Magnetostatics

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Electromagnetism
Electricity · Magnetism
Electrostatics
 · Electric charge · Coulomb’s law · Electric field · Electric flux · Gauss’ law · Electric potential · Electrostatic induction · Electric dipole moment ·
Magnetostatics
 · Ampère’s law · Electric current · Magnetic field · Magnetic flux · Biot–Savart law · Magnetic dipole moment · Gauss’s law for magnetism ·
Electrodynamics
 · Free space · Lorentz force law · EMF · Electromagnetic induction · Faraday’s law · Displacement current · Maxwell’s equations · EM field · Electromagnetic radiation · Liénard-Wiechert Potentials · Maxwell tensor · Eddy current ·
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 · Electrical conduction · Electrical resistance · Capacitance · Inductance · Impedance · Resonant cavities · Waveguides ·
Covariant formulation
 · Electromagnetic tensor · EM Stress-energy tensor · Four-current · Four-potential ·
Scientists
 · Ampere · Coulomb · Faraday · Heaviside · Henry · Hertz · Lorentz · Maxwell · Tesla · Weber ·
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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.

Contents

Applications

Magnetostatics as a special case of Maxwell's equations

Starting from Maxwell's equations, the following simplifications can be made:

\vec{D} = 0\vec{D} = 0\vec{\nabla} \cdot \vec{B} = 0\oint_A \vec{B} \cdot \mathrm{d}\vec{A} = 0\vec{E} = 0\vec{E} = 0\vec{\nabla} \times \vec{H} = \vec{J}\oint_S \vec{H} \cdot \mathrm{d}\vec{l} = I_{\mathrm{enc}}
Name Partial differential form Integral form
presumption
Gauss's law for magnetism:
presumption
Ampère's law:
\vec{J}

Re-introducing Faraday's law

\frac{\partial \vec{B}} {\partial t}

Solving magnetostatic problems

\vec{J}\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).

\vec{B}