Supriya Ghosh (Editor)

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 gradient is a given vector field.

Contents

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

v = × A .

Consequence

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

( × A ) = 0

(divergence of the curl is zero) one obtains

v = ( × A ) = 0 ,

which implies that v must be a solenoidal vector field.

Theorem

Let

v : R 3 R 3

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

A ( x ) = 1 4 π R 3 y × v ( y ) x y d 3 y .

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

× A = 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

A + 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.

References

Vector potential Wikipedia


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