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    Magnetic vector potential

    Magnetic vector potential

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    In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field:

    {\textstyle \nabla \times \mathbf {A} =\mathbf {B} }

    . Together with the electric potential , the magnetic vector potential can be used to specify the electric field E as well. Therefore, many equations of electromagnetism can be written either in terms of the fields E and B, or equivalently in terms of the potentials and A. In more advanced theories such as quantum mechanics, most equations use potentials rather than fields.

    Magnetic vector potential was first introduced by Franz Ernst Neumann and Wilhelm Eduard Weber in 1845 and in 1846, respectively. Lord Kelvin also introduced vector potential in 1847, along with the formula relating it to the magnetic field.[1][2]

    Contents

    1 Magnetic vector potential

    1.1 Gauge choices

    1.2 Maxwell's equations in terms of vector potential

    1.3 Calculation of potentials from source distributions

    1.4 Depiction of the A-field

    1.5 Electromagnetic four-potential

    2 See also 3 Notes 4 References 5 External links

    Magnetic vector potential[edit]

    The magnetic vector potential A is a vector field, defined along with the electric potential (a scalar field) by the equations:[3]

    {\displaystyle \mathbf {B} =\nabla \times \mathbf {A} \,,\quad \mathbf {E} =-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}\,,}

    where B is the magnetic field and E is the electric field. In magnetostatics where there is no time-varying charge distribution, only the first equation is needed. (In the context of electrodynamics, the terms and are used for and , 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. For example, if A is continuous and well-defined everywhere, then it is guaranteed not to result in magnetic monopoles. (In the mathematical theory of magnetic monopoles, A 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:

    {\displaystyle {\begin{aligned}\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{aligned}}}

    Alternatively, the existence of A and is guaranteed from these two laws using Helmholtz's theorem. For example, since the magnetic field is divergence-free (Gauss's law for magnetism; i.e., B = 0), A always exists that satisfies the above definition.

    The vector potential A is used when studying the Lagrangian in classical mechanics and in quantum mechanics (see Schrödinger equation for charged particles, Dirac equation, Aharonov–Bohm effect).

    In the SI system, the units of A are V·s·m−1 and are the same as that of momentum per unit charge, or force per unit current. In minimal coupling, qA is called the potential momentum, and is part of the canonical momentum.

    The line integral of A over a closed loop, Γ, is equal to the magnetic flux, ΦB, through a surface, , that it encloses:

    {\displaystyle \oint _{\Gamma }\mathbf {A} \,\cdot \,d{\mathbf {\Gamma } }=\iint _{S}\nabla \times \mathbf {A} \,\cdot \,d\mathbf {S} =\Phi _{\mathbf {B} }.}

    Therefore, the units of A are also equivalent to Weber per metre. The above equation is useful in the flux quantization of superconducting loops.

    Although the magnetic field B is a pseudovector (also called axial vector), the vector potential A is a polar vector.[4] This means that if the right-hand rule for cross products were replaced with a left-hand rule, but without changing any other equations or definitions, then B 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.[4]

    Gauge choices[edit]

    Main article: Gauge fixing

    The above definition does not define the magnetic vector potential uniquely because, by definition, we can arbitrarily add curl-free components to the magnetic potential without changing the observed magnetic field. Thus, there is a degree of freedom available when choosing A. This condition is known as gauge invariance.

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