Questions: Classification of Boundary Value Problems

Question 1

A physicist is solving Laplace's equation inside a spherical cavity and knows only the surface charge density (not the potential) on every part of the boundary. Which type of boundary condition is this, and what does the uniqueness theorem say about the solution?

• A. Dirichlet; the solution is unique
• B. Neumann; the solution is unique up to an additive constant
• C. Mixed; no unique solution exists without additional constraints
• D. Dirichlet; the solution is unique up to an additive constant

Answer: (B)

Knowing the surface charge density is equivalent to knowing σ = ε₀E_n, the normal component of the electric field, which equals −ε₀(∂V/∂n). This is a Neumann boundary condition — the normal derivative of V is specified. The uniqueness theorem for Neumann problems states that the solution is unique up to an additive constant: you can always add a global constant to V without changing E (since E = −∇V). This ambiguity is physical — the absolute potential is arbitrary; only differences matter.

Question 2

The method of images replaces a physical configuration (a point charge near a grounded conducting plane) with a simpler equivalent (a point charge and its mirror image, no plane). Why is this substitution valid?

• A. Because the image charge produces the same total charge as the original problem
• B. Because the method of images only works when the boundary is an equipotential surface
• C. Because the uniqueness theorem guarantees that any solution satisfying the boundary conditions is the unique physical solution — finding it by any method is sufficient
• D. Because both configurations satisfy Gauss's law globally

Answer: (C)

The uniqueness theorem is exactly what justifies the method of images. The grounded plane requires V = 0 on the plane (Dirichlet condition). The image charge configuration produces V = 0 everywhere on the plane and satisfies Laplace's equation in the region above the plane. Since those are exactly the boundary conditions of the original problem, and since the uniqueness theorem says only one solution can satisfy those conditions, the image configuration must be the right answer. It doesn't matter how you found the solution — any valid solution is the solution.

Question 3

Specifying the normal derivative ∂V/∂n on a closed boundary is equivalent to specifying the potential V itself on that boundary.

• T. True
• F. False

Answer: False

These are two fundamentally different types of boundary conditions with different physical content. A Dirichlet condition specifies V directly — as on a conductor held at a fixed voltage. A Neumann condition specifies ∂V/∂n, which corresponds to the normal electric field and thus to surface charge density. They are not interchangeable: Dirichlet problems have a unique solution, while Neumann problems are unique only up to an additive constant. Knowing the surface charge density tells you about E, not V.

Question 4

The uniqueness theorem justifies solving electrostatic boundary value problems by any means necessary — including clever tricks, symmetry arguments, or images — because once you find a solution satisfying the boundary conditions, you know no other solution exists.

• T. True
• F. False

Answer: True

This is the practical power of the uniqueness theorem. It is proved by assuming two solutions exist, taking their difference (which satisfies Laplace's equation with zero boundary conditions), and showing by energy arguments that this difference must be zero. The conclusion is that any valid solution is the unique physical solution. This is what licenses the method of images, Fourier series methods, and other non-obvious approaches: you don't need to derive the solution from first principles — you just need to find it.

Question 5

A Neumann boundary value problem has a solution that is 'unique up to an additive constant.' What does this mean physically, and why doesn't it affect the ability to compute the electric field?

Model answer: It means any two solutions to a Neumann problem can differ only by a global constant: if V(r) is one solution, then V(r) + C is another for any constant C. Physically, only potential differences matter — the absolute value of the potential at a single point is arbitrary and has no physical meaning. The electric field E = −∇V is unaffected by adding a constant because the gradient of a constant is zero. So the non-uniqueness is physically trivial: both solutions give the same electric field, the same forces, and the same energy differences.

This is why in problems where you only know the surface charge (Neumann), you can still uniquely determine the electric field even though the potential is only determined up to a constant. The ambiguity is absorbed by the freedom to choose a reference point for the zero of potential — a choice that has no physical consequence.