Magnetostatics

 ( Electric And Magnetic Fields In Matter ) 

Magnetostatics is the study of in systems where the currents are steady current (not changing with time). It is the magnetic analogue of electrostatics, where the electric charge are stationary. The magnetization need not be static; the equations of magnetostatics can be used to predict fast magnetic switching events that occur on time scales of nanoseconds or less. Magnetostatics is even a good approximation when the currents are not static – as long as the currents do not alternate rapidly. Magnetostatics is widely used in applications of micromagnetics such as models of magnetic storage devices as in computer memory.

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Applications

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Magnetostatics as a special case of Maxwell's equations 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. 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.

Gauss's law for magnetism	$\backslash mathbf\{\backslash nabla\}\; \backslash cdot\; \backslash mathbf\{B\}\; =\; 0$	$\backslash oint\_S\; \backslash mathbf\{B\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{S\}\; =\; 0$
Ampère's law	$\backslash mathbf\{\backslash nabla\}\; \backslash times\; \backslash mathbf\{H\}\; =\; \backslash mathbf\{J\}$	$\backslash oint\_C\; \backslash mathbf\{H\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{l\}\; =\; I\_\{\backslash mathrm\{enc\}\}$

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 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.

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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 $\backslash partial\; \backslash mathbf\{B\}\; /\; \backslash partial\; t$. 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.

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Solving for the magnetic field

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Current sources 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:

(1975). 047143132X, Wiley. . 047143132X

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

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 $\backslash mathbf\{B\}$ 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, $$\backslash mathbf\{B\}\; =\; \backslash nabla\; \backslash times\; \backslash mathbf\{A\},$$ and the relation of the vector potential to current is: $$\backslash mathbf\{A\}(\backslash mathbf\{r\})\; =\; \backslash frac\{\backslash mu\_\{0\}\}\{4\backslash pi\}\; \backslash int\{\; \backslash frac\{\backslash mathbf\{J(\backslash mathbf\{r\}\text{'})\}\; \}\; \{$$

\mathrm{d}^3\mathbf{r}'}.

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Magnetization Strongly magnetic materials (i.e., Ferromagnetism, Ferrimagnetism or Paramagnetism) 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. 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

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

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See also

• Darwin Lagrangian

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External links

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