magnetic vector potential

This article uses the SI system.

In the SI system, the units of A are V·s·m⁻¹ or Wb·m⁻¹ and are the same as that of momentum per unit charge, or force per unit current.

The magnetic vector potential, $\backslash mathbf\{A\}$, is a vector field, and the electric potential, $\backslash phi$, is a scalar field such that:{{harvp|Feynman|1964|loc=chpt. 15}}

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

where $\backslash mathbf\{B\}$ is the magnetic field and $\backslash mathbf\{E\}$ 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 $\backslash mathbf\{A\}$ is continuous and well-defined everywhere, then it is guaranteed not to result in magnetic monopoles. (In the mathematical theory of magnetic monopoles, $\backslash mathbf\{A\}$ 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 divergence of the curl is zero and the curl of the gradient is the zero vector:

$$\backslash begin\{align\}$$

\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{align}

Alternatively, the existence of $\backslash mathbf\{A\}$ and $\backslash phi$ 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., $\backslash nabla\; \backslash cdot\; \backslash mathbf\{B\}\; =\; 0$), $\backslash mathbf\{A\}$ always exists that satisfies the above definition.

The vector potential $\backslash mathbf\{A\}$ 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 minimal coupling, $q\; \backslash mathbf\{A\}$ is called the potential momentum, and is part of the canonical momentum.

The line integral of $\backslash mathbf\{A\}$ over a closed loop, $\backslash Gamma$, is equal to the magnetic flux, $\backslash Phi\_\{\backslash mathbf\{B\}\}$, through a surface, $S$, that it encloses:

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

Therefore, the units of $\backslash mathbf\{A\}$ are also equivalent to weber per metre. The above equation is useful in the flux quantization of superconducting loops.

In the Coulomb gauge $\backslash nabla\; \backslash cdot\; \backslash mathbf\{A\}\; =\; 0$, there is a formal analogy between the relationship between the vector potential and the magnetic field to Ampere's law $\backslash nabla\; \backslash times\; \backslash mathbf\{B\}\; =\; \backslash mu\_0\; \backslash mathbf\{J\}$. Thus, when finding the vector potential of a given magnetic field, one can use the same methods one uses when finding the magnetic field given a current distribution.

Although the magnetic field, $\backslash mathbf\{B\}$, is a pseudovector (also called axial vector), the vector potential, $\backslash mathbf\{A\}$, is a polar vector.{{cite web |first=Richard |last=Fitzpatrick |title=Tensors and pseudo-tensors |type=lecture notes |publisher=University of Texas |place=Austin, TX |url=http://farside.ph.utexas.edu/teaching/em/lectures/node120.html }} 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 $\backslash mathbf\{B\}$ 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.

In magnetostatics, if the Coulomb gauge $\backslash \; \backslash nabla\; \backslash cdot\; \backslash mathbf\{A\}\; =\; 0$ is imposed, then there is an analogy between $\backslash mathbf\{A\},\; \backslash mathbf\{J\}$ and $V,\; \backslash rho$ in electrostatics:{{cite journal

| author = Mark D. Semon and John R. Taylor

| title = Thoughts on the magnetic vector potential

| journal = American Journal of Physics

| volume = 64

| issue = 11

| pages = 1361–1369

| year = 1996

| doi = 10.1119/1.18400

| bibcode = 1996AmJPh..64.1361S

| url = https://doi.org/10.1119/1.18400

| url-access = subscription

}}

$$\backslash nabla^2\; \backslash mathbf\{A\}\; =\; -\backslash mu\_0\; \backslash mathbf\{J\}$$

just like the electrostatic equation

$$\backslash nabla^2\; V\; =\; -\backslash frac\{\backslash rho\}\{\backslash epsilon\_0\}$$

Likewise one can integrate to obtain the potentials:

$$\backslash mathbf\{A\}(\backslash mathbf\{r\})\; =\; \backslash frac\{\backslash mu\_0\}\{4\; \backslash pi\}\; \backslash int\_R\; \backslash frac\{\backslash mathbf\{J\}(\backslash mathbf\{r\}\text{'})\}\{\backslash left\; |\; \backslash mathbf\{r\}\; -\; \backslash mathbf\{r\}\text{'}\; \backslash right\; |\}\; \backslash mathrm\{d\}^3\; r\text{'}$$

just like the equation for the electric potential:

$$V(\backslash mathbf\{r\})\; =\; \backslash frac\{1\}\{4\backslash pi\backslash varepsilon\_0\}\; \backslash int\_R\; \backslash frac\{\backslash rho(\backslash mathbf\{r\}\text{'})\}$$

{{context|date=April 2025}}

By equating Newton's second law with the Lorentz force law we can obtain

m\frac{\mathrm{d}v}{\mathrm{d}t}=q \left(\mathbf{E} + \mathbf{v}\times\mathbf{B} \right).

