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Mathematics for Physics

90 Vector Potential

A vector field whose curl is the magnetic field that is used, with the scalar potential, to express electromagnetic fields.

\(\mathbf{B}=\nabla\times\mathbf{A}\). Because the magnetic field is divergenceless, it is the curl of a vector potential. This principle is used to replace \(\mathbf{B}\) by \(\mathbf{A}\) in calculations.

The magnetic field from the vector potential is

\[ \mathbf{B} = \nabla\times\mathbf{A} \]

where

  • \(\mathbf{B}\) is the magnetic field.
  • \(\mathbf{A}\) is the vector potential.
  • \(\nabla\times\) is the curl.

The Coulomb-gauge Poisson equation. In magnetostatics with the Coulomb gauge the vector potential obeys a Poisson equation sourced by the current. The Coulomb gauge is the condition \(\nabla\cdot\mathbf{A}=0\). This principle is used to compute \(\mathbf{A}\) from a known steady current.

The magnetostatic Poisson equation for \(\mathbf{A}\) is

\[ \nabla^{2}\mathbf{A} = -\mu_{0}\mathbf{J} \]

where

  • \(\nabla^{2}\) is the Laplacian.
  • \(\mathbf{A}\) is the vector potential.
  • \(\mathbf{J}\) is the volume current density.
  • \(\mu_{0}\) is the permeability of free space.

The four-potential. The vector potential is the spatial part of the electromagnetic four-potential. This principle is used to write \(V\) and \(\mathbf{A}\) as one spacetime vector.

The four-potential is

\[ A^{\alpha} = \Bigl(\dfrac{V}{c},\,\mathbf{A}\Bigr) \]

where

  • \(A^{\alpha}\) is the four-potential.
  • \(V\) is the electric scalar potential.
  • \(\mathbf{A}\) is the magnetic vector potential.
  • \(c\) is the speed of light.

Note: Also denoted \(\mathbf{A}\). Also called the magnetic vector potential.