See Feynman page 15-11 for a diagram of the magnetic vector potential around a long thin solenoid which also exhibits total internal confinement of the "'B "'field, at least in the infinite limit.
22.
The LP effect can be seen as a result of the requirement that quantum physics be invariant with respect to the gauge choice for the electromagnetic potential, of which the magnetic vector potential "'A "'forms part.
23.
The magnetic Aharonov Bohm effect can be seen as a result of the requirement that quantum physics be invariant with respect to the gauge choice for the electromagnetic potential, of which the magnetic vector potential "'A "'forms part.
24.
Therefore, many equations of electromagnetism can be written either in terms of the "'E "'and "'B "', " or " in terms of the magnetic vector potential and electric potential.
25.
For one charged particle of electric charge in an electromagnetic field, given by the magnetic vector potential defined by the magnetic field & nabla; & times; "'A "'} }, and electric scalar potential, this is:
26.
In magnetostatics where there is no time-varying charge distribution, only the first equation is needed . ( In the context of electrodynamics, the terms " vector potential " and " scalar potential " are used for " magnetic vector potential " and " electric potential ", respectively.
27.
With practical railgun geometries, much more accurate two or three dimensional models of the rail and armature current distributions ( and the associated forces ) can be computed, e . g ., by using finite element methods to solve formulations based on either the scalar magnetic potential or the magnetic vector potential.
28.
However, there is at least one experiment in which value of the \ mathbf { E } and \ mathbf { B } are both zero at the location of a charged particle, but it is nevertheless affected by the presence of a local magnetic vector potential; see the Aharonov Bohm effect for details.
29.
Where \ vec { E } ^ i is an incident electric field, \ vec { J } is a current density, \ vec { A } is the magnetic vector potential, \ phi is the scalar electric potential, and \ sigma the electrical conductivity all at observation point \ vec { r }.
30.
The solutions of Maxwell's equations in the Lorenz gauge ( see Feynman ) with the boundary condition that both potentials go to zero sufficiently fast as they approach infinity are called the retarded potentials, which are the magnetic vector potential and the electric scalar potential due to a current distribution of current density, charge density, and volume ?, within which " ? " and "'J "'are non-zero at least sometimes and some places ):
