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 negative gradient is a given vector field.

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

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

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.

An interesting question is then if any solenoidal vector field admits a vector potential. The answer is affirmative, if the vector field satisfies certain conditions.

Theorem

Let

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

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

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

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.

Nonuniqueness

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.

See also

• Fundamental theorem of vector analysis
• Magnetic potential
• Solenoid

References

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