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

Applications of Magnetostatics

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Magnetostatics as a special case of Maxwell's equations

Starting from Maxwell's equations, the following approximations can be made:

• ignore any electrostatic charge
• ignore the electric field
• presume the magnetic field is constant with respect to time

Name	Partial differential form	Integral form
presumption	$\backslash mathbf\{D\}\; =\; 0$	$\backslash mathbf\{D\}\; =\; 0$
Gauss's law for magnetism:	$\backslash nabla\; \backslash cdot\; \backslash mathbf\{B\}\; =\; 0$	$\backslash oint\_A\; \backslash mathbf\{B\}\; \backslash cdot\; d\backslash mathbf\{A\}\; =\; 0$
presumption	$\backslash mathbf\{E\}\; =\; 0$	$\backslash mathbf\{E\}\; =\; 0$
Ampere's law:	$\backslash nabla\; \backslash times\; \backslash mathbf\{H\}\; =\; \backslash mathbf\{J\}$	$\backslash oint\_S\; \backslash mathbf\{H\}\; \backslash cdot\; d\backslash mathbf\{s\}\; =\; I\_\{\backslash mathrm\{enc\}\}\{dt\}$

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

Solving Magnetostatic Problems

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

$\backslash mathbf\{B\}=\; \backslash frac\{\backslash mu\_\{0\}\}\{4\backslash pi\}\; \backslash int\{\backslash frac\{d\backslash mathbf\{I\}\; \backslash times\; \backslash mathbf\{\backslash hat\; r\}\}\{r^2\}\}$

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.

One pitfall in the use of the Biot-Savart equation 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 Biot-Savart 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 $\backslash mathbf\{B\}$ can be found from the magnetic potential.