# Uniqueness given specified surface charges and voltages

• komdu
In summary, the author is thinking about a boundary value problem for voltage that involves two conductors with different surface charges. One side of the boundary has a voltage value and the other side has a surface charge. There are uniqueness theorems that allow for this problem to be solved. However, one of the potentials that results does not satisfy the boundary conditions.
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.

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

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?

## 1. What is uniqueness in the context of specified surface charges and voltages?

Uniqueness refers to the concept that for a given set of specified surface charges and voltages, there is only one possible solution for the resulting electric field and potential distribution. This means that the electric field and potential at any point on the surface can be determined uniquely based on the given surface charges and voltages.

## 2. How is uniqueness determined in a system with specified surface charges and voltages?

Uniqueness is determined by solving the governing equations of electrostatics, namely Gauss's law and the Laplace equation, for the given boundary conditions of the specified surface charges and voltages. The resulting solution will be unique, assuming the system has well-defined boundary conditions and no singularities.

## 3. What role do surface charges and voltages play in determining uniqueness?

Surface charges and voltages serve as boundary conditions for the electrostatic system, and they are necessary for determining uniqueness. Without these specified values, there would be an infinite number of possible solutions for the electric field and potential distribution.

## 4. Can uniqueness be violated in a system with specified surface charges and voltages?

Yes, uniqueness can be violated in certain cases, such as when there are overlapping conductors or when the system contains a singularity. In these situations, there may be multiple solutions for the electric field and potential distribution, and uniqueness cannot be guaranteed.

## 5. How does the concept of uniqueness apply to practical applications?

In practical applications, uniqueness is important because it allows us to accurately predict and control the behavior of electric fields and potentials in a given system. This is crucial in fields such as electrical engineering, where the precise manipulation of electric fields is necessary for the functioning of devices and systems.

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