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Encyclopedia > Magnetic potential

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In physics, the magnetic potential is a method of representing the magnetic field by using a potential value instead of the actual $mathbf{B}$ vector field. There are two methods of relating the magnetic field to a potential field and they give rise to two possible types of magnetic potential, used in different situations. Image File history File links Broom_icon. ... Physics (Greek: (phÃºsis), nature and (phusikÃ©), knowledge of nature) is the science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ... In physics, a magnetic field is an axial vector field that traces out solenoidal lines of force in and around closed electric circuits and bar magnets. ... It has been suggested that this article or section be merged with Scalar potential. ... In physics, a magnetic field is an axial vector field that traces out solenoidal lines of force in and around closed electric circuits and bar magnets. ...

1 Magnetic vector potential
2 Magnetic scalar potential
3 Four dimensional potentials
4 Reality of potential fields
5 See also

Magnetic vector potential

This is the most popular method of defining a magnetic vector potential and used in most physics text books.

The vector potential $mathbf{A}$ used extensively when studying the Lagrangian in classical mechanics (see Lagrangian#Special relativistic test particle with electromagnetism), and in Quantum Mechanics, such as the Schrödinger equation for charged particles or the Dirac equation. A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... Fig. ... The Pauli equation is a SchrÃ¶dinger equation which handles spin. ... In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary spin-Â½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. ...

The magnetic vector potential $mathbf{A(mathbf{x})}$ is a three-dimensional vector field whose curl is the magnetic field in the theory of electromagnetism: Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In vector calculus, curl is a vector operator that shows a vector fields rate of rotation: the direction of the axis of rotation and the magnitude of the rotation. ... In physics, a magnetic field is an axial vector field that traces out solenoidal lines of force in and around closed electric circuits and bar magnets. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ...

$mathbf{B} = nabla times mathbf{A}$

Since the magnetic field is divergence free (i.e. $nabla cdot mathbf{B} = 0$ ), this guarentees that $mathbf{A}$ always exists. (But is not unique, see below on Gauge choices.) In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...

Further, the electric field is related to the magnetic potential (and electric potential) as: This article or section does not cite any references or sources. ...

$mathbf{E} = - nabla Phi - frac { partial mathbf{A} } { partial t }$

Starting with the above definitions:

$nabla cdot mathbf{B} = nabla cdot (nabla times mathbf{A}) = 0$

$nabla times mathbf{E} = nabla times left( - nabla Phi - frac { partial mathbf{A} } { partial t } right) = - frac { partial } { partial t } nabla times mathbf{A} = - frac { partial mathbf{B} } { partial t }$

Note that the divergence of a curl will always give zero. Conveniently, this solves the second and third of Maxwell's Equations automatically, which is to say that a continuous magnetic vector potential field is guaranteed not to result in magnetic monopoles. In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ... In physics, magnetic monopole is a term describing a hypothetical particle that could be quickly clarified to a person familiar with magnets but not electromagnetic theory as a magnet with only one pole. In more accurate terms, it would have net magnetic charge. Interest in the concept stems from particle...

Gauge choices

It should be noted that the above definition does not define the magnetic vector potential uniquely because the divergence might be anything and still have no effect on the magnetic field. Thus, there is a degree of freedom available when choosing a definition. This condition is known as gauge invariance. In physics, a magnetic field is an axial vector field that traces out solenoidal lines of force in and around closed electric circuits and bar magnets. ... Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ...

One possible choice is the Coulomb gauge: In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes the act of removing redundant field variables. ...

$nabla cdot mathbf{A} = 0$ .

Another is the Lorenz gauge (often misspelled "Lorentz"): The Lorenz gauge (or Lorenz gauge condition) was published by the Danish physicist Ludwig Lorenz. ...

$nabla cdot mathbf{A} = - frac { 1 } { c^2 } frac { partial phi } { partial t }$

See Gauge fixing. In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes the act of removing redundant field variables. ...

Decoupling Maxwell's Equations

In the Lorenz Gauge, the remaining of the two Maxwell's equation (in a vacuum) become:

$frac{rho}{varepsilon _0} = nabla cdot mathbf{E} = nabla cdot left( -nabla Phi - frac{ partialmathbf{A} }{partial t} right) = - nabla^2 Phi - frac{partial}{partial t} ( nabla cdot mathbf{A} )$

$= - nabla^2 Phi + frac{1}{c^2} frac{partial^2 Phi}{partial t^2}$

$mu_0 J = nabla times mathbf{B} - mu_0 varepsilon _0 frac{partialmathbf{E}}{partial t} = nabla times ( nabla times mathbf{A} ) + frac{1}{c^2} frac{partial}{partial t} left( nabla Phi + frac{partialmathbf{A}}{partial t} right)$

$= left( nabla (nabla cdot mathbf{A}) - nabla^2 mathbf{A} right) + frac{1}{c^2}nablafrac{partial Phi}{partial t} + frac{1}{c^2}frac{partial^2 mathbf{A}}{partial t^2}$

$= - nabla^2 mathbf{A} + frac{1}{c^2} frac{partial^2 mathbf{A}}{partial t^2}$

Notice that there are now 4 equations (one from the electric potential $Φ$ , three from the components of $mathbf{A}$ ) which are all decoupled from each other - there are now 4 separate differential equations. This contrasts to Maxwell's equation in its original form: 8 coupled equations with 6 unknowns (3 E fields, 3 B fields).

