Magnetostatics

Starting from Maxwell's equations and assuming that charges are either fixed or move as a steady current $\backslash mathbf\{J\}$, the equations separate into two equations for the electric field (see electrostatics) and two for the magnetic field.[https://feynmanlectures.caltech.edu/II_13.html The Feynman Lectures on Physics Vol. II Ch. 13: Magnetostatics] The fields are independent of time and each other. The magnetostatic equations, in both differential and integral forms, are shown in the table below.

Where ∇ with the dot denotes divergence, and B is the magnetic flux density, the first integral is over a surface $S$ with oriented surface element $d\backslash mathbf\{S\}$. Where ∇ with the cross denotes curl, J is the current density and {{math|H}} is the magnetic field intensity, the second integral is a line integral around a closed loop $C$ with line element $\backslash mathbf\{l\}$. The current going through the loop is $I\_\backslash text\{enc\}$.

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 $\backslash mathbf\{J\}$ term against the $\backslash partial\; \backslash mathbf\{D\}\; /\; \backslash partial\; t$ term. If the $\backslash mathbf\{J\}$ term is substantially larger, then the smaller term may be ignored without significant loss of accuracy.

A common technique is to solve a series of magnetostatic problems at incremental time steps and then use these solutions to approximate the term $\backslash partial\; \backslash mathbf\{B\}\; /\; \backslash partial\; t$. 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.{{Citation needed|date=October 2010}}

File:Magnetostatics_relation_triangle.svg

If all currents in a system are known (i.e., if a complete description of the current density $\backslash mathbf\{J\}(\backslash mathbf\{r\})$ is available) then the magnetic field can be determined, at a position r, from the currents by the Biot–Savart equation:{{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 }}{{rp|174}}

$$\backslash mathbf\{B\}(\backslash mathbf\{r\})\; =\; \backslash frac\{\backslash mu\_0\}\{4\backslash pi\}\; \backslash int\{\backslash frac\{\backslash mathbf\{J\}(\backslash mathbf\{r\}\text{'})\; \backslash times\; \backslash left(\backslash mathbf\{r\}\; -\; \backslash mathbf\{r\}\text{'}\backslash right)\}$$

{{further|Demagnetizing field|Micromagnetics}}

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

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

Except in the case of conductors, electric currents can be ignored. Then Ampère's law is simply

$$\backslash nabla\backslash times\backslash mathbf\{H\}\; =\; 0.$$

This has the general solution

$$\backslash mathbf\{H\}\; =\; -\backslash nabla\; \backslash Phi\_M,$$

where $\backslash Phi\_M$ is a scalar potential.{{rp|192}} Substituting this in Gauss's law gives

$$\backslash nabla^2\; \backslash Phi\_M\; =\; \backslash nabla\backslash cdot\backslash mathbf\{M\}.$$

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

The vector potential method can also be employed with an effective current density

$$\backslash mathbf\{J\_M\}\; =\; \backslash nabla\; \backslash times\; \backslash mathbf\{M\}.$$

• Darwin Lagrangian

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Category:Electric and magnetic fields in matter

Category:Potentials