KNOWPIA
WELCOME TO KNOWPIA

In classical electromagnetism, **magnetic vector potential** (often called **A**) is the vector quantity 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.

Magnetic vector potential was first introduced by Franz Ernst Neumann and Wilhelm Eduard Weber in 1845 and in 1846, respectively. William Thomson also introduced vector potential in 1847, along with the formula relating it to the magnetic field.^{[1]}^{[2]}

This article uses the SI system.

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.

The magnetic vector potential is a vector field, defined along with the electric potential (a scalar field) by the equations:^{[3]}
where is the magnetic field and 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 can be generalized to higher dimensions.)

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 is continuous and well-defined everywhere, then it is guaranteed not to result in magnetic monopoles. (In the mathematical theory of magnetic monopoles, is allowed to be either undefined or multiple-valued in some places; see magnetic monopole for details).

Starting with the above definitions and remembering that the divergence of the curl is zero and the curl of the gradient is the zero vector:

Alternatively, the existence of and is guaranteed from these two laws using Helmholtz's theorem. For example, since the magnetic field is divergence-free (Gauss's law for magnetism; i.e., ), always exists that satisfies the above definition.

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

In minimal coupling, is called the potential momentum, and is part of the canonical momentum.

The line integral of over a closed loop, is equal to the magnetic flux, through a surface, that it encloses:

Therefore, the units of are also equivalent to Weber per metre. The above equation is useful in the flux quantization of superconducting loops.

Although the magnetic field is a pseudovector (also called axial vector), the vector potential is a polar vector.^{[4]} 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 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.^{[4]}

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 This condition is known as gauge invariance.

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 is chosen to satisfy:^{[3]}

Using the Lorenz gauge, Maxwell's equations can be written compactly in terms of the magnetic vector potential and the electric scalar potential ^{[3]}

In other gauges, the equations are different. A different notation to write these same equations (using four-vectors) is shown below.

In a field with electric potential and magnetic potential the Lagrangian ( ) and the Hamiltonian ( ) of a particle with mass and charge are

The solutions of Maxwell's equations in the Lorenz gauge (see Feynman^{[3]} and Jackson^{[5]}) 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 and the electric scalar potential due to a current distribution of current density charge density and volume within which and are non-zero at least sometimes and some places):
where the fields at position vector and time are calculated from sources at distant position at an earlier time The location is a source point in the charge or current distribution (also the integration variable, within volume ). The earlier time is called the *retarded time*, and calculated as

There are a few notable things about and calculated in this way:

- The Lorenz gauge condition is satisfied:

- The position of the point at which values for and are found, only enters the equation as part of the scalar distance from to The direction from to 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*, 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 and from remote location must also be at some prior time - The equation for is a vector equation. In Cartesian coordinates, the equation separates into three scalar equations:
^{[6]}

In this form it is easy to see that the component of in a given direction depends only on the components of that are in the same direction. If the current is carried in a long straight wire, points in the same direction as the wire.

In other gauges, the formula for and is different; for example, see Coulomb gauge for another possibility.

See Feynman^{[7]} for the depiction of the field around a long thin solenoid.

Since assuming quasi-static conditions, i.e.

- and

the lines and contours of relate to like the lines and contours of relate to Thus, a depiction of the field around a loop of flux (as would be produced in a toroidal inductor) is qualitatively the same as the field around a loop of current.

The figure to the right is an artist's depiction of the 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 field.

The drawing tacitly assumes true under any one of the following assumptions:

- the Coulomb gauge is assumed
- the Lorenz gauge is assumed and there is no distribution of charge,
- the Lorenz gauge is assumed and zero frequency is assumed
- the Lorenz gauge is assumed and a non-zero frequency, but still assumed sufficiently low to neglect the term

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 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.

**^**Neumann, Franz Ernst (January 1, 1846). "Allgemeine Gesetze der induzirten elektrischen Ströme (General laws of induced electrical currents)".*Annalen der Physik*.**143**(11): 31–34. doi:10.1002/andp.18461430103.**^**Yang, ChenNing (2014). "The conceptual origins of Maxwell's equations and gauge theory".*Physics Today*.**67**(11): 45–51. Bibcode:2014PhT....67k..45Y. doi:10.1063/PT.3.2585.- ^
^{a}^{b}^{c}^{d}Feynman (1964), p. 15 - ^
^{a}^{b}Fitzpatrick, Richard. "Tensors and pseudo-tensors" (lecture notes). Austin, TX: University of Texas. **^**Jackson (1999), p. 246**^**Kraus (1984), p. 189**^**Feynman (1964), p. 11, cpt 15

- Duffin, W.J. (1990).
*Electricity and Magnetism, Fourth Edition*. McGraw-Hill. - Feynman, Richard P; Leighton, Robert B; Sands, Matthew (1964).
*The Feynman Lectures on Physics Volume 2*. Addison-Wesley. ISBN 0-201-02117-X. - Jackson, John David (1999).
*Classical Electrodynamics*(3rd ed.). John Wiley & Sons. ISBN 0-471-30932-X. - Kraus, John D. (1984).
*Electromagnetics*(3rd ed.). McGraw-Hill. ISBN 0-07-035423-5.

- Media related to Magnetic vector potential at Wikimedia Commons