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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.

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

Consequence

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If a vector field admits a vector potential , then from the equality (divergence of the curl is zero) one obtains which implies that must be a solenoidal vector field.

Theorem

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Let be a solenoidal vector field which is twice continuously differentiable. Assume that decreases at least as fast as for . Define where denotes curl with respect to variable . Then is a vector potential for . That is,

The integral domain can be restricted to any simply connected region . That is, also is a vector potential of , where

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

By analogy with the Biot-Savart law, also qualifies as a vector potential for , where

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Substituting (current density) for and (H-field) for , yields the Biot-Savart law.

Let be a star domain centered at the point , where . Applying Poincaré's lemma for differential forms to vector fields, then also is a vector potential for , where

Nonuniqueness

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

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References

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