Vector potential

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 $𝐯 = \nabla \times 𝐀 .$

Consequence

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

Theorem

Let $𝐯 : ℝ^{3} \to ℝ^{3}$ be a solenoidal vector field which is twice continuously differentiable. Assume that $𝐯 (𝐱)$ decreases at least as fast as $1 / ‖ 𝐱 ‖$ for $‖ 𝐱 ‖ \to \infty$ . Define $𝐀 (𝐱) = \frac{1}{4 π} \int_{ℝ^{3}} \frac{\nabla_{𝐬} \times 𝐯 (𝐬)}{‖ 𝐱 - 𝐬 ‖} d^{3} 𝐬$ where $\nabla_{𝐬} \times$ denotes curl with respect to variable $𝐬$ . Then $𝐀$ is a vector potential for $𝐯$ . That is, $\nabla \times 𝐀 = 𝐯 .$

The integral domain can be restricted to any simply connected region $Ω$ . That is, $𝐀^{'}$ also is a vector potential of $𝐯$ , where $𝐀^{'} (𝐱) = \frac{1}{4 π} \int_{Ω} \frac{\nabla_{𝐬} \times 𝐯 (𝐬)}{‖ 𝐱 - 𝐬 ‖} d^{3} 𝐬 .$

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 $𝐯$ , where

$𝐀^{″} (𝐱) = \int_{Ω} \frac{𝐯 (𝐬) \times (𝐱 - 𝐬)}{4 π {| 𝐱 - 𝐬 |}^{3}} d^{3} 𝐬$

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

Let $Ω$ 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 $𝐯$ , where

$𝐀^{‴} (𝐱) = \int_{0}^{1} s [(𝐱 - 𝐩) \times 𝐯 (s 𝐱 + (1 - s) 𝐩)] d s$

Nonuniqueness

The vector potential admitted by a solenoidal field is not unique. If $𝐀$ is a vector potential for $𝐯$ , then so is $𝐀 + \nabla f,$ 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.

References

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

Consequence

Theorem

Nonuniqueness

See also

References