Magnetic vector potential

A magnetic vector potential satisfies B equal to its curl and is defined up to a gradient.

For a smooth divergence-free field on ${\mathbb{R}}^3$, the differential form $F={\iota }_{\mathbf{B}}(d{x}^1\wedge d{x}^2\wedge d{x}^3)$ is closed. The radial homotopy formula gives a global magnetic vector potential $\mathbf{A}(\mathbf{x})={\int }_0^{1}s\mathbf{B}(s\mathbf{x})\times \mathbf{x}ds$ with curl $\mathbf{B}$. For constant $\mathbf{B}$, this reduces to $\mathbf{A}=\frac{1}{2}\mathbf{B}\times \mathbf{x}$.

For thin closed wire loops $C_1,C_2$, the mutual inductance is

The Coulomb gauge imposeson the magnetic vector potential. A gauge transformation $A\mapsto A+\nabla \chi$ reaches this gauge when $\chi$ solves

For a free electromagnetic field with suitable boundary conditions, combine Coulomb gauge ${\partial }_i{A}_i=0$ with $A_0=0$. Gauss's law makes the second condition consistent in the source-free problem. The remaining field operator is spatially transverse and describes the two radiative photon polarizations. A constrained scalar potential must be restored when charges are present.

The transformation $A\mapsto A+\nabla \chi$ leaves $B=\nabla \times A$ unchanged because the curl of a gradient vanishes.

For the Maxwell field, the variation $\delta A_{\nu }=\alpha F_{0\nu }=\alpha ({\partial }_0{A}_{\nu }-{\partial }_{\nu }{A}_0)$ is time translation combined with a gauge transformation. With constant $\alpha$, $\delta F_{\mu \nu }=\alpha {\partial }_0{F}_{\mu \nu }$, and the Lagrangian changes by $\alpha {\partial }_0\mathcal{L}$. Its Noether current is directly the time component of the gauge-invariant Maxwell stress-energy tensor, providing the physical energy density without a potential-dependent improvement step.

For a localized steady current density,It has dimensions $\mathrm{A}{\mathrm{m}}^2$.

In an external field that varies slowly across a localized magnetic dipole,

The Bohr magneton,is the natural unit of an electron's orbital and spin magnetic moment.

If $\nabla \cdot J=0$ and the current decays enough for boundary terms to vanish, thensatisfies $\nabla \cdot A=0$. Its far field is

A circular charge distribution rotating with angular speed $\omega$ can be decomposed into current loops. A hoop of line density $\eta$ and radius $R$ has $m=\pi \eta \omega R^3\hat{n}$; a disc of surface density $\sigma$ has $m=\pi \sigma \omega R^4\hat{n}/4$.