Magnetic vector potential

The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential (often called simply the vector potential) and the magnetic scalar potential. (However, the magnetic vector potential is more commonly encountered than the magnetic scalar potential.) The magnetic scalar potential is analogous to the electric potential which defines the electric field in electrostatics and is used to specify the H-field when there are no free currents. The magnetic vector potential performs a similar role for the B-field. Together with the electric potential it can be used to specify both the E-field and the B-field. Advanced theories such as relativity and quantum mechanics use the magnetic vector potential and the electric scalar potential instead of the electric and magnetic fields.

Magnetic vector potential

The magnetic vector potential A is a three-dimensional vector field whose curl is the magnetic field, i.e.:

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

Since the magnetic field is divergence-free (Gauss's law), i.e. ▽ · B = 0, A always exists (Helmholtz's theorem).

Unlike the magnetic field, the electric field is derived from both the scalar and vector potentials:

$\mathbf {E} =-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}.$

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.

The vector potential A 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 the SI system, the units of A are volt-seconds per metre (V·s·m⁻¹).

Gauge choices

Main article: Gauge fixing

It should be noted that 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 A. This condition is known as gauge invariance.

Maxwel equations in terms of vector potential

$\nabla ^{2}\psi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\psi }{\partial t^{2}}}=-\rho /\epsilon _{0}$

$\nabla ^{2}{\textbf {A}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}{\textbf {A}}}{\partial t^{2}}}=-\mu _{0}{\textbf {J}}$

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 the absence of currents.

The magnetic scalar potential is defined by the equation:

$\mathbf {B} =-\mu _{0}\nabla \mathbf {\psi } .$

Applying Ampère's law to the above definition we get:

$\mathbf {J} ={\frac {1}{\mu _{0}}}\nabla \times \mathbf {B} =-\nabla \times \nabla \mathbf {\psi } =0.$

Solenoidality of the magnetic field leads to Laplace's equation for potential:

$\nabla ^{2}\mathbf {\psi } =0.$

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 discontinuity). These discontinuities are also known as "cuts". When solving magnetostatics problems using magnetic scalar potential, the source currents must be applied at the discontinuity.

Electromagnetic four-potential

Main article: Electromagnetic four-potential

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:

$\partial ^{\mu }A_{\mu }=0\,$

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

where □ is the d'Alembertian and J is the four-current. The first equation is the Lorenz gauge condition while the second contains Maxwell's equations.

Yet another motivation for creating the electromagnetic four-potential is that it plays a very important role in quantum electrodynamics.

References

• Ulaby, Fawwaz (2007). Fundamentals of Applied Electromagnetics, Fifth Edition. Pearson Prentice Hall. pp. 226–228. ISBN 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.