Magnetostatics

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. Magnetostatic focussing can be achieved either by a permanent magnet or by passing current through a coil of wire whose axis coincides with the beam axis.
Applications.
Magnetostatics as a special case of Maxwell's equations.
Starting from Maxwell's equations and assuming that charges are either fixed or move as a steady current formula_1, 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, the first integral is over a surface formula_2 with oriented surface element formula_3. Where ∇ with the cross denotes curl, J is the current density and is the magnetic field intensity, the second integral is a line integral around a closed loop formula_4 with line element formula_5. The current going through the loop is formula_6.
The quality of this approximation may be guessed by comparing the above equations with the full version of Maxwell's equations and considering the importance of the terms that have been removed. Of particular significance is the comparison of the formula_7 term against the formula_8 term. If the formula_7 term is substantially larger, then the smaller term may be ignored without significant loss of accuracy.
Re-introducing Faraday's law.
A common technique is to solve a series of magnetostatic problems at incremental time steps and then use these solutions to approximate the term formula_10. Plugging this result into Faraday's Law finds a value for formula_11 (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.
If all currents in a system are known (i.e., if a complete description of the current density formula_12 is available) then the magnetic field can be determined, at a position r, from the currents by the Biot–Savart equation:
Magnetization.
Strongly magnetic materials (i.e., ferromagnetic, ferrimagnetic or paramagnetic) have a magnetization that is primarily due to electron spin. In such materials the magnetization must be explicitly included using the relation
Except in the case of conductors, electric currents can be ignored. Then Ampère's law is simply
This has the general solution
where formula_17 is a scalar potential. Substituting this in Gauss's law gives
Thus, the divergence of the magnetization, formula_19 has a role analogous to the electric charge in electrostatics and is often referred to as an effective charge density formula_20.
The vector potential method can also be employed with an effective current density