Quasistatic approximation

In electromagnetism, magnetostatics equations such as Ampère's Law or the more general Biot–Savart law allow one to solve for the magnetic fields produced by steady electrical currents. Often, however, one may want to calculate the magnetic field due to time varying currents (accelerating charge) or other forms of moving charge. Strictly speaking, in these cases the aforementioned equations are invalid, as the field measured at the observer must incorporate distances measured at the retarded time, that is the observation time minus the time it took for the field (traveling at the speed of light) to reach the observer. It is important to realize that the retarded time is different for every point of a charged object, hence the resulting equations are quite complicated; often it is easier to formulate the problem in terms of potentials; see retarded potential and Jefimenko's equations.

In most situations, however, provided that the velocities involved are small compared to the speed of light, one may invoke the quasistatic approximation. This simply permits one to assume that the magnetostatic equations will yield approximately correct values provided the v/c fraction remains small. (Indeed, to first order, the mistake of using only Biot–Savart's law rather than both terms of Jefimenko's magnetic field equation and using time instead of retarded time will fortuitously cancel).[1]

More generally, the quasistatic approximation can be understood through the idea that the sources in the problem change sufficiently slowly that the system can be taken to be in equilibrium at all times. This approximation can then be applied to areas beyond classical electromagnetism such as fluid mechanics, magnetohydrodynamics, thermodynamics, and generally, any hyperbolic partial differential equation. In this case, the typical spatial scale divided by the typical temporal scale is much smaller than the characteristic velocity with which information is propagated along the characteristics.

Quasi-static approximations of Maxwell equations[2]

Notes

  1. ↑ Griffiths, David J., Introduction to Electrodynamics -3rd Ed., 1999.
  2. ↑ G. Rubinacci, F. Villone March 2002: link for download