# Uniqueness given specified surface charges and voltages

1. May 13, 2014

### komdu

Suppose we have a collection of conductors for which the voltage is specified on some conductors and the surface charge is specified on others. Is there a coherent way to specify this as a boundary value problem for the voltage (satisfying Laplace's, or in the presence of charge density, Poisson's equation). Note that here I am only interested in electrostatics.

My first thought is that this corresponds to a Dirichlet BC on parts of the boundary and a Von Neumann BC on other parts of the boundary (since the normal derivative of voltage is controlled by the surface charge). I gather from reading online that even with these "mixed" conditions, Laplace's equation satisfies some uniqueness theorems.

Thinking in this way seems like a neat, principled approach to solving many problems, but what if both sides of a metal plate are involved as BCs? Suppose we have an infinite metal plate with surface charge sigma and V --> 0 far away. The normal derivative is now specified on both sides of the plate as dV/dn = sigma/e_0, which appears to give twice the field strength derived from Gauss' law. Is it that really we should model the plate as two plates close together each with half the charge? This seems like an "ad hoc", unsatisfying solution.

To give a concrete example, suppose we have two infinite plates, one with surface charge \sigma and the other held at constant voltage V=0. I can think of two ways to satisfy these boundary conditions: (a) the V=0 plate acquires a surface charge -\sigma, producing a field of magnitude \sigma/e_0 between the plates (apparently what really happens) and zero elsewhere. Or (b) the V=0 plate remains electrically neutral, a field of magnitude \sigma/(2 e_0) fills all space (changing direction at the other plate). Both seem to satisfy the condition on dV/dn at the charged plate, and by adding constants to the potential, can satisfy V=0 also. How can this be?

I'm not interested in alternative ways to derive the answer, using Gauss' law, etc... I'm interested specifically in whether there is a way to formulate these kinds of questions as BVP for which we can exploit a uniqueness theorem.

2. May 13, 2014

### UltrafastPED

I believe that this is gone over in some detail in Griffith's Introduction to Electrodynamics - you might take a look.

3. May 13, 2014

### komdu

Yes, I've looked there. In Chapter 3: Special Techniques, Griffith's states and proves uniqueness theorems both for specified voltage and specified surface charge (although in the latter case only for electric field as potential only unique up to constants). The mixed case is stated to be true and left as an exercise.

But in this case, one of the two fields (a) and (b) (and therefore potentials) I've described above fails to satisfy the boundary conditions. I can't tell which one. Something confuses me perhaps because the normal derivative of the potential is discontinuous at surface charges?