Dotting this with the velocity yields

\frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{1}{2}m v^2\right) = q\mathbf{v} \cdot \left(\mathbf{E} + \mathbf{v}\times\mathbf{B} \right).

With the dot product of the cross product being zero, substituting

$$\backslash mathbf\{E\}\; =\; -\; \backslash nabla\; \backslash phi\; -\; \backslash frac\; \{\; \backslash partial\; \backslash mathbf\{A\}\; \}\; \{\; \backslash partial\; t\; \},$$

and the convective derivative of $\backslash phi$ in the above equation then gives

$$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}\; t\}\; \backslash left\; (\; \backslash frac\{1\}\{2\}\; mv^2\; +\; q\; \backslash phi\; \backslash right\; )\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; q\; \backslash left\; (\; \backslash phi\; -\; \backslash mathbf\{v\}\; \backslash cdot\; \backslash mathbf\{A\}\; \backslash right\; )$$

which tells us the time derivative of the "generalized energy" $\backslash frac\{1\}\{2\}\; mv^2\; +\; q\; \backslash phi$ in terms of a velocity dependent potential $q\; \backslash left\; (\; \backslash phi\; -\; \backslash mathbf\{v\}\; \backslash cdot\; \backslash mathbf\{A\}\; \backslash right\; )$, and

$$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}\; t\}\; \backslash left\; (\; mv\; +\; q\; \backslash mathbf\{A\}\; \backslash right\; )\; =\; -\; \backslash nabla\; q\; \backslash left\; (\; \backslash phi\; -\; \backslash mathbf\{v\}\; \backslash cdot\; \backslash mathbf\{A\}\; \backslash right\; )$$

which gives the time derivative of the generalized momentum $m\; \backslash mathbf\{v\}\; +\; q\; \backslash mathbf\{A\}$ in terms of the (minus) gradient of the same velocity dependent potential.

Thus, when the (partial) time derivative of the velocity dependent potential $q\; (\backslash phi\; -\; \backslash mathbf\{v\}\; \backslash cdot\; \backslash mathbf\{A\})$ is zero, the generalized energy is conserved, and likewise when the gradient is zero, the generalized momentum is conserved. As a special case, if the potentials are time or space symmetric, then the generalized energy or momentum respectively will be conserved. Likewise the fields contribute $q\; \backslash mathbf\{r\}\; \backslash times\; \backslash mathbf\{A\}$ to the generalized angular momentum, and rotational symmetries will provide conservation laws for the components.

Relativistically, we have the single equation

$$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}\; \backslash tau\}\; \backslash left\; (\; p^\{\backslash mu\}\; +\; qA^\{\backslash mu\}\; \backslash right\; )\; =\; \backslash partial\_\{\backslash nu\}\; \backslash left\; (\; U^\{\backslash mu\}\; \backslash cdot\; A^\{\backslash mu\}\; \backslash right\; )$$

where

• $\backslash tau$ is the proper time,
• $p^\{\backslash mu\}$ is the four momentum $(E/c,\; \backslash gamma\; m\; \backslash mathbf\{v\})$
• $U^\{\backslash mu\}$ is the four velocity $\backslash gamma\; (c,\; \backslash mathbf\{v\})$
• $A^\{\backslash mu\}$ is the four potential $(\backslash phi/c,\; \backslash mathbf\{A\})$
• $\backslash partial\_\{\backslash nu\}$ is the four gradient $(\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash left\; (\; ct\; \backslash right\; )\},\; -\backslash nabla)$

In a field with electric potential $\backslash \; \backslash phi\backslash$ and magnetic potential $\backslash \; \backslash mathbf\{A\}$, the Lagrangian ($\backslash \; \backslash mathcal\{L\}\backslash$) and the Hamiltonian ($\backslash \; \backslash mathcal\{H\}\backslash$) of a particle with mass $\backslash \; m\backslash$ and charge $\backslash \; q\backslash$ are$$\backslash begin\{aligned\}$$

\mathcal{L} &= \frac{1}{2} m\ \mathbf v^2 + q\ \mathbf v \cdot \mathbf A - q\ \phi\ ,\\

\mathcal{H} &= \frac{1}{2m}\left( \mathbf{p} - q\mathbf A \right)^2 + q\ \phi ~.

