On an identity for the volume integral of the square of a vector field

Abstract

A proof is given of the vector identity proposed by Gubarev, Stodolsky and Zakarov that relates the volume integral of the square of a 3-vector field to non-local integrals of the curl and divergence of the field. The identity is applied to the case of the magnetic vector potential and magnetic field of a rotating charged shell. The latter provides a straightforward exercise in the use of the addition theorem of spherical harmonics.

Based upon the paper in Am. J. Phys., but contains also a derivation of the central result that is valid for the scalar product of two different vector fields. Also, a proof is given that the general expression for the vector potential in the Coulomb gauge satisfies the central result. A discussion is given of the requirements on the gauge function. 12 pages pdf

References in corpus (2)

• On the Significance of the Quantity "A Squared"
• Comments on "On vector potential of the Coulomb gauge"

Cited by in corpus (4)

• Role of the non-locality of the vector potential in the Aharonov-Bohm effect
• Does the Helmholtz theorem of vector decomposition apply to the wave fields of electromagnetic radiation?
• Angular momentum of the physical electron
• Comment on "On an identity for the volume integral of the square of a vector field"