- #1

Hak

- 709

- 56

How can we try to show that the solution of the electrostatics problem in the presence of dielectric bodies is unique, i.e., that given real charges, set the potential at 0 at infinity, the polarization of the dielectric, the field outside and the field inside are uniquely determined? This would authorize us to search for "solutions of a particular form" that satisfies the fitting conditions, and once such a solution is found, reassure ourselves that it is precisely the solution sought, as the only one that satisfies these conditions.

Suppose we have a surface ##S_0## with zero potential. We have a ##\rho## distribution of real charges around space, and there is a dielectric of any shape (I call ##S_d## the closed surface enclosing the dielectric. Let us further assume that the dielectric is linear, so it is ##D=\epsilon E##.

The problem to be solved is as follows:

$$\nabla^2 u = -4\pi\rho$$ (1)

$$u(S_0) = 0$$ (2)

$$\epsilon \partial_n u_{int} = \epsilon_0 \partial_n u_{ext}$$ along ##S_d## (3)

(3) expresses the discontinuity of fields along the dielectric shell (these are boundary conditions).

One of the problems is due to the fact that the uniqueness theorem holds only for "##C^2## fields" (it is among the assumptions).

The second problem is that (1) and (2) already give a unique solution, so if I found a solution (1)-(2)-(3) this would have to coincide with the unique solution of (1)-(2) (i.e., it is as if the dielectric is not there).

A professor suggested that I add Dirac deltas to the second member of (1) to express the discontinuity given in (3), but I have no idea how to do that (having to be expressed with deltas here is not a discontinuity of ##u##, but the normal derivative!)

Do you have any ideas for tracing problem (1)-(2)-(3) back to a Poisson problem with Dirichlet or Neumann edge conditions?