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Magnetic vector potential
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== Definition == The magnetic vector potential, <math>\mathbf{A}</math>, is a [[vector field]], and the [[electric potential]], <math>\phi</math>, is a [[scalar field]] such that:<ref name=Feynman1515>{{harvp|Feynman|1964|loc=chpt. 15}}</ref> <math display="block">\mathbf{B} = \nabla \times \mathbf{A}\ , \quad \mathbf{E} = -\nabla \phi - \frac{ \partial \mathbf{A} }{ \partial t },</math> where <math>\mathbf{B}</math> is the [[magnetic field]] and <math>\mathbf{E} </math> is the [[electric field]]. In [[magnetostatics]] where there is no time-varying current or [[charge distribution]], only the first equation is needed. (In the context of [[electrodynamics]], the terms ''vector potential'' and ''scalar potential'' are used for ''magnetic vector potential'' and ''[[electric potential]]'', respectively. In mathematics, [[vector potential]] and [[scalar potential]] can be generalized to higher dimensions.) If electric and magnetic fields are defined as above from potentials, they automatically satisfy two of [[Maxwell's equations]]: [[Gauss's law for magnetism]] and [[Faraday's law of induction|Faraday's law]]. For example, if <math>\mathbf{A}</math> is continuous and well-defined everywhere, then it is guaranteed not to result in [[magnetic monopole]]s. (In the mathematical theory of magnetic monopoles, <math>\mathbf{A}</math> is allowed to be either undefined or multiple-valued in some places; see [[magnetic monopole]] for details). Starting with the above definitions and remembering that the divergence of the curl is zero and the curl of the gradient is the zero vector: <math display="block">\begin{align} \nabla \cdot \mathbf{B} &= \nabla \cdot \left(\nabla \times \mathbf{A}\right) = 0\ ,\\ \nabla \times \mathbf{E} &= \nabla \times \left( -\nabla\phi - \frac{ \partial\mathbf{A} }{ \partial t } \right) = -\frac{ \partial }{ \partial t } \left(\nabla \times \mathbf{A}\right) = -\frac{ \partial \mathbf{B} }{ \partial t } ~. \end{align}</math> Alternatively, the existence of <math>\mathbf{A}</math> and <math>\phi</math> is guaranteed from these two laws using [[Helmholtz decomposition|Helmholtz's theorem]]. For example, since the magnetic field is [[divergence]]-free (Gauss's law for magnetism; i.e., <math>\nabla \cdot \mathbf{B} = 0</math>), <math> \mathbf{A}</math> always exists that satisfies the above definition. The vector potential <math>\mathbf{A} </math> is used when studying the [[Lagrangian mechanics|Lagrangian]] in [[classical mechanics]] and in [[quantum mechanics]] (see [[Pauli equation|Schrödinger equation for charged particles]], [[Dirac equation]], [[Aharonov–Bohm effect]]). In [[minimal coupling]], <math>q \mathbf{A} </math> is called the potential momentum, and is part of the [[canonical momentum]]. The [[line integral]] of <math>\mathbf{A}</math> over a closed loop, <math>\Gamma</math>, is equal to the [[magnetic flux]], <math>\Phi_{\mathbf{B}}</math>, through a surface, <math>S</math>, that it encloses: <math display="block">\oint_\Gamma \mathbf{A}\, \cdot\ d{\mathbf{\Gamma}} = \iint_S \nabla\times\mathbf{A}\ \cdot\ d \mathbf{S} = \Phi_\mathbf{B} ~.</math> Therefore, the units of <math>\mathbf{A}</math> are also equivalent to [[weber (unit)|weber]] per [[metre]]. The above equation is useful in the [[flux quantization]] of [[Superconductor|superconducting loops]]. In the Coulomb gauge <math> \nabla \cdot \mathbf{A} = 0 </math>, there is a formal analogy between the relationship between the vector potential and the magnetic field to [[Ampere's law]] <math> \nabla \times \mathbf{B} = \mu_0 \mathbf{J} </math>. Thus, when finding the vector potential of a given magnetic field, one can use the same methods one uses when finding the magnetic field given a current distribution. Although the magnetic field, <math>\mathbf{B}</math>, is a [[pseudovector]] (also called [[axial vector]]), the vector potential, <math>\mathbf{A}</math>, is a [[polar vector]].<ref name=Fitzpatrick>{{cite web |first=Richard |last=Fitzpatrick |title=Tensors and pseudo-tensors |type=lecture notes |publisher=[[University of Texas]] |place=Austin, TX |url=http://farside.ph.utexas.edu/teaching/em/lectures/node120.html }}</ref> This means that if the [[right-hand rule]] for [[cross product]]s were replaced with a left-hand rule, but without changing any other equations or definitions, then <math>\mathbf{B}</math> would switch signs, but '''A''' would not change. This is an example of a general theorem: The curl of a polar vector is a pseudovector, and vice versa.<ref name=Fitzpatrick/>
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