The vector potential is often used when finding the magnetic / electric field of a certain charge / current distribution. First find the potentials, then differentiate to get the desired fields.

Magnetostatic integral formulation

For magnetostatics, the follwing vector integral also defines magnetic vector potential in terms of current density: This article needs to be cleaned up to conform to a higher standard of quality. ...

$mathbf{ A(mathbf{r}) } = frac {mu_0} {4 pi } int frac { mathbf{ J(mathbf{r}') } dV } { |mathbf{r}-mathbf{r}'| }$

This definition uses Green's function and is equivalent to the above definition. Note that the Gauge is fixed in this definition. This form is useful when computing the vector potential $mathbf{A}$ and subsequently $mathbf{B}$ from the current sources $mathbf{J}$ . In mathematics, a Greens function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. ...

Magnetic scalar potential

The magnetic scalar potential is another useful tool in describing the magnetic field around a current source. It is only defined in regions of space in absence of (but could be near) currents.

The magnetic scalar potential is defined by the equation:

$mathbf{H} = - nabla mathbf{psi}$

Applying Ampere's Law to the above definition we get: In physics, Ampères law is the magnetic equivalent of Gausss law, discovered by André-Marie Ampère. ...

$mathbf{J} = nabla times mathbf{H} = - nabla times nabla mathbf{psi} = 0$

Since in any continuous field, the curl of a gradient is zero, this would suggest that magnetic scalar potential fields cannot support any sources. In fact, sources can be supported by applying discontinuities to the potential field (thus the same point can have two values for points along the disconuity). These discontinuities are also known as "cuts". When solving magnetostatics problems using magnetic scalar potential, the source currents must be applied at the discontinuity. This article needs to be cleaned up to conform to a higher standard of quality. ...

The magnetic scalar potential is suited to use around lines/loops of currents, but not a region of space with finite current density. The use of magnetic potential reduces the three components of the magnetic field $mathbf{H}$ to one component $mathbf{psi}$ , making computations and algebraic manipulations easier. It is often used in magnetostatics, but rarely used in other applications.

Four dimensional potentials

In special relativity, the magnetic potential joins with the electric potential into the electromagnetic potential. This may be done by joining a scalar electric potential with a vector magnetic potential or by joining a scalar magnetic potential with a vector electric potential. Either way, the final result must have 4 dimensions. The former method is more popular because the scalar electric potential is widely familiar as voltage and because "the concept of vector electric potential is just too weird to exist in the same universe as decent common-sense folks." The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... This article or section does not cite any references or sources. ... In theoretical physics, the electromagnetic potential is a physical quantity that unifies the electric potential and the vector potential (see also magnetic potential) into a single quantity with four components (four is the dimension of the spacetime). ... International safety symbol Caution, risk of electric shock (ISO 3864), colloquially known as high voltage symbol. ...

In four dimensional notation, the Lorenz gauge may be written more concisely by using the D'Alembertian and the four-current, J: In special relativity, electromagnetism and wave theory, the dAlembert operator, also called dAlembertian, is the Laplace operator of Minkowski space. ... In special and general relativity, the four-current is the Lorentz covariant four-vector that replaces the electromagnetic current density where c is the speed of light, ρ the charge density, and j the conventional current density. ...

$Box^2 A_mu = frac{4 pi}{c} J_mu$

in Gaussian units. This equation can be expanded to yield Maxwell's equations, and by extension the rest of classical electrodynamics. CGS is an acronym for centimetre-gram-second. ...

Reality of potential fields

Since the magnetic field may be defined in terms of the magnetic vector potential field, which one of them is the "real" field? Presuming reality is what can be measured, it is possible to measure $mathbf{B}$ using the Hall effect, while measuring $mathbf{A}$ in a direct way is quite difficult. In physics, a magnetic field is an axial vector field that traces out solenoidal lines of force in and around closed electric circuits and bar magnets. ... Hall effect diagram, showing electron flow (rather than conventional current). ...

The interesting situation occurs that just outside a long solenoid, the value of $mathbf{B}$ is quite small, whereas the value of $mathbf{A}$ in the same region is comparatively large. The Aharonov-Bohm effect was first described as a thought experiment in 1956 and involves making an interference pattern using a stream of electrons passing through a double slit. Placing a magnetised iron whisker between the slits simulates the effect of a long, thin solenoid. In 1985 the experiment was constructed and it was observed that the interference pattern did shift as a result of the solenoid. This suggests that the $mathbf{A}$ field can act in a region where $mathbf{B} = 0$ and thus we can conclude that $mathbf{A}$ is the "real" field. The Aharonov-Bohm effect, sometimes called the Ehrenberg-Siday-Aharonov-Bohm effect, is a quantum mechanical phenomenon by which a charged particle is affected by electromagnetic fields in regions from which the particle is excluded. ... Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ...

Contents

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