Magnetic moment

The magnetic moment m is the vector that fully characterises a magnetic dipole. For a current loop, m equals the current I times the loop's area A, pointing along the normal n̂ defined by the right-hand rule (curl fingers along the current direction, thumb gives m). For a magnetised lump of material, m is the volume integral of the magnetisation M (which is itself the magnetic moment per unit volume).

Magnetic moment determines everything about how a dipole interacts with an external magnetic field. The torque on a dipole is τ = m × B (which is why a compass needle rotates to align with Earth's field). The potential energy is U = −m·B (which is minimised when m is parallel to B, maximised when antiparallel — the basis of every NMR experiment and every MRI scan). The force on a dipole in a non-uniform field is F = ∇(m·B), which is why a strong magnet attracts iron filings: the filings polarise into induced dipoles, and the field gradient pulls them toward the source.

The natural unit for atomic magnetic moments is the Bohr magneton, μ_B = eℏ/(2m_e) ≈ 9.274 × 10⁻²⁴ J/T — the magnetic moment of an electron orbiting the proton in the ground state of hydrogen, and a useful scale for any electronic moment. The electron's intrinsic spin moment is about 1.001 μ_B (with QED corrections); proton magnetic moments are about a thousand times smaller (m_p ≈ 1.41 × 10⁻²⁶ J/T) because nuclear masses are much larger. A typical bar magnet's macroscopic moment is on the order of 0.1–1 A·m², built from the cumulative alignment of about 10²² atomic moments.