In classical electromagnetism, magnetic vector potential (often denoted A) is the vector quantity defined so that its curl is equal to the magnetic field, B: ∇ × A = B {\textstyle \nabla \times \mathbf {A} =\mathbf {B} } . Together with the electric potential φ, 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 φ and A. In more advanced theories such as quantum mechanics, most equations use potentials rather than fields. Magnetic vector potential was independently introduced by Franz Ernst Neumann and Wilhelm Eduard Weber in 1845 and in 1846, respectively to discuss Ampère's circuital law. William Thomson also introduced the modern version of the vector potential in 1847, along with the formula relating it to the magnetic field.
Unit conventions This article uses the SI system. In the SI system, the units of A are V·s·m−1 or Wb·m−1 and are the same as that of momentum per unit charge, or force per unit current.
Definition The magnetic vector potential, A {\displaystyle \mathbf {A} } , is a vector field, and the electric potential, ϕ {\displaystyle \phi } , is a scalar field such that:
B = ∇ × A , E = − ∇ ϕ − ∂ A ∂ t , {\displaystyle \mathbf {B} =\nabla \times \mathbf {A} \ ,\quad \mathbf {E} =-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}},}
where B {\displaystyle \mathbf {B} } is the magnetic field and E {\displaystyle \mathbf {E} } 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. For example, if A {\displaystyle \mathbf {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 {\displaystyle \mathbf {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:
∇ ⋅ B = ∇ ⋅ ( ∇ × A ) = 0 , ∇ × E = ∇ × ( − ∇ ϕ − ∂ A ∂ t ) = − ∂ ∂ t ( ∇ × A ) = − ∂ B ∂ t . {\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}}}
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