\end{aligned}

The generalized momentum $\backslash mathbf\{p\}$ is $\backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; v\}\; =\; m\; \backslash mathbf\{v\}\; +\; q\; \backslash mathbf\{A\}$. The generalized force is $\backslash nabla\; \backslash mathcal\{L\}\; =\; -q\; \backslash nabla\; \backslash left\; (\; \backslash phi\; -\; \backslash mathbf\{v\}\; \backslash cdot\; \backslash mathbf\{A\}\; \backslash right\; )$. These are exactly the quantities from the previous section. It this framework, the conservation laws come from Noether's theorem.

Consider a charged particle of charge $q$ located distance $r$ outside a solenoid oriented on the $z$ that is suddenly turned off. By Faraday's law of induction, an electric field will be induced that will impart an impulse to the particle equal to $q\; \backslash Phi\_0/2\; \backslash pi\; r\; \backslash hat\{\backslash phi\}$ where $\backslash Phi\_0$ is the initial magnetic flux through a cross section of the solenoid. {{cite book | last1=Feynman | first1=Richard P. | author-link1=Richard Feynman | last2=Leighton | first2=Robert B. | author-link2=Robert B. Leighton | last3=Sands | first3=Matthew | author-link3=Matthew Sands | title=The Feynman Lectures on Physics | volume=2 | chapter=17 | chapter-url=https://www.feynmanlectures.caltech.edu/II_17.html | publisher=Addison-Wesley | year=1964 | isbn=978-0-201-02115-8 }}

We can analyze this problem from the perspective of generalized momentum conservation. Using the analogy to Ampere's law, the magnetic vector potential is $\backslash mathbf\{A\}(r)\; =\; \backslash Phi\_0/2\; \backslash pi\; r\; \backslash hat\{\backslash phi\}$. Since $\backslash mathbf\{p\}\; +\; q\backslash mathbf\{A\}$ is conserved, after the solenoid is turned off the particle will have momentum equal to $q\; \backslash mathbf\{A\}\; =\; q\; \backslash Phi\_0/2\; \backslash pi\; r\; \backslash hat\{\backslash phi\}$

Additionally, because of the symmetry, the $z$ component of the generalized angular momentum is conserved. By looking at the Poynting vector of the configuration, one can deduce that the fields have nonzero total angular momentum pointing along the solenoid. This is the angular momentum transferred to the fields.

{{Main|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 $\backslash mathbf\{A\}$. This condition is known as gauge invariance.

Two common gauge choices are

• The Lorenz gauge: $$\backslash \; \backslash nabla\; \backslash cdot\; \backslash mathbf\{A\}\; +\; \backslash frac\{1\}\{\backslash \; c^2\}\; \backslash frac\{\backslash partial\; \backslash phi\}\{\backslash partial\; t\}\; =\; 0$$
• The Coulomb gauge: $$\backslash \; \backslash nabla\; \backslash cdot\; \backslash mathbf\{A\}\; =\; 0$$

{{see also|Mathematical descriptions of the electromagnetic field#Potential field approach|l1=Potential formulation of electromagnetic field}}

{{Main|Retarded potential}}

In other gauges, the formulas for $\backslash mathbf\{A\}$ and $\backslash phi$ are different; for example, see Coulomb gauge for another possibility.

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 $\backslash mathbf\{A\}$ is chosen to satisfy:

$$\backslash \; \backslash nabla\; \backslash cdot\; \backslash mathbf\{A\}\; +\; \backslash frac\{1\}\{\backslash \; c^2\}\; \backslash frac\{\backslash partial\; \backslash phi\}\{\backslash partial\; t\}\; =\; 0$$

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

• Wave equation of the scalar potential $$\backslash begin\{align\}$$

\nabla^2\phi - \frac{1}{c^2} \frac{\partial^2 \phi}{\ \partial t^2} &= - \frac{\rho}{\epsilon_0} \\[2.734ex]

\end{align}

• Wave equation of the vector potential $$\backslash begin\{align\}$$

\nabla^2\mathbf{A} - \frac{1}{\ c^2} \frac{\partial^2 \mathbf{A}}{\ \partial t^2} &= - \mu_0\ \mathbf{J}

