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

 .

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.