Vector potential

Mathematical concept in vector calculus

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 {\displaystyle C^{2}} vector field A such that

v = ∇ × A . {\displaystyle \mathbf {v} =\nabla \times \mathbf {A} .}

Consequence

If a vector field v admits a vector potential A, then from the equality

∇ ⋅ ( ∇ × A ) = 0 {\displaystyle \nabla \cdot (\nabla \times \mathbf {A} )=0}
(divergence of the curl is zero) one obtains
∇ ⋅ v = ∇ ⋅ ( ∇ × A ) = 0 , {\displaystyle \nabla \cdot \mathbf {v} =\nabla \cdot (\nabla \times \mathbf {A} )=0,}
which implies that v must be a solenoidal vector field.

Theorem

Let

v : R 3 → R 3 {\displaystyle \mathbf {v} :\mathbb {R} ^{3}\to \mathbb {R} ^{3}}
be a solenoidal vector field which is twice continuously differentiable. Assume that v(x) decreases at least as fast as 1 / ‖ x ‖ {\displaystyle 1/\|\mathbf {x} \|} for ‖ x ‖ → ∞ {\displaystyle \|\mathbf {x} \|\to \infty } . Define
A ( x ) = 1 4 π ∫ R 3 ∇ y × v ( y ) ‖ x − y ‖ d 3 y . {\displaystyle \mathbf {A} (\mathbf {x} )={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}{\frac {\nabla _{y}\times \mathbf {v} (\mathbf {y} )}{\left\|\mathbf {x} -\mathbf {y} \right\|}}\,d^{3}\mathbf {y} .}

Then, A is a vector potential for v, that is,　

∇ × A = v . {\displaystyle \nabla \times \mathbf {A} =\mathbf {v} .}
Here, ∇ y × {\displaystyle \nabla _{y}\times } is curl for variable y. Substituting curl[v] for the current density j of the retarded potential, you will get this formula. In other words, v corresponds to the H-field.

You can restrict the integral domain to any single-connected region Ω. That is, A' below is also a vector potential of v;

A ′ ( x ) = 1 4 π ∫ Ω ∇ y × v ( y ) ‖ x − y ‖ d 3 y . {\displaystyle \mathbf {A'} (\mathbf {x} )={\frac {1}{4\pi }}\int _{\Omega }{\frac {\nabla _{y}\times \mathbf {v} (\mathbf {y} )}{\left\|\mathbf {x} -\mathbf {y} \right\|}}\,d^{3}\mathbf {y} .}

A generalization of this theorem is the Helmholtz decomposition 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 Biot-Savart's law, the following A ″ ( x ) {\displaystyle {\boldsymbol {A''}}({\textbf {x}})} is also qualify as a vector potential for v.

A ″ ( x ) = ∫ Ω v ( y ) × ( x − y ) 4 π | x − y | 3 d 3 y {\displaystyle {\boldsymbol {A''}}({\textbf {x}})=\int _{\Omega }{\frac {{\boldsymbol {v}}({\boldsymbol {y}})\times ({\boldsymbol {x}}-{\boldsymbol {y}})}{4\pi |{\boldsymbol {x}}-{\boldsymbol {y}}|^{3}}}d^{3}{\boldsymbol {y}}}

Substitute j (current density) for v and H (H-field)for A, we will find the Biot-Savart law.

Let p ∈ R {\displaystyle {\textbf {p}}\in \mathbb {R} } and let the Ω be a star domain centered on the p then, translating Poincaré's lemma for differential forms into vector fields world, the following A ‴ ( x ) {\displaystyle {\boldsymbol {A'''}}({\boldsymbol {x}})} is also a vector potential for the v {\displaystyle {\boldsymbol {v}}}

A ‴ ( x ) = ∫ 0 1 s ( ( x − p ) × ( v ( s x + ( 1 − s ) p ) )   d s {\displaystyle {\boldsymbol {A'''}}({\boldsymbol {x}})=\int _{0}^{1}s(({\boldsymbol {x}}-{\boldsymbol {p}})\times ({\boldsymbol {v}}(s{\boldsymbol {x}}+(1-s){\boldsymbol {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

A + ∇ f , {\displaystyle \mathbf {A} +\nabla f,}
where 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.


See also

References

  • Fundamentals of Engineering Electromagnetics by David K. Cheng, Addison-Wesley, 1993.
Authority control databases: National Edit this at Wikidata
  • Germany