magnetostatic

Magnetostatics is the study of magnetic fields in systems where currents and magnetization are steady in time, so that time-varying electric fields and displacement currents are negligible. It is a quasi-static approximation of classical electromagnetism used when frequencies are low and electrical conductors carry constant currents.

In magnetostatics, Maxwell's equations reduce to ∇·B = 0 and ∇×H = Jf, with B = μ0(H + M). In regions without free current (Jf = 0), ∇×H = 0 and H can be expressed as the gradient of a scalar potential: H = -∇ΦM, while B remains divergence-free.

The magnetization M describes how a material responds to the magnetic field; the total magnetic field B includes contributions from B = μ0(H + M). Bound currents Jb = ∇×M and surface bound currents Kb = M × n account for magnetic effects due to magnetization, while free currents Jf are sources of magnetic fields as well.

Across interfaces, B and H satisfy boundary conditions: the normal component of B is continuous, and the tangential component of H is continuous except for any surface free current. These conditions guide the analysis of magnetic circuits and material interfaces.

Applications of magnetostatics include the design of electromagnets, magnetic circuits and cores, permanent magnets, magnetic shielding, and sensors such as Hall effect devices and fluxgate magnetometers. Typical problems involve computing magnetic fields in iron cores, solenoids, and toroidal geometries, and understanding how material properties influence field distribution.

Magnetostatics applies when currents are steady and displacement currents are negligible. At higher frequencies or rapidly changing fields, full electrodynamics or quasi-static approximations are required, where time dependence cannot be ignored.