Magnetostatics

Magnetostatics is the branch of classical electromagnetism that studies magnetic fields produced by steady electric currents and time-invariant magnetic phenomena. It assumes currents are constant in time and charges do not accumulate, so the fields it describes do not depend on time. This regime is valid for slowly varying currents or systems where changes occur on timescales much longer than the transit time of signals.

The fundamental equations of magnetostatics arise from Maxwell’s equations in the static limit. The magnetic field B and the auxiliary field H satisfy ∇×B = μ0 J and ∇·B = 0, where J is the total current density. In media with magnetization M, B = μ0(H + M) and ∇×H = J_free, with J_free the free current density. The bound current density is J_bound = ∇×M, and on material surfaces there is a bound surface current K_bound = M × n. The magnetic field from steady currents can be computed with the Biot–Savart law: B(r) = μ0/4π ∫ J(r') × (r − r') / |r − r'|^3 d^3r', including contributions from both volume and surface currents.

Potential formulations arise in regions without free current: since ∇×H = 0 there, H can be written as H = −∇φ_m, and B = μ0(H + M). Boundary conditions link fields across interfaces: the normal component of B is continuous, and the tangential component of H has a jump equal to any free surface current.

Materials commonly encountered include paramagnetic, diamagnetic, and ferromagnetic media, with qualitatively different magnetization responses and, in ferromagnets, phenomena such as hysteresis and saturation. Magnetostatics provides a foundation for understanding permanent magnets, magnetic circuits, and steady-state magnetic fields, while more general time-varying situations require the full Maxwell equations.