\end{align}

The solutions of Maxwell's equations in the Lorenz gauge (see Feynman and Jackson{{harvp|Jackson|1999|p=246}}) 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 $\backslash mathbf\{A\}(\backslash mathbf\{r\},\; t)$ and the electric scalar potential $\backslash phi(\backslash mathbf\{r\},\; t)$ due to a current distribution of current density $\backslash mathbf\{J\}(\backslash mathbf\{r\},\; t)$, charge density $\backslash rho(\backslash mathbf\{r\},\; t)$, and volume $\backslash Omega$, within which $\backslash rho$ and $\backslash mathbf\{J\}$ are non-zero at least sometimes and some places):

• Solutions $$\backslash begin\{align\}$$

\mathbf{A}\!\left(\mathbf{r}, t\right) &= \frac{\mu_0}{\ 4\pi\ } \int_\Omega \frac{ \mathbf{J}\left(\mathbf{r}' , t'\right) } R\ 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) } R\ d^3\mathbf{r}'

\end{align}

where the fields at position vector $\backslash mathbf\{r\}$ and time $t$ are calculated from sources at distant position $\backslash mathbf\{r\}\text{'}$ at an earlier time $t\text{'}\; .$ The location $\backslash mathbf\{r\}\text{'}$ is a source point in the charge or current distribution (also the integration variable, within volume $\backslash Omega$). The earlier time $t\text{'}$ is called the retarded time, and calculated as

$$R\; =\; \backslash bigl\backslash |\backslash mathbf\{r\}\; -\; \backslash mathbf\{r\}\text{'}\; \backslash bigr\backslash |\; ~.$$

$$t\text{'}\; =\; t\; -\; \backslash frac\{\backslash \; R\; \backslash \; \}\{c\}\; ~.$$

With these equations:

• The Lorenz gauge condition is satisfied:

: $$\backslash \; \backslash nabla\; \backslash cdot\; \backslash mathbf\{A\}\; +\; \backslash frac\{1\}\{\backslash \; c^2\}\backslash frac\{\backslash partial\backslash phi\}\{\backslash partial\; t\}\; =\; 0\; ~.$$

• The position of $\backslash mathbf\{r\}$, the point at which values for $\backslash phi$ and $\backslash mathbf\{A\}$ are found, only enters the equation as part of the scalar distance from $\backslash mathbf\{r\}\text{'}$ to $\backslash mathbf\{r\}\; .$ The direction from $\backslash mathbf\{r\}\text{'}$ to $\backslash mathbf\{r\}$ 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, $t\text{'}\; .$ 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 $\backslash mathbf\{r\}$ and $t$, from remote location $\backslash mathbf\{r\}\text{'}$ must also be at some prior time $t\text{'}.$
• The equation for $\backslash mathbf\{A\}$ is a vector equation. In Cartesian coordinates, the equation separates into three scalar equations:{{harvp|Kraus|1984|p=189}} $$\backslash begin\{align\}$$

A_x\left(\mathbf{r}, t\right) &= \frac{\mu_0}{\ 4\pi\ } \int_\Omega\frac{J_x\left(\mathbf{r}', t'\right)}R\ d^3\mathbf{r}'\ , \qquad

A_y\left(\mathbf{r}, t\right) &= \frac{\mu_0}{\ 4\pi\ } \int_\Omega\frac{J_y\left(\mathbf{r}', t'\right)}R\ d^3\mathbf{r}'\ ,\qquad

A_z\left(\mathbf{r}, t\right) &= \frac{\mu_0}{\ 4\pi\ } \int_\Omega\frac{J_z\left(\mathbf{r}', t'\right)}R\ d^3\mathbf{r}' ~.

\end{align} In this form it is apparent that the component of $\backslash mathbf\{A\}$ in a given direction depends only on the components of $\backslash mathbf\{J\}$ that are in the same direction. If the current is carried in a straight wire, $\backslash mathbf\{A\}$ points in the same direction as the wire.

