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Inclassical electromagnetism,magnetic vector potential (often denotedA) is thevector quantity defined so that itscurl is equal to themagnetic field,B:$\mathrm{\nabla }\times \mathbf{A}=\mathbf{B}$. Together with theelectric potentialφ, the magnetic vector potential can be used to specify theelectric fieldE as well. Therefore, many equations of electromagnetism can be written either in terms of the fieldsE andB, or equivalently in terms of the potentialsφ andA. In more advanced theories such asquantum mechanics, most equations use potentials rather than fields.

Magnetic vector potential was independently introduced byFranz Ernst Neumann^[1] andWilhelm Eduard Weber^[2] in 1845 and in 1846, respectively to discussAmpère's circuital law.^[3]William Thomson also introduced the modern version of the vector potential in 1847, along with the formula relating it to the magnetic field.^[4]

This article uses the SI system.

In theSI system, the units ofA areV·s·m⁻¹ orWb·m⁻¹ and are the same as that ofmomentum per unitcharge, orforce per unitcurrent.

The magnetic vector potential,${\displaystyle \mathbf{A}}$, is avector field, and theelectric potential,${\displaystyle \varphi }$, is ascalar field such that:^[5]$${\displaystyle \mathbf{B}=\mathrm{\nabla }\times \mathbf{A},\mathbf{E}=-\mathrm{\nabla }\varphi -\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t},}$$where${\displaystyle \mathbf{B}}$ is themagnetic field and${\displaystyle \mathbf{E}}$ is theelectric field. Inmagnetostatics where there is no time-varying current orcharge distribution, only the first equation is needed. (In the context ofelectrodynamics, the termsvector potential andscalar potential are used formagnetic vector potential andelectric potential, respectively. In mathematics,vector potential andscalar potential can be generalized to higher dimensions.)

If electric and magnetic fields are defined as above from potentials, they automatically satisfy two ofMaxwell's equations:Gauss's law for magnetism andFaraday's law. For example, if${\displaystyle \mathbf{A}}$ is continuous and well-defined everywhere, then it is guaranteed not to result inmagnetic monopoles. (In the mathematical theory of magnetic monopoles,${\displaystyle \mathbf{A}}$ is allowed to be either undefined or multiple-valued in some places; seemagnetic 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${\displaystyle \mathbf{A}}$ and${\displaystyle \varphi }$ is guaranteed from these two laws usingHelmholtz's theorem. For example, since the magnetic field isdivergence-free (Gauss's law for magnetism; i.e.,${\displaystyle \mathrm{\nabla }\cdot \mathbf{B}=0}$),${\displaystyle \mathbf{A}}$ always exists that satisfies the above definition.

The vector potential${\displaystyle \mathbf{A}}$ is used when studying theLagrangian inclassical mechanics and inquantum mechanics (seeSchrödinger equation for charged particles,Dirac equation,Aharonov–Bohm effect).

Inminimal coupling,${\displaystyle q\mathbf{A}}$ is called the potential momentum, and is part of thecanonical momentum.

Theline integral of${\displaystyle \mathbf{A}}$ over a closed loop,${\displaystyle \mathrm{\Gamma }}$, is equal to themagnetic flux,${\displaystyle {\mathrm{\Phi }}_{\mathbf{B}}}$, through a surface,${\displaystyle S}$, that it encloses:$${\displaystyle {\oint }_{\mathrm{\Gamma }}\mathbf{A}\cdot d\mathbf{\Gamma }={\iint }_S\mathrm{\nabla }\times \mathbf{A}\cdot d\mathbf{S}={\mathrm{\Phi }}_{\mathbf{B}}.}$$

Therefore, the units of${\displaystyle \mathbf{A}}$ are also equivalent toweber permetre. The above equation is useful in theflux quantization ofsuperconducting loops.

In the Coulomb gauge${\displaystyle \mathrm{\nabla }\cdot \mathbf{A}=0}$, there is a formal analogy between the relationship between the vector potential and the magnetic field toAmpere's law${\displaystyle \mathrm{\nabla }\times \mathbf{B}={\mu }_0\mathbf{J}}$. 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,${\displaystyle \mathbf{B}}$, is apseudovector (also calledaxial vector), the vector potential,${\displaystyle \mathbf{A}}$, is apolar vector.^[6] This means that if theright-hand rule forcross products were replaced with a left-hand rule, but without changing any other equations or definitions, then${\displaystyle \mathbf{B}}$ would switch signs, butA would not change. This is an example of a general theorem: The curl of a polar vector is a pseudovector, and vice versa.^[6]

