Editing Magnetic field (section)

==== Magnetic vector potential ====
{{Main|Magnetic vector potential}}

In advanced topics such as [[quantum mechanics]] and [[Theory of relativity|relativity]] it is often easier to work with a potential formulation of electrodynamics rather than in terms of the electric and magnetic fields. In this representation, the ''[[magnetic vector potential]]'' {{math|'''A'''}}, and the [[electric potential|electric scalar potential]] {{math|''φ''}}, are defined using [[gauge fixing]] such that:

<math display="block">\begin{align}
  \mathbf{B} &=  \nabla \times \mathbf{A}, \\
  \mathbf{E} &= -\nabla \varphi - \frac{ \partial \mathbf{A} }{ \partial t }.
\end{align}</math>

The vector potential, ''{{math|'''A'''}}'' given by this form may be interpreted as a ''generalized potential [[momentum]] per unit charge'' <ref>{{cite journal | author = E. J. Konopinski | author-link = Emil Konopinski | year =  1978 | title = What the electromagnetic vector potential describes | journal = Am. J. Phys. | volume = 46 | issue = 5| pages = 499–502 | doi = 10.1119/1.11298 | bibcode = 1978AmJPh..46..499K }}</ref> just as {{math|''φ''}} is interpreted as a ''generalized [[potential energy]] per unit charge''. There are multiple choices one can make for the potential fields that satisfy the above condition. However, the choice of potentials is represented by its respective gauge condition.

Maxwell's equations when expressed in terms of the potentials in [[Lorenz gauge condition|Lorenz gauge]] can be cast into a form that agrees with [[special relativity]].<ref>{{harvnb|Griffiths|1999|p=422}}</ref> In relativity, {{math|'''A'''}} together with {{math|''φ''}} forms a [[four-potential]] regardless of the gauge condition, analogous to the [[Four-vector#Four-momentum|four-momentum]] that combines the momentum and energy of a particle. Using the four potential instead of the electromagnetic tensor has the advantage of being much simpler—and it can be easily modified to work with quantum mechanics.