Introduction to Electrodynamics · Problem 5.56

Magnetic Fields in Matter — David J. Griffiths

Physics · uniqueness-magnetostatics

Problem

Prove the following uniqueness theorem: If the current density

\mathbf{J}

is specified throughout a volume

\mathcal{V}

, and either the potential

\mathbf{A}

or the magnetic field

\mathbf{B}

is specified on the surface

S

bounding

\mathcal{V}

, then the magnetic field itself is uniquely determined throughout

\mathcal{V}

. [Hint: First use the divergence theorem to show that

\int\{(\nabla \times \mathbf{U})\cdot(\nabla \times \mathbf{V}) - \mathbf{U}\cdot[\nabla \times (\nabla \times \mathbf{V})]\}\,d\tau = \oint[\mathbf{U} \times (\nabla \times \mathbf{V})]\cdot d\mathbf{a},

for arbitrary vector functions

\mathbf{U}

and

\mathbf{V}