# PlanetPhysics/Magnetostatics

Magnetostatics is the study of static magnetic fields. In electrostatics the charges were stationary but in magnetostatics the currents are steady. As it turns out magnetostatics is a good approximation even when the currents are not steady as long as the currents do not alternate "too fast".

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

${\mathbf {B} }={\frac {\mu _{0}}{4\pi }}\int {\frac {I{\mathbf {d} l}\times {\mathbf {\hat {r}} }}{r^{2}}}$ This technique works well for problems where the medium is a vacuum, 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.

One pitfall in using the Biot-Savart law is that it does not implicitly enforce Gauss's Law for magnetism so it is possible to come up with an answer that includes magnetic monopoles. This will occur if some section of the current path has not been included in the integral (implying that electrons are being continuously created in one place and destroyed in another).

Using the Biot-Savart law in the presence of Ferromagnetic, Ferrimagnetic or Paramagnetic materials is difficult because the external current induces a surface current in the magnetic material which in turn must be included in the integral. The value of the surface current depends on the magnetic field which was what you were trying to calculate in the first place. For these problems, using Ampere's law (usually in integral form) is a better choice. 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 {\mathbf B} can be found from the magnetic potential.