Vector potential

In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose gradient is a given vector field.

Formally, given a vector field v, a vector potential is a vector field A such that

\mathbf {v} =\nabla \times \mathbf {A} .

Consequence

If a vector field v admits a vector potential A, then from the equality

\nabla \cdot (\nabla \times \mathbf {A} )=0

(divergence of the curl is zero) one obtains

\nabla \cdot \mathbf {v} =\nabla \cdot (\nabla \times \mathbf {A} )=0,

which implies that v must be a solenoidal vector field.

Let

\mathbf {v} :\mathbb {R} ^{3}\to \mathbb {R} ^{3}

be a solenoidal vector field which is twice continuously differentiable. Assume that v(x) decreases sufficiently fast as ||x||→∞. Define

\mathbf {A} (\mathbf {x} )={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}{\frac {\nabla _{y}\times \mathbf {v} (\mathbf {y} )}{\left\|\mathbf {x} -\mathbf {y} \right\|}}\,d^{3}\mathbf {y} .

Then, A is a vector potential for v, that is,

\nabla \times \mathbf {A} =\mathbf {v} .

A generalization of this theorem is the Helmholtz decomposition which states that any vector field can be decomposed as a sum of a solenoidal vector field and an irrotational vector field.

The vector potential admitted by a solenoidal field is not unique. If A is a vector potential for v, then so is

\mathbf {A} +\nabla m

where m is any continuously differentiable scalar function. This follows from the fact that the curl of the gradient is zero.

This nonuniqueness leads to a degree of freedom in the formulation of electrodynamics, or gauge freedom, and requires choosing a gauge.

Fundamentals of Engineering Electromagnetics by David K. Cheng, Addison-Wesley, 1993.

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