Source: https://en.wikipedia.org/wiki/Magnetic_vector_potential
In classical electromagnetism, magnetic vector potential (often denoted A) is the vector quantity defined so that its curl is equal to the magnetic field, 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.
Magnetic vector potential was independently introduced by Franz Ernst Neumann[1] and Wilhelm Eduard Weber[2] in 1845 and in 1846, respectively to discuss Ampère'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 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, , is a vector field, and the electric potential, , is a scalar field such that:[5]
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:
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.
In the Coulomb gauge , there is a formal analogy between the relationship between the vector potential and the magnetic field to Ampere's law . 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, , is a pseudovector (also called axial vector), the vector potential, , is a polar vector.[6] 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.[6]
Magnetostatics in the Coulomb gauge[edit]
In magnetostatics, if the Coulomb gauge is imposed, then there is an analogy between and in electrostatics:[7]
just like the electrostatic equation
Likewise one can integrate to obtain the potentials:
just like the equation for the electric potential:
Interpretation as potential momentum[edit]
By equating Newton's second law with the Lorentz force law we can obtain[7]
Dotting this with the velocity yields
With the dot product of the cross product being zero, substituting
and the convective derivative of in the above equation then gives
which tells us the time derivative of the "generalized energy" in terms of a velocity dependent potential , and
which gives the time derivative of the generalized momentum in terms of the (minus) gradient of the same velocity dependent potential.
Thus, when the (partial) time derivative of the velocity dependent potential 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 to the generalized angular momentum, and rotational symmetries will provide conservation laws for the components.
Relativistically, we have the single equation
where
Analytical mechanics of a charged particle[edit]
In a field with electric potential and magnetic potential , the Lagrangian () and the Hamiltonian () of a particle with mass and charge are
The generalized momentum is . The generalized force is . These are exactly the quantities from the previous section. In this framework, the conservation laws come from Noether's theorem.
Consider a charged particle of charge located distance outside a solenoid oriented on the 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 where is the initial magnetic 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 . Since is conserved, after the solenoid is turned off the particle will have momentum equal to
Additionally, because of the symmetry, the 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.
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
In other gauges, the formulas for and 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 is chosen to satisfy:[5]
Using the Lorenz gauge, the electromagnetic 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 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):
Solutions
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
where
With these equations:
The preceding time domain equations can be expressed in the frequency domain.[11]: 139
where
There are a few notable things about and calculated in this way:
Depiction of the A-field[edit]
Representing the Coulomb gauge magnetic vector potential , magnetic flux density and current density 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 field coming out of the picture, plus signs represent field going into the picture. has been assumed.
See Feynman[12] 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:
Electromagnetic four-potential[edit]
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.
By the Helmholtz theorem, a vector field is described completely by its divergence and curl. As A was initially defined solely by its curl (), we are justified by choosing any definition of , provided that we consistently use this definition in all subsequent analysis. All such definitions are valid, and lead to different sets of equations which describe the same phenomena, and the solutions of the equations for any choice of lead to the same electromagnetic fields, and the same physical predictions about the fields and charges.
It is natural to think that if a quantity exhibits this degree of freedom in its choice, then it should not be interpreted as a real physical quantity. After all, if we can freely choose to be anything, then is not unique. One may ask: what is the "true" value of measured in an experiment? If is not unique, then the only logical answer must be that we can never measure the value of . On this basis, it is often stated that it is not a real physical quantity and it is believed that the fields and are the true physical quantities.
However, there is at least one experiment in which value of the and are both zero at the location of a charged particle, but it is nevertheless affected by the presence of a local magnetic vector potential; see the Aharonov–Bohm effect for details. Nevertheless, even in the Aharonov–Bohm experiment, the divergence never enters the calculations; only along the path of the particle determines the measurable effect.
Magnetic scalar potential
Aharonov–Bohm effect
Gluon field
^ 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.
^ W. E. Weber, Elektrodymische Maassbestimungen, uber ein allgemeines Grundgesetz der elektrischen Wirkung, Abhandlungen bei Begrund der Koniglichen Sachsischen Gesellschaft der Wissenschaften (Leipzig, 1846), pp. 211–378 [W. E. Weber, Wilhelm Weber’s Werkes, Vols. 1–6 (Berlin, 1892–1894); Vol. 3, pp. 25–214].
^ Wu, A. C. T.; Yang, Chen Ning (2006-06-30). "Evolution of the Concept of the Vector Potential in the Description of Fundamental Interactions". International Journal of Modern Physics A. 21 (16): 3235–3277. Bibcode:2006IJMPA..21.3235W. doi:10.1142/S0217751X06033143. ISSN 0217-751X.
^ 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.
^ Jump up to: a b c d Feynman (1964), chpt. 15
^ Jump up to: a b Fitzpatrick, Richard. "Tensors and pseudo-tensors" (lecture notes). Austin, TX: University of Texas.
^ Jump up to: a b c Mark D. Semon and John R. Taylor (1996). "Thoughts on the magnetic vector potential". American Journal of Physics. 64 (11): 1361–1369. Bibcode:1996AmJPh..64.1361S. doi:10.1119/1.18400.
^ Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1964). "17". The Feynman Lectures on Physics. Vol. 2. Addison-Wesley. ISBN 978-0-201-02115-8.
^ Jackson (1999), p. 246
^ Jump up to: a b Kraus (1984), p. 189
^ Balanis, Constantine A. (2005), Antenna Theory (third ed.), John Wiley, ISBN 047166782X
^ 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.
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