Inmagnetostatics, if the Coulomb gauge${\displaystyle \mathrm{\nabla }\cdot \mathbf{A}=0}$ is imposed, then there is an analogy between${\displaystyle \mathbf{A},\mathbf{J}}$ and${\displaystyle V,\rho }$ inelectrostatics:^[7]$${\displaystyle {\mathrm{\nabla }}^2\mathbf{A}=-{\mu }_0\mathbf{J}}$$just like the electrostatic equation$${\displaystyle {\mathrm{\nabla }}^{2}V=-\frac{\rho }{{ϵ}_0}}$$

Likewise one can integrate to obtain the potentials:$${\displaystyle \mathbf{A}(\mathbf{r})=\frac{{\mu }_0}{4\pi }{\int }_R\frac{\mathbf{J}({\mathbf{r}}^{\prime })}{\left|\mathbf{r}-{\mathbf{r}}^{\prime }\right|}{\mathrm{d}}^3{r}^{\prime }}$$just like the equation for theelectric potential:$${\displaystyle V(\mathbf{r})=\frac{1}{4\pi {\epsilon }_0}{\int }_R\frac{\rho ({\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}{\mathrm{d}}^3{r}^{\prime }}$$

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By equatingNewton's second law with theLorentz force law we can obtain^[7]$${\displaystyle m\frac{\mathrm{d}v}{\mathrm{d}t}=q\left(\mathbf{E}+\mathbf{v}\times \mathbf{B}\right).}$$Dotting this with the velocity yields$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{2}m{v}^2\right)=q\mathbf{v}\cdot \left(\mathbf{E}+\mathbf{v}\times \mathbf{B}\right).}$$With thedot product of thecross product being zero, substituting$${\displaystyle \mathbf{E}=-\mathrm{\nabla }\varphi -\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t},}$$and theconvective derivative of${\displaystyle \varphi }$ in the above equation then gives$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{2}m{v}^2+q\varphi \right)=\frac{\mathrm{\partial }}{\mathrm{\partial }t}q\left(\varphi -\mathbf{v}\cdot \mathbf{A}\right)}$$which tells us the time derivative of the "generalized energy"${\displaystyle \frac{1}{2}m{v}^2+q\varphi }$ in terms of a velocity dependent potential${\displaystyle q\left(\varphi -\mathbf{v}\cdot \mathbf{A}\right)}$, and$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\left(mv+q\mathbf{A}\right)=-\mathrm{\nabla }q\left(\varphi -\mathbf{v}\cdot \mathbf{A}\right)}$$which gives the time derivative of thegeneralized momentum${\displaystyle m\mathbf{v}+q\mathbf{A}}$ in terms of the (minus) gradient of the same velocity dependent potential.

Thus, when the (partial) time derivative of the velocity dependent potential${\displaystyle q(\varphi -\mathbf{v}\cdot \mathbf{A})}$ 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${\displaystyle q\mathbf{r}\times \mathbf{A}}$ to the generalized angular momentum, and rotational symmetries will provide conservation laws for the components.

Relativistically, we have the single equation$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}\tau }\left(p^{\mu }+q{A}^{\mu }\right)={\mathrm{\partial }}_{\nu }\left(U^{\mu }\cdot A^{\mu }\right)}$$where

• ${\displaystyle \tau }$ is theproper time,
• ${\displaystyle p^{\mu }}$ is thefour momentum${\displaystyle (E/c,\gamma m\mathbf{v})}$
• ${\displaystyle U^{\mu }}$ is thefour velocity${\displaystyle \gamma (c,\mathbf{v})}$
• ${\displaystyle A^{\mu }}$ is thefour potential${\displaystyle (\varphi /c,\mathbf{A})}$
• ${\displaystyle {\mathrm{\partial }}_{\nu }}$ is thefour gradient${\displaystyle (\frac{\mathrm{\partial }}{\mathrm{\partial }\left(ct\right)},-\mathrm{\nabla })}$

In a field with electric potential${\displaystyle \varphi }$ and magnetic potential${\displaystyle \mathbf{A}}$, theLagrangian (${\displaystyle \mathcal{L}}$) and theHamiltonian (${\displaystyle \mathcal{H}}$) of a particle with mass${\displaystyle m}$ and charge${\displaystyle q}$ are

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

Consider a charged particle of charge${\displaystyle q}$ located distance${\displaystyle r}$ outside a solenoid oriented on the${\displaystyle z}$ that is suddenly turned off. ByFaraday's law of induction, an electric field will be induced that will impart an impulse to the particle equal to${\displaystyle q{\mathrm{\Phi }}_0/2\pi r\hat{\varphi }}$ where${\displaystyle {\mathrm{\Phi }}_0}$ is the initialmagnetic flux through a cross section of the solenoid.^[8]

We can analyze this problem from the perspective of generalized momentum conservation.^[7] Using the analogy to Ampere's law, the magnetic vector potential is${\displaystyle \mathbf{A}(r)={\mathrm{\Phi }}_0/2\pi r\hat{\varphi }}$. Since${\displaystyle \mathbf{p}+q\mathbf{A}}$ is conserved, after the solenoid is turned off the particle will have momentum equal to${\displaystyle q\mathbf{A}=q{\mathrm{\Phi }}_0/2\pi r\hat{\varphi }}$

Additionally, because of the symmetry, the${\displaystyle z}$ component of the generalized angular momentum is conserved. By looking at thePoynting 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.

