Vector potential
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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
If a vector field v admits a vector potential A, then from the equality
(divergence of the curl is zero) one obtains
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 potential satisfies certain conditions.
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[edit] Theorem
Let
be solenoidal vector field which is twice continuously differentiable. Assume that v(x) decreases sufficiently fast as ||x||→∞. Define
Then, A is a vector potential for v, that is,
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.
[edit] Nonuniqueness
The vector potential admitted by a solenoidal field is not unique. If A is a vector potential for v, then so is
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, and is referred to as choosing a gauge.
[edit] See also
[edit] References
- Fundamentals of Engineering Electromagnetics by David K. Cheng, Addison-Wesley, 1993.
