Magnetic vector potential explained

In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: $\nabla \times \mathbf = \mathbf$ . 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]

Unit conventions

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.

Magnetic vector potential

The magnetic vector potential,

, is a vector field, and the electric potential,

\phi

, is a scalar field such that:

\mathbf = \nabla \times \mathbf\, \quad \mathbf = -\nabla \phi - \frac

,where

is the magnetic field and

is the electric field. In magnetostatics where there is no time-varying current or 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: $\begin \nabla \cdot \mathbf &= \nabla \cdot \left(\nabla \times \mathbf\right) = 0\,\\ \nabla \times \mathbf &= \nabla \times \left(-\nabla\phi - \frac \right) = -\frac \left(\nabla \times \mathbf\right) = -\frac ~.\end$

Alternatively, the existence of

and

\phi

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.,

\nabla ⋅ B=0

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,

\Gamma

, is equal to the magnetic flux,

\Phi_B

, through a surface,

, that it encloses:

\oint_\Gamma \mathbf\, \cdot\ d = \iint_S \nabla\times\mathbf\ \cdot\ d \mathbf = \Phi_\mathbf ~.

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.^[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

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

See main article: Gauge fixing.

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.

Two common gauge choices are

The Lorenz gauge: $\ \nabla \cdot \mathbf + \frac \frac = 0$
The Coulomb gauge: $\ \nabla \cdot \mathbf = 0$

Lorenz gauge

See main article: Retarded potential.

In other gauges, the formulas for

and

\phi

are different; for example, see Coulomb gauge for another possibility.

Time domain

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:

\ \nabla \cdot \mathbf + \frac \frac = 0

Using the Lorenz gauge, the electromagnetic wave equations can be written compactly in terms of the potentials,

Wave equation of the scalar potential

$\begin \nabla^2\phi - \frac \frac &= - \frac \\[2.734ex]
\end$

Wave equation of the vector potential

$\begin \nabla^2\mathbf - \frac \frac &= - \mu_0\ \mathbf\end$

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

\phi(r,t)

due to a current distribution of current density

J(r,t)

, charge density

\rho(r,t)

, and volume

\Omega

, within which

\rho

and

are non-zero at least sometimes and some places):

Solutions

and time

are calculated from sources at distant position

at an earlier time

t'.

The location

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

\Omega

). The earlier time

is called the retarded time, and calculated as

R = \bigl\|\mathbf - \mathbf' \bigr\| ~.

t' = t - \frac ~.

time domain notes

The Lorenz gauge condition is satisfied:

$\ \nabla \cdot \mathbf + \frac\frac = 0 ~.$

The position of

, the point at which values for

\phi

and

are found, only enters the equation as part of the scalar distance from

The direction from

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 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

t'.

The equation for

is a vector equation. In Cartesian coordinates, the equation separates into three scalar equations:

$\begin A_x\left(\mathbf, t\right) &= \frac \int_\Omega\fracR\ d^3\mathbf'\, \qquad A_y\left(\mathbf, t\right) &= \frac \int_\Omega\fracR\ d^3\mathbf'\,\qquad A_z\left(\mathbf, t\right) &= \frac \int_\Omega\fracR\ d^3\mathbf' ~.\end$ In this form it is apparent 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 straight wire,

points in the same direction as the wire.

Frequency domain

The preceding time domain equations can be expressed in the frequency domain.

Lorenz gauge

\nabla ⋅ A+

	j\omega
	c²

\phi=0

\phi=

	j\omega
	k²

\nabla ⋅ A

Solutions

A\left(r,\omega\right)=

	\mu₀
	4\pi

\int_\Omega

	J\left(r',\omega\right)
	R

e^-jkRd³r' \phi\left(r,\omega\right)=

	1
	4\pi\epsilon₀

\int_\Omega

	\rho\left(r',\omega\right)
	R

e^-jkRd³r'

Wave equations

\nabla²\phi+k²\phi=-

	\rho
	\epsilon₀

\nabla²A+k²A=-\mu_0 J.

Electromagnetic field equations

B=\nabla x A E=-\nabla\phi-j\omegaA=-j\omegaA-j

	\omega
	k²

\nabla(\nabla ⋅ A)

where

\phi

and

\rho

are scalar phasors.

A,B,E,

and

are vector phasors.

	\omega
	c

frequency domain notes

There are a few notable things about

and

\phi

calculated in this way:

The Lorenz gauge condition is satisfied:

\phi=-

	c²
	j\omega

\nabla ⋅ A.

This implies that the electric potential,

\phi

, can be computed entirely from the current density distribution,

The position of

r ,

the point at which values for

\phi

and

are found, only enters the equation as part of the scalar distance from

r~.

The direction from

does not enter into the equation. The only thing that matters about a source point is how far away it is.

The integrand uses the phase shift term

e^-jkR

which plays a role equivalent to retarded time. This reflects the fact that changes in the sources propagate at the speed of light; propagation delay in the time domain is equivalent to a phase shift in the frequency domain.

The equation for

is a vector equation. In Cartesian coordinates, the equation separates into three scalar equations:

$\begin \mathbf_x\!\left(\mathbf, \omega \right) &= \frac \int_\Omega \fracR\ e^ \ d^3\mathbf', \qquad \mathbf_y\!\left(\mathbf, \omega \right) &= \frac \int_\Omega \fracR\ e^ \ d^3\mathbf', \qquad \mathbf_z\!\left(\mathbf, \omega \right) &= \frac \int_\Omega \fracR\ e^ \ d^3\mathbf' \end$

In this form it is apparent 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 straight wire,

points in the same direction as the wire.

Depiction of the A-field

See Feynman for the depiction of the

field around a long thin solenoid.

Since $\nabla \times \mathbf = \mu_0\ \mathbf$ assuming quasi-static conditions, i.e.

	\partialE
	\partialt

\to0

and

\nabla x A=B

,the lines and contours of

relate to

like the lines and contours of

relate to

J~.

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

The drawing tacitly assumes

\nabla ⋅ A=0

, 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,

\rho=0

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

	1
	c

	\partial\phi
	\partialt

Electromagnetic four-potential

See main article: 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: $\begin \partial^\nu A_\nu &= 0 \\ \Box^2 A_\nu &= \frac\ J_\nu\end$ where

\Box²

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.

Charged particle in a field

In a field with electric potential

\phi

and magnetic potential

, the Lagrangian (

l{L}

) and the Hamiltonian (

l{H}

) of a particle with mass

and charge

are

\begin\mathcal &= \frac m\ \mathbf v^2 + q\ \mathbf v \cdot \mathbf A - q\ \phi\,\\\mathcal &= \frac\left(q\ \mathbf A- \mathbf\right)^2 + q\ \phi ~.\end

References

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

Notes and References

Neumann . Franz Ernst . Allgemeine Gesetze der induzirten elektrischen Ströme (General laws of induced electrical currents) . Annalen der Physik . 143 . 11 . 31–34 . January 1, 1846. 10.1002/andp.18461430103 .
Yang . ChenNing . The conceptual origins of Maxwell's equations and gauge theory . Physics Today . 67 . 11 . 45–51 . 2014 . 10.1063/PT.3.2585 . 2014PhT....67k..45Y .
Web site: Richard . Fitzpatrick . Tensors and pseudo-tensors . lecture notes . . Austin, TX .

Magnetic vector potential explained

Unit conventions

Magnetic vector potential

Gauge choices

Lorenz gauge

Time domain

time domain notes

Frequency domain

frequency domain notes

Depiction of the A-field

Electromagnetic four-potential

Charged particle in a field

See also

References

Notes and References