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. Starting from Maxwell's equations and assuming that charges are either fixed or move as a steady current J {displaystyle scriptstyle mathbf {J} } , 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 S {displaystyle scriptstyle S} with oriented surface element d S {displaystyle scriptstyle dmathbf {S} } . Where ∇ with the cross denotes curl, J is the current density and H is the magnetic field intensity, the second integral is a line integral around a closed loop C {displaystyle scriptstyle C} with line element l {displaystyle scriptstyle mathbf {l} } . The current going through the loop is I enc {displaystyle scriptstyle I_{ ext{enc}}} . 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 J {displaystyle scriptstyle mathbf {J} } term against the ∂ D / ∂ t {displaystyle scriptstyle partial mathbf {D} /partial t} term. If the J {displaystyle scriptstyle mathbf {J} } term is substantially larger, then the smaller term may be ignored without significant loss of accuracy. A common technique is to solve a series of magnetostatic problems at incremental time steps and then use these solutions to approximate the term ∂ B / ∂ t {displaystyle scriptstyle partial mathbf {B} /partial t} . Plugging this result into Faraday's Law finds a value for E {displaystyle scriptstyle mathbf {E} } (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. If all currents in a system are known (i.e., if a complete description of the current density J ( r ) {displaystyle scriptstyle mathbf {J} (mathbf {r} )} is available) then the magnetic field can be determined, at a position r, from the currents by the Biot–Savart equation::174 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 inductors 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. The value of B {displaystyle scriptstyle mathbf {B} } can be found from the magnetic potential.