Magnetic vector potential

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

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[1] and Wilhelm Eduard Weber[2] in 1845 and in 1846, respectively to discuss Ampère's circuital law.

[3] William Thomson also introduced the modern version of the vector potential in 1847, along with the formula relating it to the magnetic field.

In magnetostatics where there is no time-varying current or charge distribution, only the first equation is needed.

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.

is continuous and well-defined everywhere, then it is guaranteed not to result in magnetic monopoles.

is allowed to be either undefined or multiple-valued in some places; see magnetic monopole for details).

is guaranteed from these two laws using Helmholtz's theorem.

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

The above equation is useful in the flux quantization of superconducting loops.

[6] 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

This is an example of a general theorem: The curl of a polar vector is a pseudovector, and vice versa.

[6] 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.

Using the above definition of the potentials and applying it to the other two Maxwell's equations (the ones that are not automatically satisfied) results in a complicated differential equation that can be simplified using the Lorenz gauge where

Using the Lorenz gauge, the electromagnetic wave equations can be written compactly in terms of the potentials, [5] The solutions of Maxwell's equations in the Lorenz gauge (see Feynman[5] and Jackson[7]) with the boundary condition that both potentials go to zero sufficiently fast as they approach infinity are called the retarded potentials, which are the magnetic vector potential

are non-zero at least sometimes and some places): where the fields at position vector

are calculated from sources at distant position

is a source point in the charge or current distribution (also the integration variable, within volume

is called the retarded time, and calculated as

field around a long thin solenoid.

assuming quasi-static conditions, i.e. the lines and contours of

flux (as would be produced in a toroidal inductor) is qualitatively the same as the

The lines are drawn to (aesthetically) impart the general look of the

, true under any one of the following assumptions: In the context of special relativity, it is natural to join the magnetic vector potential together with the (scalar) electric potential into the electromagnetic potential, also called four-potential.

One motivation for doing so is that the four-potential is a mathematical four-vector.

Thus, using standard four-vector transformation rules, if the electric and magnetic potentials are known in one inertial reference frame, they can be simply calculated in any other inertial reference frame.

Another, related motivation is that the content of classical electromagnetism can be written in a concise and convenient form using the electromagnetic four potential, especially when the Lorenz gauge is used.

In particular, in abstract index notation, the set of Maxwell's equations (in the Lorenz gauge) may be written (in Gaussian units) as follows:

The four-potential also plays a very important role in quantum electrodynamics.