Wednesday, October 14, 2009

Magnetostatics

Magnetostatics is the study of static magnetic fields. In electrostatics, the charges are stationary, whereas here, the currents are stationary or dc(direct current). As it turns out magnetostatics is a good approximation even when the currents are not static as long as the currents do not alternate rapidly.

Applications

Magnetostatics as a special case of Maxwell's equations

Starting from Maxwell's equations, the following simplifications 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 \vec{D} = 0 \vec{D} = 0
Gauss's law for magnetism: \vec{\nabla} \cdot \vec{B} = 0 \oint_A \vec{B} \cdot \mathrm{d}\vec{A} = 0
presumption \vec{E} = 0 \vec{E} = 0
Ampère's law: \vec{\nabla} \times \vec{H} = \vec{J} \oint_S \vec{H} \cdot \mathrm{d}\vec{l} = I_{\mathrm{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 \vec{J} term against the \frac{\partial \vec{D}} {\partial t} term. If the \vec{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 \frac{\partial \vec{B}} {\partial t}. Plugging this result into Faraday's Law finds a value for \vec{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

If all currents in a system are known (i.e. if a complete description of \vec{J} is available) then the magnetic field can be determined from the currents by the Biot-Savart equation:
\vec{B}= \frac{\mu_{0}}{4\pi}I \int{\frac{\mathrm{d}\vec{l} \times \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 Ampère'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 \vec{B} can be found from the magnetic potential.
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