Vector potential of a solenoidal field

Abstract

The article discusses examples of determining the vector potential of a solenoidal field, which is defined by the relation F = ∇ × a, where a is the vector potential and ∇ is the Hamiltonian operator, as well as some of its applications. Unlike the case of a potential field, for which determining the scalar potential u = u(x, y, z) is related to the path-independence of the corresponding line integral — a well-studied issue — the determination of a vector potential is significantly more complex and requires solving an appropriate system of first-order linear partial differential equations. It is noted that such a potential is defined up to the gradient of an arbitrary scalar function. The possibility of decomposing an arbitrary vector field into the sum of its potential and solenoidal components is also demonstrated. Examples of determining the vector potential for solenoidal and harmonic fields are provided. The obtained results can be applied to solving problems in electromagnetic field theory, mathematical physics, and applied mathematics.