# Magnetic vector potential

Last updated

Magnetic vector potential, A, is the vector quantity in classical electromagnetism defined so that its curl is equal to the magnetic field: ${\textstyle \nabla \times \mathbf {A} =\mathbf {B} \,}$. 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.

## Contents

Historically, Lord Kelvin first introduced vector potential in 1851, along with the formula relating it to the magnetic field. [1]

## 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: [2]

${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} \,,\quad \mathbf {E} =-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}\,,}$

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 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 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 and remembering that the curl of the gradient is zero:

{\displaystyle {\begin{aligned}\nabla \cdot \mathbf {B} &=\nabla \cdot \left(\nabla \times \mathbf {A} \right)=0\\\nabla \times \mathbf {E} &=\nabla \times \left(-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}\right)=-{\frac {\partial }{\partial t}}\left(\nabla \times \mathbf {A} \right)=-{\frac {\partial \mathbf {B} }{\partial t}}.\end{aligned}}}

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

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 closed loop, Γ, is equal to the magnetic flux, ΦB, through a surface, S, that it encloses:

${\displaystyle \oint _{\Gamma }\mathbf {A} \,\cdot \,d{\mathbf {\Gamma } }=\iint _{S}\nabla \times \mathbf {A} \,\cdot \,d\mathbf {S} =\Phi _{\mathbf {B} }.}$

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 (also called axial vector), the vector potential A is a polar vector. [3] 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. [3]

### 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 (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:

${\displaystyle \nabla \cdot \mathbf {A} +{\frac {1}{c^{2}}}{\frac {\partial \phi }{\partial t}}=0}$ [2]

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: [2]

{\displaystyle {\begin{aligned}\nabla ^{2}\phi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\phi }{\partial t^{2}}}&=-{\frac {\rho }{\epsilon _{0}}}\\\nabla ^{2}\mathbf {A} -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}&=-\mu _{0}\mathbf {J} \end{aligned}}}

In other 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 [2] and Jackson [4] ) 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):

{\displaystyle {\begin{aligned}\mathbf {A} \left(\mathbf {r} ,t\right)&={\frac {\mu _{0}}{4\pi }}\int _{\Omega }{\frac {\mathbf {J} \left(\mathbf {r} ',t'\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\,\mathrm {d} ^{3}\mathbf {r} '\\\phi \left(\mathbf {r} ,t\right)&={\frac {1}{4\pi \epsilon _{0}}}\int _{\Omega }{\frac {\rho \left(\mathbf {r} ',t'\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\,\mathrm {d} ^{3}\mathbf {r} '\end{aligned}}}

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

${\displaystyle t'=t-{\frac {\left|\mathbf {r} -\mathbf {r} '\right|}{c}}}$.

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

• The Lorenz gauge condition: ${\textstyle \nabla \cdot \mathbf {A} +{\frac {1}{c^{2}}}{\frac {\partial \phi }{\partial t}}=0}$ 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: [5]
{\displaystyle {\begin{aligned}A_{x}\left(\mathbf {r} ,t\right)&={\frac {\mu _{0}}{4\pi }}\int _{\Omega }{\frac {J_{x}\left(\mathbf {r} ',t'\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\,\mathrm {d} ^{3}\mathbf {r} '\\A_{y}\left(\mathbf {r} ,t\right)&={\frac {\mu _{0}}{4\pi }}\int _{\Omega }{\frac {J_{y}\left(\mathbf {r} ',t'\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\,\mathrm {d} ^{3}\mathbf {r} '\\A_{z}\left(\mathbf {r} ,t\right)&={\frac {\mu _{0}}{4\pi }}\int _{\Omega }{\frac {J_{z}\left(\mathbf {r} ',t'\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\,\mathrm {d} ^{3}\mathbf {r} '\end{aligned}}}
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, 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

See Feynman [6] for the depiction of the A field around a long thin solenoid.

Since

${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} }$

assuming quasi-static conditions, i.e.

${\displaystyle {\frac {\partial E}{\partial t}}\rightarrow 0\,\quad \nabla \times \mathbf {A} =\mathbf {B} \,,}$

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:

• the Coulomb gauge is assumed
• the Lorenz gauge is assumed and there is no distribution of charge, ρ = 0
• the Lorenz gauge is assumed and zero frequency is assumed
• the Lorenz gauge is assumed and a non-zero but sufficiently low frequency to neglect ${\textstyle {\frac {1}{c}}{\frac {\partial \phi }{\partial t}}}$ is assumed

### Electromagnetic four-potential

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:

{\displaystyle {\begin{aligned}\partial ^{\mu }A_{\mu }&=0\\\Box A_{\mu }&={\frac {4\pi }{c}}J_{\mu }\end{aligned}}}

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.

