Second Uniqueness Theorem

The second uniqueness theorem states that if the total charge on each conductor and the charge density in the region between conductors are known, the electric field can be uniquely determined. This means there is only one possible electric field configuration that satisfies these boundary conditions, ensuring a unique solution to electrostatic problems involving multiple conductors.

The proof applies Gauss's law in differential form to the region between conductors and in integral form to surfaces enclosing each conductor. By assuming two different electric field solutions exist and defining a third field as their difference, the divergence of this third field becomes zero, leading to the conclusion that both original fields must be identical.

The divergence theorem converts a volume integral of the square of the electric field magnitude into a surface integral. Since the surface integral of the third field equals zero at conductor boundaries, the volume integral must also be zero, proving the third field magnitude is zero everywhere and confirming the two original fields are identical.

Defining a third field as the difference between two assumed solutions allows the proof to show this difference field has zero divergence and zero surface integral. This mathematical approach demonstrates that if two solutions existed, they would be identical, establishing uniqueness by contradiction and confirming only one electric field solution is possible.

To uniquely determine the electric field, you must know the total charge on each conductor and the charge density in the region between the conductors. These boundary conditions and source specifications are sufficient to guarantee a unique electric field solution throughout the entire volume, eliminating ambiguity in electrostatic calculations.

The product rule is applied to the divergence of the third field multiplied by its associated potential. By rewriting the potential gradient as the electric field, this yields the square of the field magnitude. Integrating this expression over the volume and applying the divergence theorem demonstrates the field magnitude must be zero everywhere.

When the magnitude of the third field is zero everywhere, it means the difference between the two assumed electric field solutions is zero at every point in the volume. This proves the two solutions are identical, establishing that the electric field is uniquely determined by the given boundary conditions and charge distribution.