The above definition does not define the magnetic vector potential uniquely because, by definition, we can arbitrarily addcurl-free components to the magnetic potential without changing the observed magnetic field. Thus, there is adegree of freedom available when choosing${\displaystyle \mathbf{A}}$. This condition is known asgauge invariance.

Two common gauge choices are

• TheLorenz gauge:$${\displaystyle \mathrm{\nabla }\cdot \mathbf{A}+\frac{1}{c^2}\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }t}=0}$$
• TheCoulomb gauge:$${\displaystyle \mathrm{\nabla }\cdot \mathbf{A}=0}$$

In other gauges, the formulas for${\displaystyle \mathbf{A}}$ and${\displaystyle \varphi }$ are different; for example, seeCoulomb 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 theLorenz gauge where${\displaystyle \mathbf{A}}$ is chosen to satisfy:^[5]$${\displaystyle \mathrm{\nabla }\cdot \mathbf{A}+\frac{1}{c^2}\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }t}=0}$$

Using the Lorenz gauge, theelectromagnetic wave equations can be written compactly in terms of the potentials,^[5]

• Wave equation of the scalar potential
• Wave equation of the vector potential

The solutions of Maxwell's equations in the Lorenz gauge (see Feynman^[5] and Jackson^[9]) with the boundary condition that both potentials go to zero sufficiently fast as they approach infinity are called theretarded potentials, which are the magnetic vector potential${\displaystyle \mathbf{A}(\mathbf{r},t)}$ and the electric scalar potential${\displaystyle \varphi (\mathbf{r},t)}$ due to a current distribution ofcurrent density${\displaystyle \mathbf{J}(\mathbf{r},t)}$,charge density${\displaystyle \rho (\mathbf{r},t)}$, andvolume${\displaystyle \mathrm{\Omega }}$, within which${\displaystyle \rho }$ and${\displaystyle \mathbf{J}}$ are non-zero at least sometimes and some places):

• Solutions

where the fields atposition vector${\displaystyle \mathbf{r}}$ and time${\displaystyle t}$ are calculated from sources at distant position${\displaystyle {\mathbf{r}}^{\prime }}$ at an earlier time${\displaystyle t^{\prime }.}$ The location${\displaystyle {\mathbf{r}}^{\prime }}$ is a source point in the charge or current distribution (also the integration variable, within volume${\displaystyle \mathrm{\Omega }}$). The earlier time${\displaystyle t^{\prime }}$ is called theretarded time, and calculated as$${\displaystyle R=\Vert \mathbf{r}-{\mathbf{r}}^{\prime }\Vert .}$$$${\displaystyle t^{\prime }=t-\frac{R}{c}.}$$

With these equations:

• TheLorenz gauge condition is satisfied:

$${\displaystyle \mathrm{\nabla }\cdot \mathbf{A}+\frac{1}{c^2}\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }t}=0.}$$

• The position of${\displaystyle \mathbf{r}}$, the point at which values for${\displaystyle \varphi }$ and${\displaystyle \mathbf{A}}$ are found, only enters the equation as part of the scalar distance from${\displaystyle {\mathbf{r}}^{\prime }}$ to${\displaystyle \mathbf{r}.}$ The direction from${\displaystyle {\mathbf{r}}^{\prime }}$ to${\displaystyle \mathbf{r}}$ does not enter into the equation. The only thing that matters about a source point is how far away it is.
• The integrand usesretarded time,${\displaystyle t^{\prime }.}$ 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${\displaystyle \mathbf{r}}$ and${\displaystyle t}$, from remote location${\displaystyle {\mathbf{r}}^{\prime }}$ must also be at some prior time${\displaystyle t^{\prime }.}$
• The equation for${\displaystyle \mathbf{A}}$ is a vector equation. In Cartesian coordinates, the equation separates into three scalar equations:^[10] In this form it is apparent that the component of${\displaystyle \mathbf{A}}$ in a given direction depends only on the components of${\displaystyle \mathbf{J}}$ that are in the same direction. If the current is carried in a straight wire,${\displaystyle \mathbf{A}}$ points in the same direction as the wire.

