• Source: 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



{\displaystyle \mathbf {v} }

, a vector potential is a




C

2




{\displaystyle C^{2}}

vector field




A



{\displaystyle \mathbf {A} }

such that





v

=
∇
×

A

.


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





Consequence


If a vector field




v



{\displaystyle \mathbf {v} }

admits a vector potential




A



{\displaystyle \mathbf {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



{\displaystyle \mathbf {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

)


{\displaystyle \mathbf {v} (\mathbf {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} }


where




∇

y


×


{\displaystyle \nabla _{y}\times }

denotes curl with respect to variable




y



{\displaystyle \mathbf {y} }

. Then




A



{\displaystyle \mathbf {A} }

is a vector potential for




v



{\displaystyle \mathbf {v} }

. That is,




∇
×

A

=

v

.


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


The integral domain can be restricted to any simply connected region




Ω



{\displaystyle \mathbf {\Omega } }

. That is,





A
′




{\displaystyle \mathbf {A'} }

also is a vector potential of




v



{\displaystyle \mathbf {v} }

, where






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 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

)


{\displaystyle \mathbf {A''} (\mathbf {x} )}

also qualifies as a vector potential for




v



{\displaystyle \mathbf {v} }

, where






A
″


(

x

)
=

∫

Ω






v

(

y

)
×
(

x

−

y

)


4
π

|


x

−

y



|


3






d

3



y



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

.
Substituting




j



{\displaystyle \mathbf {j} }

(current density) for




v



{\displaystyle \mathbf {v} }

and




H



{\displaystyle \mathbf {H} }

(H-field) for




A



{\displaystyle \mathbf {A} }

, yields the Biot-Savart law.
Let




Ω



{\displaystyle \mathbf {\Omega } }

be a star domain centered at the point




p



{\displaystyle \mathbf {p} }

, where




p

∈


R


3




{\displaystyle \mathbf {p} \in \mathbb {R} ^{3}}

. Applying Poincaré's lemma for differential forms to vector fields, then





A
‴


(

x

)


{\displaystyle \mathbf {A'''} (\mathbf {x} )}

also is a vector potential for




v



{\displaystyle \mathbf {v} }

, where






A
‴


(

x

)
=

∫

0


1


s
(
(

x

−

p

)
×
(

v

(
s

x

+
(
1
−
s
)

p

)
)

d
s


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





Nonuniqueness


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




A



{\displaystyle \mathbf {A} }

is a vector potential for




v



{\displaystyle \mathbf {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


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.

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