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

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

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

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

$${\displaystyle \mathbf {A''} (\mathbf {x} )=\int _{\Omega }{\frac {\mathbf {v} (\mathbf {s} )\times (\mathbf {x} -\mathbf {s} )}{4\pi \left|\mathbf {x} -\mathbf {s} \right|^{3}}}d^{3}\mathbf {s} }$${\displaystyle \mathbf {A''} (\mathbf {x} )=\int _{\Omega }{\frac {\mathbf {v} (\mathbf {s} )\times (\mathbf {x} -\mathbf {s} )}{4\pi \left|\mathbf {x} -\mathbf {s} \right|^{3}}}d^{3}\mathbf {s} }

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

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

$${\displaystyle \mathbf {A'''} (\mathbf {x} )=\int _{0}^{1}s\left[(\mathbf {x} -\mathbf {p} )\times \mathbf {v} (s\mathbf {x} +(1{-}s)\mathbf {p} )\right]ds}$${\displaystyle \mathbf {A'''} (\mathbf {x} )=\int _{0}^{1}s\left[(\mathbf {x} -\mathbf {p} )\times \mathbf {v} (s\mathbf {x} +(1{-}s)\mathbf {p} )\right]ds}

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

• 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.