Poisson’s equation and the promise of uniqueness

In electrostatics the electric field is conservative and can be written as the negative gradient of a scalar potential. Combining this statement with Gauss’s law produces Poisson’s equation. Positive charge gives negative potential curvature when summed over all coordinate directions. A positive second derivative in one direction does not alone imply negative charge, because the other directions also contribute. Reducing the Laplacian to one dimension therefore requires translational symmetry, not merely choosing an axis.

A particular solution accounts for the specified volume charge. Adding any harmonic function leaves that volume charge unchanged, while changing the field and usually the boundary potential. A uniform applied field is a simple example of this freedom. Thus the Coulomb integral over charges inside a finite region is generally incomplete: external electrodes contribute a harmonic correction inside that region. The correction encodes the apparatus rather than an additional interior source.

Suppose two sufficiently regular potentials solve the same Poisson problem in one bounded, connected region, with equal potential everywhere on its boundary. Their difference has zero Laplacian and zero boundary value. Multiply its Laplace equation by the difference itself and integrate by parts. The remaining volume integral is the squared magnitude of its gradient. Every contribution is nonnegative, so cancellation cannot conceal a nonzero difference field.

The boundary term vanishes because the difference potential is zero there. Consequently its gradient vanishes throughout the connected region, making the difference constant. The boundary condition fixes that constant to zero. Connectedness matters: a constant could otherwise be chosen separately in each disconnected component. For point sources one compares solutions with the same singularities and applies the argument to their regular difference, rather than integrating divergent point self-energies.

Uniqueness is conditional: it says at most one solution satisfies these requirements. It does not construct that solution or establish that arbitrary inconsistent data admit one. Likewise, checking boundary values only at a few points is insufficient. Exterior problems also require an admissible condition at infinity; otherwise a growing harmonic field can slip into a proposed answer. These distinctions make uniqueness useful when validating images, separated series, or numerical approximations.

Inspect the field lab as a diagnostic of the conservative-field assumption. Reset and set source strength to 1 and rotation to 0; predict radial arrows. For a fresh comparison, Reset and set source strength to 1 and rotation to 1; predict an added tangential component. Rotation introduces circulation and disqualifies the field from this electrostatic gradient argument. The softened planar model neither solves three-dimensional Poisson boundary data nor proves uniqueness.