The preceding time domain equations can be expressed in the frequency domain.^[11]: 139

• Lorenz gauge${\displaystyle \mathrm{\nabla }\cdot \mathbf{A}+\frac{j\omega }{c^2}\varphi =0}$ or${\displaystyle \varphi =\frac{j\omega }{k^2}\mathrm{\nabla }\cdot \mathbf{A}}$
• Solutions${\displaystyle \mathbf{A}\left(\mathbf{r},\omega \right)=\frac{{\mu }_0}{4\pi }{\int }_{\mathrm{\Omega }}\frac{\mathbf{J}\left({\mathbf{r}}^{\prime },\omega \right)}{R}{e}^{-jkR}{d}^3{\mathbf{r}}^{\prime }\varphi \left(\mathbf{r},\omega \right)=\frac{1}{4\pi {ϵ}_0}{\int }_{\mathrm{\Omega }}\frac{\rho \left({\mathbf{r}}^{\prime },\omega \right)}{R}{e}^{-jkR}{d}^3{\mathbf{r}}^{\prime }}$
• Wave equations${\displaystyle {\mathrm{\nabla }}^2\varphi +k^2\varphi =-\frac{\rho }{{ϵ}_0}{\mathrm{\nabla }}^2\mathbf{A}+k^2\mathbf{A}=-{\mu }_0\mathbf{J}.}$
• Electromagnetic field equations${\displaystyle \mathbf{B}=\mathrm{\nabla }\times \mathbf{A}\mathbf{E}=-\mathrm{\nabla }\varphi -j\omega \mathbf{A}=-j\omega \mathbf{A}-j\frac{\omega }{k^2}\mathrm{\nabla }(\mathrm{\nabla }\cdot \mathbf{A})}$

where

${\displaystyle \varphi }$ and${\displaystyle \rho }$ are scalarphasors.

${\displaystyle \mathbf{A},\mathbf{B},\mathbf{E},}$ and${\displaystyle \mathbf{J}}$ are vectorphasors.

${\displaystyle k=\frac{\omega }{c}}$

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

• TheLorenz gauge condition is satisfied:${\displaystyle \varphi =-\frac{c^2}{j\omega }\mathrm{\nabla }\cdot \mathbf{A}.}$ This implies that the frequency domain electric potential,${\displaystyle \varphi }$, can be computed entirely from the current density distribution,${\displaystyle \mathbf{J}}$.
• The position of${\displaystyle \mathbf{r},}$ the point at which values for${\displaystyle \varphi }$ and${\displaystyle \mathbf{A}}$ are found, only enters the equation as part of the scalar distance from${\displaystyle {\mathbf{r}}^{\prime }}$ to${\displaystyle \mathbf{r}.}$ The direction from${\displaystyle {\mathbf{r}}^{\prime }}$ to${\displaystyle \mathbf{r}}$ does not enter into the equation. The only thing that matters about a source point is how far away it is.
• The integrand uses thephase shift term${\displaystyle e^{-jkR}}$ which plays a role equivalent toretarded 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${\displaystyle \mathbf{A}}$ is a vector equation. In Cartesian coordinates, the equation separates into three scalar equations:^[10] In this form it is apparent that the component of${\displaystyle \mathbf{A}}$ in a given direction depends only on the components of${\displaystyle \mathbf{J}}$ that are in the same direction. If the current is carried in a straight wire,${\displaystyle \mathbf{A}}$ points in the same direction as the wire.

See Feynman^[12] for the depiction of the${\displaystyle \mathbf{A}}$ field around a long thinsolenoid.

Since$${\displaystyle \mathrm{\nabla }\times \mathbf{B}={\mu }_0\mathbf{J}}$$assuming quasi-static conditions, i.e.

${\displaystyle \frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\to 0}$ and${\displaystyle \mathrm{\nabla }\times \mathbf{A}=\mathbf{B}}$,

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

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

The drawing tacitly assumes${\displaystyle \mathrm{\nabla }\cdot \mathbf{A}=0}$, true under any one of the following assumptions:

• theCoulomb gauge is assumed
• theLorenz gauge is assumed and there is no distribution of charge,${\displaystyle \rho =0}$
• theLorenz gauge is assumed and zero frequency is assumed
• theLorenz gauge is assumed and a non-zero frequency, but still assumed sufficiently low to neglect the term${\displaystyle \frac{1}{c}\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }t}}$

In the context ofspecial relativity, it is natural to join the magnetic vector potential together with the (scalar)electric potential into theelectromagnetic potential, also calledfour-potential.

One motivation for doing so is that the four-potential is a mathematicalfour-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 theLorenz gauge is used. In particular, inabstract index notation, the set ofMaxwell's equations (in the Lorenz gauge) may be written (inGaussian units) as follows:where${\displaystyle {◻}^2}$ is thed'Alembertian and${\displaystyle J}$ is thefour-current. The first equation is theLorenz gauge condition while the second contains Maxwell's equations. The four-potential also plays a very important role inquantum electrodynamics.

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

• 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.{{cite book}}:ISBN / Date incompatibility (help)
• 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.

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