The preceding time domain equations can be expressed in the frequency domain.{{Citation |last=Balanis |first= Constantine A. |year= 2005 |title= Antenna Theory |edition=third |publisher= John Wiley |isbn= 047166782X |author-link=Constantine A. Balanis}}{{rp|139}}

• Lorenz gauge $\backslash nabla\; \backslash cdot\; \backslash mathbf\{A\}\; +\; \backslash frac\{j\; \backslash omega\}\{c^2\}\; \backslash phi=\; 0\; \backslash qquad$ or $\backslash qquad\; \backslash phi\; =\; \backslash frac\; \{j\; \backslash omega\}\{k^2\}\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{A\}$
• Solutions $\backslash mathbf\{A\}\backslash !\backslash left(\backslash mathbf\{r\},\; \backslash omega\; \backslash right)\; =\; \backslash frac\{\backslash mu\_0\}\{\backslash \; 4\backslash pi\backslash \; \}\; \backslash int\_\backslash Omega\; \backslash frac\{\; \backslash mathbf\{J\}\backslash left(\backslash mathbf\{r\}\text{'}\; ,\; \backslash omega\; \backslash right)\; \}R\backslash \; e^\{-jkR\}\; d^3\; \backslash mathbf\{r\}\text{'}\; \backslash qquad$

\phi\!\left(\mathbf{r}, \omega \right) = \frac{1}{4\pi\epsilon_0} \int_\Omega \frac{ \rho \left(\mathbf{r}', \omega \right) } R \ e^{-jkR} d^3 \mathbf{r}'

• Wave equations $\backslash nabla^2\; \backslash phi\; +\; k^2\; \backslash phi\; =\; -\; \backslash frac\{\backslash rho\}\{\backslash epsilon\_0\}\; \backslash qquad\; \backslash nabla^2\; \backslash mathbf\{A\}\; +\; k^2\; \backslash mathbf\{A\}\; =\; -\; \backslash mu\_0\backslash \; \backslash mathbf\{J\}\; .$
• Electromagnetic field equations $\backslash mathbf\{B\}\; =\; \backslash nabla\; \backslash times\; \backslash mathbf\{A\}\backslash \; \backslash qquad\; \backslash mathbf\{E\}\; =\; -\backslash nabla\; \backslash phi\; -\; j\; \backslash omega\; \backslash mathbf\{A\}\; =\; -\; j\; \backslash omega\; \backslash mathbf\{A\}\; -j\; \backslash frac\; \{\backslash omega\}\{k^2\}\; \backslash nabla\; (\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{A\}\; )$

where

: $\backslash phi$ and $\backslash rho$ are scalar phasors.

: $\backslash mathbf\{A\},\; \backslash mathbf\{B\},\; \backslash mathbf\{E\},$ and $\backslash mathbf\{J\}$ are vector phasors.

: $k\; =\; \backslash frac\; \backslash omega\; c$

There are a few notable things about $\backslash mathbf\{A\}$ and $\backslash phi$ calculated in this way:

• The Lorenz gauge condition is satisfied: $\backslash textstyle\; \backslash phi\; =\; -\backslash frac\; \{c^2\}\{j\; \backslash omega\}\backslash nabla\; \backslash cdot\; \backslash mathbf\{A\}.$ This implies that the frequency domain electric potential, $\backslash phi$, can be computed entirely from the current density distribution, $\backslash mathbf\{J\}$.
• The position of $\backslash mathbf\{r\},$ the point at which values for $\backslash phi$ and $\backslash mathbf\{A\}$ are found, only enters the equation as part of the scalar distance from $\backslash mathbf\{r\}\text{'}$ to $\backslash \; \backslash mathbf\{r\}.$ The direction from $\backslash mathbf\{r\}\text{'}$ to $\backslash mathbf\{r\}$ does not enter into the equation. The only thing that matters about a source point is how far away it is.
• The integrand uses 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 $\backslash mathbf\{A\}$ is a vector equation. In Cartesian coordinates, the equation separates into three scalar equations:{{harvp|Kraus|1984|p=189}} $$\backslash begin\{align\}$$

\mathbf{A}_x\!\left(\mathbf{r}, \omega \right) &= \frac{\mu_0}{4\pi} \int_\Omega \frac{ \mathbf{J}_x \left(\mathbf{r}' , \omega \right) }R\ e^{-jkR} \ d^3\mathbf{r}',

\qquad \mathbf{A}_y\!\left(\mathbf{r}, \omega \right) &= \frac{\mu_0}{4\pi} \int_\Omega \frac{ \mathbf{J}_y \left(\mathbf{r}' , \omega \right) }R\ e^{-jkR} \ d^3\mathbf{r}',

\qquad \mathbf{A}_z\!\left(\mathbf{r}, \omega \right) &= \frac{\mu_0}{4\pi} \int_\Omega \frac{ \mathbf{J}_z \left(\mathbf{r}' , \omega \right) }R\ e^{-jkR} \ d^3\mathbf{r}'

\end{align} In this form it is apparent that the component of $\backslash mathbf\{A\}$ in a given direction depends only on the components of $\backslash \; \backslash mathbf\{J\}\backslash$ that are in the same direction. If the current is carried in a straight wire, $\backslash mathbf\{A\}$ points in the same direction as the wire.

