Vector potential

 ( Concepts In Physics ) 

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

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

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Theorem Let $$\backslash mathbf\{v\}\; :\; \backslash R^3\; \backslash to\; \backslash R^3$$ be a solenoidal vector field which is twice smooth function. Assume that $\backslash mathbf\{v\}(\backslash mathbf\{x\})$ decreases at least as fast as $1/\backslash |\backslash mathbf\{x\}\backslash |$ for $\backslash |\; \backslash mathbf\{x\}\backslash |\; \backslash to\; \backslash infty$. Define $$\backslash mathbf\{A\}\; (\backslash mathbf\{x\})\; =\; \backslash frac\{1\}\{4\; \backslash pi\}\; \backslash int\_\{\backslash mathbb\; R^3\}\; \backslash frac\{\; \backslash nabla\_y\; \backslash times\; \backslash mathbf\{v\}\; (\backslash mathbf\{y\})\}\{\backslash left\backslash |\backslash mathbf\{x\}\; -\backslash mathbf\{y\}\; \backslash right\backslash |\}\; \backslash ,\; d^3\backslash mathbf\{y\}$$ where $\backslash nabla\_y\; \backslash times$ denotes curl with respect to variable $\backslash mathbf\{y\}$. Then $\backslash mathbf\{A\}$ is a vector potential for $\backslash mathbf\{v\}$. That is, $$\backslash nabla\; \backslash times\; \backslash mathbf\{A\}\; =\backslash mathbf\{v\}.$$

The integral domain can be restricted to any simply connected region $\backslash mathbf\{\backslash Omega\}$. That is, $\backslash mathbf\{A\text{'}\}$ also is a vector potential of $\backslash mathbf\{v\}$, where $$\backslash mathbf\{A\text{'}\}\; (\backslash mathbf\{x\})\; =\; \backslash frac\{1\}\{4\; \backslash pi\}\; \backslash int\_\{\backslash Omega\}\; \backslash frac\{\; \backslash nabla\_y\; \backslash times\; \backslash mathbf\{v\}\; (\backslash mathbf\{y\})\}\{\backslash left\backslash |\backslash mathbf\{x\}\; -\backslash mathbf\{y\}\; \backslash right\backslash |\}\; \backslash ,\; d^3\backslash mathbf\{y\}.$$

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, $\backslash mathbf\{A\text{'}\text{'}\}(\backslash mathbf\{x\})$ also qualifies as a vector potential for $\backslash mathbf\{v\}$, where

$\backslash mathbf\{A\text{'}\text{'}\}(\backslash mathbf\{x\})\; =\backslash int\_\backslash Omega\; \backslash frac\{\backslash mathbf\{v\}(\backslash mathbf\{y\})\; \backslash times\; (\backslash mathbf\{x\}\; -\; \backslash mathbf\{y\})\}\{4\; \backslash pi\; |\backslash mathbf\{x\}\; -\; \backslash mathbf\{y\}|^3\}\; d^3\; \backslash mathbf\{y\}$.

Substituting $\backslash mathbf\{j\}$ (current density) for $\backslash mathbf\{v\}$ and $\backslash mathbf\{H\}$ (H-field) for $\backslash mathbf\{A\}$, yields the Biot-Savart law.

Let $\backslash mathbf\{\backslash Omega\}$ be a star domain centered at the point $\backslash mathbf\{p\}$, where $\backslash mathbf\{p\}\backslash in\; \backslash R^3$. Applying Poincaré's lemma for differential forms to vector fields, then $\backslash mathbf\{A\text{'}\text{'}\text{'}\}(\backslash mathbf\{x\})$ also is a vector potential for $\backslash mathbf\{v\}$, where

$\backslash mathbf\{A\text{'}\text{'}\text{'}\}(\backslash mathbf\{x\})\; =\backslash int\_0^1\; s\; ((\backslash mathbf\{x\}-\backslash mathbf\{p\})\backslash times\; (\; \backslash mathbf\{v\}(\; s\; \backslash mathbf\{x\}\; +\; (1-s)\; \backslash mathbf\{p\}\; ))\backslash \; ds$

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Nonuniqueness The vector potential admitted by a solenoidal field is not unique. If $\backslash mathbf\{A\}$ is a vector potential for $\backslash mathbf\{v\}$, then so is $$\backslash mathbf\{A\}\; +\; \backslash nabla\; f,$$ 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 Gauge fixing.

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See also

• Fundamental theorem of vector calculus
• Magnetic vector potential
• Solenoidal vector field
• Closed and Exact Differential Forms

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

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