Magnetic vector potential


Magnetic vector potential, A, is the vector quantity in classical electromagnetism defined so that its curl is equal to the magnetic field:. 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.
Historically, Lord Kelvin first introduced vector potential in 1851, along with the formula relating it to the magnetic field.

Magnetic vector potential

The magnetic vector potential A is a vector field, defined along with the electric potential ϕ by the equations:
where B is the magnetic field and E is the electric field. In magnetostatics where there is no time-varying 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. For example, if A is continuous and well-defined everywhere, then it is guaranteed not to result in magnetic monopoles..
Starting with the above definitions:
Alternatively, the existence of A and ϕ is guaranteed from these two laws using Helmholtz's theorem. For example, since the magnetic field is divergence-free, A always exists that satisfies the above definition.
The vector potential A is used when studying the Lagrangian in classical mechanics and in quantum mechanics.
In the SI system, the units of A are V·s·m−1 and are the same as that of momentum per unit charge, or force per unit current. In Minimal coupling, qA is called the potential momentum, and is part of the canonical momentum.
The line integral of A over a close loop is equal to the magnetic flux through the enclosed surface:
Therefore, the units of A are also equivalent to Weber per metre. The above equation is useful in the flux quantization of superconducting loops.
Although the magnetic field B is a pseudovector, the vector potential A is a polar vector. 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 B would switch signs, but A would not change. This is an example of a general theorem: The curl of a polar vector is a pseudovector, and vice versa.

Gauge choices

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. Thus, there is a degree of freedom available when choosing A. This condition is known as gauge invariance.

Maxwell's equations in terms of vector potential

Using the above definition of the potentials and applying it to the other two Maxwell's equations results in a complicated differential equation that can be simplified using the Lorenz gauge where A is chosen to satisfy:
Using the Lorenz gauge, Maxwell's equations can be written compactly in terms of the magnetic vector potential A and the electric scalar potential ϕ:
In other gauges, the equations are different. A different notation to write these same equations is shown below.

Calculation of potentials from source distributions

The solutions of Maxwell's equations in the Lorenz gauge :
where the fields at position vector r and time t are calculated from sources at distant position r′ at an earlier time t′. The location r′ is a source point in the charge or current distribution. The earlier time t′ is called the retarded time, and calculated as
There are a few notable things about A and ϕ calculated in this way:
In other gauges, the formula for A and ϕ is different; for example, see Coulomb gauge for another possibility.

Depiction of the A-field

See Feynman for the depiction of the A field around a long thin solenoid.
Since
assuming quasi-static conditions, i.e.
the lines and contours of A relate to B like the lines and contours of B relate to j. Thus, a depiction of the A field around a loop of B flux is qualitatively the same as the B field around a loop of current.
The figure to the right is an artist's depiction of the A field. The thicker lines indicate paths of higher average intensity. The lines are drawn to impart the general look of the A-field.
The drawing tacitly assumes, true under one of the following assumptions:
In the context of special relativity, it is natural to join the magnetic vector potential together with the 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 may be written as follows:
where □ is the d'Alembertian and J is the four-current. The first equation is the Lorenz gauge condition while the second contains Maxwell's equations. The four-potential also plays a very important role in quantum electrodynamics.