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 :$mathbf\{v\}\; =\; abla\; imes\; mathbf\{A\}.$

If a vector field v admits a vector potential A, then from the equality :$abla\; cdot\; (\; abla\; imes\; mathbf\{A\})\; =\; 0$(divergence of the curl is zero) one obtains:$abla\; cdot\; mathbf\{v\}\; =\; abla\; cdot\; (\; abla\; imes\; mathbf\{A\})\; =\; 0,$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 :$mathbf\{v\}\; :\; mathbb\; R^3\; o\; mathbb\; R^3$be solenoidal vector field which is twice continuously differentiable. Assume that v(x) decreases sufficiently fast as ||x||→∞. Define

:$mathbf\{A\}\; (mathbf\{x\})\; =\; frac\{1\}\{4\; pi\}\; abla\; imes\; int\_\{mathbb\; R^3\}\; frac\{\; mathbf\{v\}\; (mathbf\{y\})\}\{left|mathbf\{x\}\; -mathbf\{y\}\; ight\; ,\; dmathbf\{y\}.$Then, A is a vector potential for v, that is,:$abla\; imes\; mathbf\{A\}\; =mathbf\{v\}.$

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

:$mathbf\{A\}\; +\; abla\; m$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.