Magnetostatics is the study of magnetic fields in systems where the currents are steady (not changing with time). It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equations of magnetostatics can be used to predict fast magnetic switching events that occur on time scales of nanoseconds or less. Magnetostatics is even a good approximation when the currents are not static as long as the currents do not alternate rapidly. Magnetostatics is widely used in applications of micromagnetics such as models of magnetic storage devices as in computer memory.

Applications

Magnetostatics as a special case of Maxwell's equations

Starting from Maxwell's equations, the equations separate into two equations for the electric field (see electrostatics) and two for the magnetic field. The fields are independent of time and each other. The magnetostatic equations, in both differential and integral forms, are shown in the table below.Where ∇ with the dot denotes divergence, and B is the magnetic flux density. Where ∇ with the cross denotes curl, J is the current density and is the magnetic field intensity.

The quality of this approximation may be guessed by comparing the above equations with the full version of Maxwell's equations term is substantially larger, then the smaller term may be ignored without significant loss of accuracy.

Re-introducing Faraday's law

. Plugging this result into Faraday's Law (which had previously been ignored). This method is not a true solution of Maxwell's equations but can provide a good approximation for slowly changing fields.

Solving for the magnetic field

Current sources

is available) then the magnetic field can be determined, at a position r, from the currents by the Biot–Savart equation:

This technique works well for problems where the medium is a vacuum or air or some similar material with a relative permeability of 1. This includes air-core inductorss and air-core transformers. One advantage of this technique is that, if a coil has a complex geometry, it can be divided into sections and the integral evaluated for each section. Since this equation is primarily used to solve linear problems, the contributions can be added. For a very difficult geometry, numerical integration may be used.

For problems where the dominant magnetic material is a highly permeable magnetic core with relatively small air gaps, a magnetic circuit approach is useful. When the air gaps are large in comparison to the magnetic circuit length, fringing becomes significant and usually requires a finite element calculation. The finite element calculation uses a modified form of the magnetostatic equations above in order to calculate magnetic potential can be found from the magnetic potential.

The magnetic field can be derived from the vector potential and the relation of the vector potential to current is:

Magnetization

Strongly magnetic materials (i.e., ferromagnetic, ferrimagnetic or paramagnetic) have a magnetization that is primarily due to electron spin

is a scalar potential.

has a role analogous to the electric charge in electrostatics.

See also

• Darwin Lagrangian

• Electrostatics

References

External links