File:Magnetic Vector Potential Circular Toroid.svg magnetic vector potential $\backslash mathbf\{A\}$, magnetic flux density $\backslash mathbf\{B\}$ and current density $\backslash mathbf\{J\}$ fields around a toroidal inductor of circular cross section. Thicker lines ,indicate field lines of higher average intensity. Circles in the cross section of the core represent the $\backslash \; \backslash mathbf\{B\}$ field coming out of the picture, plus signs represent $\backslash mathbf\{B\}$ field going into the picture. $\backslash nabla\backslash cdot\backslash mathbf\{A\}\; =\; 0$ has been assumed.]]

See Feynman{{harvp|Feynman|1964|loc=[https://feynmanlectures.caltech.edu/II_15.html cpt 15]|p=11}} for the depiction of the $\backslash mathbf\{A\}$ field around a long thin solenoid.

Since

$$\backslash nabla\; \backslash times\; \backslash mathbf\{B\}\; =\; \backslash mu\_0\backslash \; \backslash mathbf\{J\}$$

assuming quasi-static conditions, i.e.

: $\backslash frac\{\backslash \; \backslash partial\backslash mathbf\{E\}\backslash \; \}\{\backslash partial\; t\}\; \backslash to\; 0\backslash$ and $\backslash \; \backslash nabla\; \backslash times\; \backslash mathbf\{A\}\; =\; \backslash mathbf\{B\}$,

the lines and contours of $\backslash \; \backslash mathbf\{A\}\backslash$ relate to $\backslash \; \backslash mathbf\{B\}\backslash$ like the lines and contours of $\backslash mathbf\{B\}$ relate to $\backslash \; \backslash mathbf\{J\}\; .$ Thus, a depiction of the $\backslash mathbf\{A\}$ field around a loop of $\backslash mathbf\{B\}$ flux (as would be produced in a toroidal inductor) is qualitatively the same as the $\backslash mathbf\{B\}$ field around a loop of current.

The figure to the right is an artist's depiction of the $\backslash mathbf\{A\}$ 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 {{nowrap|$\backslash mathbf\{A\}$ field.}}

The drawing tacitly assumes $\backslash nabla\; \backslash cdot\; \backslash mathbf\{A\}\; =\; 0$, 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, $\backslash rho\; =\; 0$
• 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 $\backslash textstyle\; \backslash frac\{1\}\{c\}\; \backslash frac\{\backslash partial\backslash phi\}\{\backslash partial\; t\}$

{{clear}}

{{Main|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:

$$\backslash begin\{align\}$$

\partial^\nu A_\nu &= 0 \\

\Box^2 A_\nu &= \frac{ 4\pi }{\ c\ }\ J_\nu

\end{align}

where $\backslash \; \backslash Box^2\backslash$ is the d'Alembertian and $\backslash \; J\backslash$ 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.

• Magnetic scalar potential
• Aharonov–Bohm effect
• Gluon field

{{reflist|2}}

• {{cite book

| last = Duffin

| first = W.J.

| title = Electricity and Magnetism, Fourth Edition

| publisher = McGraw-Hill

| year = 1990

}}

• {{cite book

|last1= Feynman

|first1= Richard P

|last2= Leighton

|first2= Robert B

|last3= Sands

|first3= Matthew

|year= 1964

|title= The Feynman Lectures on Physics Volume 2

|publisher= Addison-Wesley

|isbn= 0-201-02117-X

|ref = {{harvid|Feynman|1964}}

|url = https://feynmanlectures.caltech.edu/II_toc.html

}}

• {{cite book

|last=Jackson

|first= John David

|year= 1999

|title= Classical Electrodynamics

|edition= 3rd

|publisher= John Wiley & Sons

|isbn= 0-471-30932-X

}}

• {{cite book

|last= Kraus

|first= John D.

|year= 1984

|title= Electromagnetics

|edition= 3rd

|publisher= McGraw-Hill

|isbn= 0-07-035423-5

|url-access= registration

|url= https://archive.org/details/electromagnetics00krau

}}

• {{Commons category inline}}

Category:Potentials

Category:Magnetism

Category:Vector physical quantities