Magnetic vector potential
The magnetic vector potential A is a
vector field, defined along with the
electric potential ϕ (a
scalar field) 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. (In the context of
electrodynamics, the terms vector potential and scalar potential are used for magnetic vector potential and
electric potential, respectively. In mathematics,
vector potential and
scalar potential have more general meanings.)
Defining the electric and magnetic fields from potentials automatically satisfies two of
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. (In the mathematical theory of magnetic monopoles, A is allowed to be either undefined or multiple-valued in some places; see
magnetic monopole for details).
Starting with the above definitions:
Alternatively, the existence of A and ϕ is guaranteed from these two laws using the
Helmholtz's theorem. For example, since the magnetic field is
divergence-free (Gauss's law for magnetism; i.e., ∇ ⋅ B = 0), A always exists that satisfies the above definition.
The vector potential A is used when studying the
classical mechanics and in
quantum mechanics (see
Schrödinger equation for charged particles,
SI system, the units of A are
m−1 and are the same as that of
momentum per unit
Although the magnetic field B is a
pseudovector (also called
axial vector), the vector potential A is a
 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.
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
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 (the ones that are not automatically satisfied) 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:
gauges, the equations are different. A different notation to write these same equations (using
four-vectors) is shown below.
Calculation of potentials from source distributions
The solutions of Maxwell's equations in the Lorenz gauge (see Feynman
 and Jackson
) 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 A(r, t) and the electric scalar potential ϕ(r, t) due to a current distribution of
current density J(r′, t′),
charge density ρ(r′, t′), and
volume Ω, within which ρ and J are non-zero at least sometimes and some places):
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 (also the integration variable, within volume Ω). 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:
Lorenz gauge condition): is satisfied.
- The position of r, the point at which values for ϕ and A are found, only enters the equation as part of the scalar distance from r′ to r. The direction from r′ to r does not enter into the equation. The only thing that matters about a source point is how far away it is.
- The integrand uses
retarded time, t′. This simply reflects the fact that changes in the sources propagate at the speed of light. Hence the charge and current densities affecting the electric and magnetic potential at r and t, from remote location r′ must also be at some prior time t′.
- The equation for A is a vector equation. In Cartesian coordinates, the equation separates into three scalar equations:
- In this form it is easy to see that the component of A in a given direction depends only on the components of J that are in the same direction. If the current is carried in a long straight wire, the A points in the same direction as the wire.
In other gauges, the formula for A and ϕ is different; for example, see
Coulomb gauge for another possibility.
Depiction of the A-field
magnetic vector potential A
, magnetic flux density B
, and current density j
fields around a
. Thicker lines indicate field lines of higher average intensity. Circles in the cross section of the core represent the B
-field coming out of the picture, plus signs represent B
-field going into the picture. ∇ ⋅ A = 0
has been assumed.
 for the depiction of the A field around a long thin
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 (as would be produced in a
toroidal inductor) 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 (shorter paths have higher intensity so that the path integral is the same). The lines are drawn to (aesthetically) impart the general look of the A-field.
The drawing tacitly assumes ∇ ⋅ A = 0, true under one of the following assumptions:
Coulomb gauge is assumed
Lorenz gauge is assumed and there is no distribution of charge, ρ = 0,
Lorenz gauge is assumed and zero frequency is assumed
Lorenz gauge is assumed and a non-zero frequency that is low enough to neglect is assumed
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:
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