Magnetostatics

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 ${\displaystyle \mathbf{J}(\mathbf{r})}$ is available) then the magnetic field can be determined, at a position r, from the currents by the Biot–Savart equation: $${\displaystyle \mathbf{B}(\mathbf{r})=\frac{{\mu }_0}{4\pi }\int \frac{\mathbf{J}({\mathbf{r}}^{\prime })\times \left(\mathbf{r}-{\mathbf{r}}^{\prime }\right)}{|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^3}{\mathrm{d}}^3{\mathbf{r}}^{\prime }}$$

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

The magnetic field can be derived from the vector potential. Since the divergence of the magnetic flux density is always zero, $${\displaystyle \mathbf{B}=\mathrm{\nabla }\times \mathbf{A},}$$ and the relation of the vector potential to current is: $${\displaystyle \mathbf{A}(\mathbf{r})=\frac{{\mu }_0}{4\pi }\int \frac{\mathbf{J}\mathbf{(}{\mathbf{r}}^{\prime }\mathbf{)}}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}{\mathrm{d}}^3{\mathbf{r}}^{\prime }.}$$

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 $${\displaystyle \mathbf{B}={\mu }_0(\mathbf{M}+\mathbf{H}).}$$

Except in the case of conductors, electric currents can be ignored. Then Ampère's law is simply $${\displaystyle \mathrm{\nabla }\times \mathbf{H}=0.}$$

This has the general solution $${\displaystyle \mathbf{H}=-\mathrm{\nabla }{\mathrm{\Phi }}_M,}$$ where ${\displaystyle {\mathrm{\Phi }}_M}$ is a scalar potential. Substituting this in Gauss's law gives $${\displaystyle {\mathrm{\nabla }}^2{\mathrm{\Phi }}_M=\mathrm{\nabla }\cdot \mathbf{M}.}$$

Thus, the divergence of the magnetization, ${\displaystyle \mathrm{\nabla }\cdot \mathbf{M},}$ has a role analogous to the electric charge in electrostatics and is often referred to as an effective charge density ${\displaystyle {\rho }_M}$.

The vector potential method can also be employed with an effective current density $${\displaystyle {\mathbf{J}}_{\mathbf{M}}=\mathrm{\nabla }\times \mathbf{M}.}$$