Vector potential explained

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

v , a vector potential is a C ^{2} vector field A such that $$\backslash mathbf\; =\; \backslash nabla\; \backslash times\; \backslash mathbf.$$

Consequence

If a vector field

v admits a vector potential A , then from the equality $$\backslash nabla\; \backslash cdot\; (\backslash nabla\; \backslash times\; \backslash mathbf)\; =\; 0$$(divergence of the curl is zero) one obtains$$\backslash nabla\; \backslash cdot\; \backslash mathbf\; =\; \backslash nabla\; \backslash cdot\; (\backslash nabla\; \backslash times\; \backslash mathbf)\; =\; 0,$$which implies that v must be a solenoidal vector field.

Theorem

Let$$\backslash mathbf\; :\; \backslash R^3\; \backslash to\; \backslash R^3$$be a solenoidal vector field which is twice continuously differentiable. Assume that

v ( x ) decreases at least as fast as 1 /\| x \| for \| x \| \to inf ty . Define$$\backslash mathbf\; (\backslash mathbf)\; =\; \backslash frac\; \backslash int\_\; \backslash frac\; \backslash ,\; d^3\backslash mathbf$$where \nabla _{y} x denotes curl with respect to variable y . Then A is a vector potential for v . That is,$$\backslash nabla\; \backslash times\; \backslash mathbf\; =\backslash mathbf.$$

The integral domain can be restricted to any simply connected region

\Omega . That is, A' also is a vector potential of v , where$$\backslash mathbf\; (\backslash mathbf)\; =\; \backslash frac\; \backslash int\_\; \backslash frac\; \backslash ,\; d^3\backslash mathbf.$$

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,

A'' ( x ) also qualifies as a vector potential for v , whereA'' ( x ) = \int _{\Omega} (v ( y ) x ( x - y ))/(4 \pi | x - y | ^{3}) d ^{3} y .

Substituting

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

Let

\Omega be a star domain centered at the point p , where p \in \R ^{3} . Applying Poincaré's lemma for differential forms to vector fields, then A''' ( x ) also is a vector potential for v , whereA''' ( x ) = \int ^{1} _{0} s (( x - p ) x ( v ( s x + ( 1- s) p )) ds

Nonuniqueness

The vector potential admitted by a solenoidal field is not unique. If

A is a vector potential for v , then so is$$\backslash mathbf\; +\; \backslash nabla\; f,$$where f 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 calculus
• Magnetic vector potential
• Solenoidal vector field
• Closed and Exact Differential Forms

References

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