## Notes

1. 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.
2. Feynman (1964 , pp. 1515)
3. Jackson (1999 , p. 246)
4. Kraus (1984 , p. 189)
5. Feynman (1964 , p. 11, cpt 15 )

## Related Research Articles

The Klein–Gordon equation is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. It is a quantized version of the relativistic energy–momentum relation. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation. Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but because common spinless particles like the pions are unstable and also experience the strong interaction the practical utility is limited.

An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.

In physics, specifically field theory and particle physics, the Proca action describes a massive spin-1 field of mass m in Minkowski spacetime. The corresponding equation is a relativistic wave equation called the Proca equation. The Proca action and equation are named after Romanian physicist Alexandru Proca.

A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations. The term 'classical field theory' is commonly reserved for describing those physical theories that describe electromagnetism and gravitation, two of the fundamental forces of nature. Theories that incorporate quantum mechanics are called quantum field theories.

Magnetostatics is the study of magnetic fields in systems where the currents are steady. It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equations of magnetostatics can be used to predict fast magnetic switching events that occur on time scales of nanoseconds or less. Magnetostatics is even a good approximation when the currents are not static — as long as the currents do not alternate rapidly. Magnetostatics is widely used in applications of micromagnetics such as models of magnetic storage devices as in computer memory. Magnetostatic focussing can be achieved either by a permanent magnet or by passing current through a coil of wire whose axis coincides with the beam axis.

In differential geometry, the four-gradient is the four-vector analogue of the gradient from vector calculus.

In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring The name is frequently confused with Hendrik Lorentz, who has given his name to many concepts in this field. The condition is Lorentz invariant. The condition does not completely determine the gauge: one can still make a gauge transformation where is a harmonic scalar function. The Lorenz condition is used to eliminate the redundant spin-0 component in the (1/2, 1/2) representation theory of the Lorentz group. It is equally used for massive spin-1 fields where the concept of gauge transformations does not apply at all.

In the physics of gauge theories, gauge fixing denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by a gauge transformation, equivalent to a shear along unphysical axes in configuration space. Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom.

The electric-field integral equation is a relationship that allows the calculation of an electric field (E) generated by an electric current distribution (J).

The Grad–Shafranov equation is the equilibrium equation in ideal magnetohydrodynamics (MHD) for a two dimensional plasma, for example the axisymmetric toroidal plasma in a tokamak. This equation takes the same form as the Hicks equation from fluid dynamics. This equation is a two-dimensional, nonlinear, elliptic partial differential equation obtained from the reduction of the ideal MHD equations to two dimensions, often for the case of toroidal axisymmetry. Taking as the cylindrical coordinates, the flux function is governed by the equation,

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:

The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems.

In electromagnetism and applications, an inhomogeneous electromagnetic wave equation, or nonhomogeneous electromagnetic wave equation, is one of a set of wave equations describing the propagation of electromagnetic waves generated by nonzero source charges and currents. The source terms in the wave equations make the partial differential equations inhomogeneous, if the source terms are zero the equations reduce to the homogeneous electromagnetic wave equations. The equations follow from Maxwell's equations.

There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

In quantum mechanics, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic field. It is the non-relativistic limit of the Dirac equation and can be used where particles are moving at speeds much less than the speed of light, so that relativistic effects can be neglected. It was formulated by Wolfgang Pauli in 1927.

In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light c, so the delay of the fields connecting cause and effect at earlier and later times is an important factor: the signal takes a finite time to propagate from a point in the charge or current distribution to another point in space, see figure below.

In physics, a gauge theory is a type of field theory in which the Lagrangian does not change under local transformations from certain Lie groups.

In analytical mechanics and quantum field theory, minimal coupling refers to a coupling between fields which involves only the charge distribution and not higher multipole moments of the charge distribution. This minimal coupling is in contrast to, for example, Pauli coupling, which includes the magnetic moment of an electron directly in the Lagrangian.

In physics, Rosser's equation aids in understanding the role of displacement current in Maxwell's equations, given that there is no aether in empty space as initially assumed by Maxwell. Due originally to William G.V. Rosser, the equation was labeled by Selvan:

It can thus be seen that Rosser's Equation (19) in terms of transverse current density has actually hidden away the displacement current.

Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

## References

• 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.