Closed and Exact Differential Forms - Application in Electrodynamics

Application in Electrodynamics

In electrodynamics, the case of the magnetic field produced by a stationary electrical current is important. There one deals with the vector potential of this field. This case corresponds to k=2, and the defining region is the full The current-density vector is It corresponds to the current two-form

For the magnetic field one has analogous results: it corresponds to the induction two-form and can be derived from the vector potential, or the corresponding one-form ,

Thereby the vector potential corresponds to the potential one-form

The closedness of the magnetic-induction two-form corresponds to the property of the magnetic field that it is source-free: i.e. there are no magnetic monopoles.

In a special gauge, this implies for i = 1, 2, 3

(Here is a constant, the magnetic vacuum permeability.)

This equation is remarkable, because it corresponds completely to a well-known formula for the electrical field, namely for the electrostatic Coulomb potential of a charge density . At this place one can already guess that

• and

• and

• and

can be unified to quantities with six rsp. four nontrivial components, which is the basis of the relativistic invariance of the Maxwell equations.

If the condition of stationarity is left, on the l.h.s. of the above-mentioned equation one must add, in the equations for to the three space coordinates, as a fourth variable also the time t, whereas on the r.h.s., in the so-called "retarded time", must be used, i.e. it is added to the argument of the current-density. Finally, as before, one integrates over the three primed space coordinates. (As usual c is the vacuum velocity of light.)