Tweede uniciteitsstelling

The second uniqueness theorem states that in a volume comprising several conductors, if the total charge on each conductor and the charge density in the region between the conductors are known, then the electric field can be uniquely determined.

Contrarily, consider that there are two solutions. For the region between the conductors, Gauss's law in the differential form is applied, and for the surface enclosing each conductor, the integral form is applied.

If a third field is defined as the difference between these two fields, then the divergence of the field is observed to be zero. Similarly, the integral form for the third field is also zero.

Consider the divergence of this field and its associated potential. Applying the product rule and rewriting the potential gradient as the field gives the square of the magnitude of the field.

Integrating this expression over the volume and applying the divergence theorem shows that the magnitude of the third field is zero everywhere. This implies that the first two fields are equal, proving the uniqueness of the solution.