Magnetostatics is the study of magnetic fields in systems where the currents are steady (not changing with time). It is the magnetic analogue of electrostatics, where the charges 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.
Applications
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 J {\displaystyle \mathbf {J} } , the equations separate into two equations for the electric field (see electrostatics) and two for the magnetic field. 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 {\displaystyle S} with oriented surface element d S {\displaystyle d\mathbf {S} } . Where ∇ with the cross denotes curl, J is the current density and H is the magnetic field intensity, the second integral is a line integral around a closed loop C {\displaystyle C} with line element l {\displaystyle \mathbf {l} } . The current going through the loop is I enc {\displaystyle I_{\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 J {\displaystyle \mathbf {J} } term against the ∂ D / ∂ t {\displaystyle \partial \mathbf {D} /\partial t} term. If the J {\displaystyle \mathbf {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 ∂ B / ∂ t {\displaystyle \partial \mathbf {B} /\partial t} . Plugging this result into Faraday's Law finds a value for E {\displaystyle \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.
Solving for the magnetic field
Current sources
If all currents in a system are known (i.e., if a complete description of the current density J ( r ) {\displaystyle \mathbf {J} (\mathbf {r} )} is available) then the magnetic field can be determined, at a position r, from the currents by the Biot–Savart equation:
B ( r ) = μ 0 4 π ∫ J ( r ′ ) × ( r − r ′ ) | r − r ′ | 3 d 3 r ′ {\displaystyle \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} '}}
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