In physics, the magnetic potential is a method of representing the magnetic field by using a potential value instead of the actual <math>\mathbf{B}</math> 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.
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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 <math>\mathbf{A}</math> 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. The magnetic vector potential <math>\mathbf{A(\mathbf{x})}</math> is a three-dimensional vector field whose curl is the magnetic field in the theory of electromagnetism:
- <math>\mathbf{B} = \nabla \times \mathbf{A}</math>
Since the magnetic field is divergence-free (i.e. <math> \nabla \cdot \mathbf{B} = 0</math>), this guarantees that <math>\mathbf{A}</math> always exists. (But is not unique, see below on Gauge choices.) Further, the electric field is related to the magnetic potential (and electric potential) as:
- <math>\mathbf{E} = - \nabla \Phi - \frac { \partial \mathbf{A} } { \partial t }</math>
Starting with the above definitions:
- <math> \nabla \cdot \mathbf{B} = \nabla \cdot (\nabla \times \mathbf{A}) = 0</math>
- <math> \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 } </math>
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.
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. One possible choice is the Coulomb gauge:
- <math>\nabla \cdot \mathbf{A} = 0</math>.
Another is the Lorenz gauge (often misspelled "Lorentz"):
- <math>\nabla \cdot \mathbf{A} = - \frac { 1 } { c^2 } \frac { \partial \Phi } { \partial t }</math>
See Gauge fixing.
Decoupling Maxwell's equations
In the Lorenz Gauge, the remaining of the two Maxwell's equation (in a vacuum) become:
- <math> \frac{\rho}{\varepsilon _0}
= \nabla \cdot \mathbf{E} = \nabla \cdot \left( -\nabla \Phi - \frac{ \partial\mathbf{A} }{\partial t} \right) = - \nabla^2 \Phi - \frac{\partial}{\partial t} ( \nabla \cdot \mathbf{A} ) </math>
-
- <math>
= - \nabla^2 \Phi + \frac{1}{c^2} \frac{\partial^2 \Phi}{\partial t^2} </math>
- <math> \mu_0 J
= \nabla \times \mathbf{B} - \mu_0 \varepsilon _0 \frac{\partial\mathbf{E}}{\partial t} = \nabla \times ( \nabla \times \mathbf{A} ) + \frac{1}{c^2} \frac{\partial}{\partial t} \left( \nabla \Phi + \frac{\partial\mathbf{A}}{\partial t} \right) </math>
-
- <math>
= \left( \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A} \right) + \frac{1}{c^2}\nabla\frac{\partial \Phi}{\partial t} + \frac{1}{c^2}\frac{\partial^2 \mathbf{A}}{\partial t^2} </math>
-
- <math>
= - \nabla^2 \mathbf{A} + \frac{1}{c^2} \frac{\partial^2 \mathbf{A}}{\partial t^2} </math> Notice that there are now four equations (one from the electric potential <math>\Phi</math>, three from the components of <math>\mathbf{A}</math>) 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 following vector integral also defines magnetic vector potential in terms of current density:
- <math>\mathbf{ A(\mathbf{r}) } = \frac {\mu_0} {4 \pi } \int \frac { \mathbf{ J(\mathbf{r}') } dV } { |\mathbf{r}-\mathbf{r}'| }</math>
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 <math>\mathbf{A}</math> and subsequently <math>\mathbf{B}</math> from the current sources <math>\mathbf{J}</math>.
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:
- <math>\mathbf{B} = - \mu_0 \nabla \mathbf{\psi}</math>
Applying Ampere's law to the above definition we get:
- <math>\mathbf{J} = \frac{1}{\mu_0} \nabla \times \mathbf{B} = - \nabla \times \nabla \mathbf{\psi} = 0</math>
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. 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 <math>\mathbf{B}</math> to one component <math>\mathbf{\psi}</math>, 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 four 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." In four dimensional notation, the Lorenz gauge may be written more concisely by using the D'Alembertian and the four-current, J:
- <math>\Box^2 A_\mu = \frac{4 \pi}{c} J_\mu</math>
in Gaussian units. This equation can be expanded to yield Maxwell's equations, and by extension the rest of classical electrodynamics.
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 <math>\mathbf{B}</math> using the Hall effect, while measuring <math>\mathbf{A}</math> in a direct way is quite difficult. The interesting situation occurs that just outside a long solenoid, the value of <math>\mathbf{B}</math> is quite small, whereas the value of <math>\mathbf{A}</math> 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 <math>\mathbf{A}</math> field can act in a region where <math>\mathbf{B} = 0</math> and thus we can conclude that <math>\mathbf{A}</math> is the "real" field.
See also
References
- Ulaby, Fawwaz (2007). Fundamentals of Applied Electromagnetics, Fifth Edition. Pearson Prentice Hall, 226-228. 0-13-241326-4.
- Jackson, John David (1998). Classical Electrodynamics, Third Edition. John Wiley & Sons.
- Duffin, W.J. (1990). Electricity and Magnetism, Fourth Edition. McGraw-Hill.
