Editing Magnetostatics (section)

==Solving for the magnetic field==

===Current sources===
[[File:Magnetostatics_relation_triangle.svg|thumb|Summary of magnetostatic relations between magnetic vector potential, magnetic field and current density. Here, <math>\mathbf r = \mathbf x - \mathbf{x'}</math>.]]
If all currents in a system are known (i.e., if a complete description of the current density <math> \mathbf{J}(\mathbf{r})</math> is available) then the magnetic field can be determined, at a position '''r''', from the currents by the [[Biot–Savart law|Biot–Savart equation]]:<ref name="jackson75">{{cite book|last1=Jackson|first1=John David|title=Classical electrodynamics | date=1975 | publisher=Wiley | location=New York | isbn=047143132X | edition = 2nd | url=https://archive.org/details/classicalelectro00jack_0 }}</ref>{{rp|174}}
<math display="block">\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int{\frac{\mathbf{J}(\mathbf{r}') \times \left(\mathbf{r} - \mathbf{r}'\right)}{|\mathbf{r} - \mathbf{r}'|^3} \mathrm{d}^3\mathbf{r}'}</math>

This technique works well for problems where the medium is a [[vacuum]] or air or some similar material with a [[Permeability (electromagnetism)|relative permeability]] of 1. This includes [[Inductor#Air-core inductor|air-core inductor]]s and [[air-core transformer]]s. 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, [[magnetic fringing|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 scalar potential|magnetic potential]]. The value of <math> \mathbf{B}</math> can be found from the magnetic potential.

The magnetic field can be derived from the [[Magnetic vector potential|vector potential]]. Since the divergence of the magnetic flux density is always zero,
<math display="block"> \mathbf{B} = \nabla \times \mathbf{A}, </math>
and the relation of the vector potential to current is:<ref name="jackson75"/>{{rp|176}}
<math display="block"> \mathbf{A}(\mathbf{r}) = \frac{\mu_{0}}{4\pi} \int{ \frac{\mathbf{J(\mathbf{r}')} } {|\mathbf{r}-\mathbf{r}'|} \mathrm{d}^3\mathbf{r}'}. </math>

===Magnetization===
{{further|Demagnetizing field|Micromagnetics}}
Strongly magnetic materials (i.e., [[Ferromagnetism|ferromagnetic]], [[Ferrimagnetism|ferrimagnetic]] or [[Paramagnetism|paramagnetic]]) have a [[magnetization]] that is primarily due to [[Spin (physics)|electron spin]]. In such materials the magnetization must be explicitly included using the relation
<math display="block"> \mathbf{B} = \mu_0(\mathbf{M}+\mathbf{H}).</math>

Except in the case of conductors, electric currents can be ignored. Then Ampère's law is simply
<math display="block"> \nabla\times\mathbf{H} = 0.</math>

This has the general solution
<math display="block"> \mathbf{H} = -\nabla \Phi_M, </math>
where <math>\Phi_M</math> is a scalar [[potential]].<ref name="jackson75"/>{{rp|192}} Substituting this in Gauss's law gives
<math display="block"> \nabla^2 \Phi_M = \nabla\cdot\mathbf{M}.</math>

Thus, the divergence of the magnetization, <math> \nabla\cdot\mathbf{M},</math> has a role analogous to the electric charge in electrostatics<ref>{{cite book |last = Aharoni |first = Amikam |author-link = Amikam Aharoni |title = Introduction to the Theory of Ferromagnetism |publisher = [[Clarendon Press]] |year = 1996 |isbn = 0-19-851791-2 |url = https://archive.org/details/introductiontoth00ahar }}</ref> and is often referred to as an effective charge density <math>\rho_M</math>.

The vector potential method can also be employed with an effective current density
<math display="block"> \mathbf{J_M} = \nabla \times \mathbf{M}. </math>