Magnetostatics

Magnetostatics is the study of static magnetic fields. In electrostatics, the charges are stationary, whereas here, the currents are steady or dc(direct current). As it turns out magnetostatics is a good approximation even when the currents are not static as long as the currents do not alternate rapidly.

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 ${\vec {J}}$, 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.

Name	Partial differential form	Integral form
Gauss's law for magnetism:	${\vec {\nabla }}\cdot {\vec {B}}=0$	$\oint _{S}{\vec {B}}\cdot \mathrm {d} {\vec {S}}=0$
Ampère's law:	${\vec {\nabla }}\times {\vec {H}}={\vec {J}}$	$\oint _{C}{\vec {H}}\cdot \mathrm {d} {\vec {l}}=I_{\mathrm {enc} }$

The first integral is over a surface $S$ with oriented surface element $d{\vec {S}}$. The second is a line integral around a closed loop $C$ with line element ${\vec {l}}$. The current going through the loop is $I_{\text{enc}}$.

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 ${\vec {J}}$ term against the ${\frac {\partial {\vec {D}}}{\partial t}}$ term. If the ${\vec {J}}$ 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 ${\frac {\partial {\vec {B}}}{\partial t}}$. Plugging this result into Faraday's Law finds a value for ${\vec {E}}$ (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.

Solving magnetostatic problems for currents

If all currents in a system are known (i.e. if a complete description of ${\vec {J}}$ is available) then the magnetic field can be determined from the currents by the Biot-Savart equation:

${\vec {B}}={\frac {\mu _{0}}{4\pi }}I\int {\frac {\mathrm {d} {\vec {l}}\times {\hat {r}}}{r^{2}}}$

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 a complex coil geometry can be integrated in sections, or for a very difficult geometry numerical integration may be used. Since this equation is primarily used to solve linear problems, the complete answer will be a sum of the integral of each component section.

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 ${\vec {B}}$ can be found from the magnetic potential.

Strongly magnetic materials

Further information: [[:Demagnetizing field and Micromagnetics]]

Strongly magnetic materials (i.e., Ferromagnetic, Ferrimagnetic or Paramagnetic) have a magnetization that is primarily due to electron spins. In such materials the magnetization must be explicitly included using the relation

${\vec {B}}=\mu _{0}({\vec {M}}+{\vec {H}}).$

Except in metals, electric currents can be ignored. Then Ampère's law is simply

$\nabla \times {\vec {H}}=0.$

This has the general solution

${\vec {H}}=-\nabla U,$

where $U$ is a scalar potential. Substituting this in Gauss's law gives

$\nabla ^{2}U=\nabla \cdot {\vec {M}}.$

Thus, the divergence of the magnetization, $\nabla \cdot {\vec {M}},$ has a role analogous to the electric charge in electrostatics.

Here "magnetostatic" is a misnomer, since the modified magnetostatic equations can be applied even to fast magnetic switching events where the magnetization is reversing itself in nanoseconds or faster.

See also

• Darwin Lagrangian

References

• Aharoni, Amikam (1996). Introduction to the Theory of Ferromagnetism. Clarendon Press. ISBN 0198517912.
• Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (2006). The Feynman Lectures on Physics. Vol. 2. ISBN 0-8053-9045-6.