Chemistry - Magnetic potential

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.

Contents

1 Magnetic vector potential

1.1 Coulomb gauge
1.2 Magnetostatic integral formulation
1.3 Lorentz gauge

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 potential and used in most physics text books. The magnetic vector potential $\mathbf{A}$ is a three-dimensional vector field whose curl is the magnetic field in the theory of electromagnetism:

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

Starting with the above definition, calculating the divergence of both sides of the equation gives:

$\nabla \cdot \mathbf{B} = \nabla \cdot \nabla \times \mathbf{A} = 0$

Note that the divergence of a curl will always give zero. Conveniently, this solves the second of Maxwell's Equations automatically, which is to say that a continuous magnetic vector potential field is guaranteed not to result in magnetic monopoles.

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 guage invariance.

Coulomb gauge

In order to uniquely define the magnetic vector potential, the following equation constrains the divergence:

$\nabla \cdot \mathbf{A} = 0$

This was named after Charles Augustin de Coulomb.

Magnetostatic integral formulation

For magnetostatics this vector integral defines magnetic vector potential in terms of current density:

$\mathbf{A} = \frac 1 {4 \pi \epsilon_0 c^2} \int \frac { \mathbf{J} dV } { |r| }$

Lorentz gauge

The Lorentz gauge can also be used to uniquely constrain the magnetic vector potential and for magnetostatics gives the same result as the Coulomb gauge. The Lorentz gauge is:

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

This was named after Hendrik Lorentz.

Magnetic scalar potential

The magnetic scalar potential is defined by the equation:

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

Applying Ampere's Law to the above definition we get:

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

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.

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.

In four dimensional notation, the Lorentz guage may be written more concisely by using the D'Alembertian and the four-current, J:

$\Box^2 A_\mu = \frac{4 \pi}{c} J_\mu$

in Gaussian units.

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.

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.