there). If this reasoning is correct, the only thing left is to prove the case where ##\partial V/\partial n## is specified but ##V## is not.

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