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 Hfield when there are no free currents. The magnetic vector potential performs a similar role for the Bfield. Together with the electric potential it can be used to specify both the Efield and the Bfield. 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.
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The magnetic vector potential A is a threedimensional vector field whose curl is the magnetic field, i.e.:
Since the magnetic field is divergencefree (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:
Starting with the above definitions:
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, AharonovBohm effect).
In the SI system, the units of A are voltseconds per metre (V·s·m^{−1}).
It should be noted that the above definition does not define the magnetic vector potential uniquely because, by definition, we can arbitrarily add curlfree 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.
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:
Applying Ampère's law to the above definition we get:
Solenoidality of the magnetic field leads to Laplace's equation for potential:
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.
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 "fourpotential".
One motivation for doing so is that the fourpotential is a mathematical fourvector. Thus, using standard fourvector 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 J is the fourcurrent. The first equation is the Lorenz gauge condition while the second contains Maxwell's equations.
Yet another motivation for creating the electromagnetic fourpotential is that it plays a very important role in quantum electrodynamics.
Redirecting to Magnetic potential
Redirecting to Magnetic potential
