Chapter 36: Ferromagnetism – Magnetic Domains & Inductance

Ferromagnetism – Magnetic Domains & Inductance explores ferromagnetism, a crucial phenomenon in which the net magnetic effects within a material far surpass those found in paramagnetic or diamagnetic substances, raising complex technical questions about the immense strength of these induced magnetic moments. The text establishes an analogy between ferromagnetism and dielectric polarization, defining the magnetization vector (M), which represents the magnetic dipole moment per unit volume. Crucially, the macroscopic effect of magnetization is linked to microscopic atomic current loops (circulating electron currents) within the material, resulting in a magnetization current density (j mag), which is described by the curl operation acting on M. To simplify the resulting Maxwell equations, the auxiliary field H is introduced, derived from B and M, allowing the curl of H to be related primarily to the external conduction currents (j cond). A major characteristic of ferromagnets, illustrated by experiments using toroidal coils, is the non-linear relationship between B and H, which is dependent not just on the instantaneous field strength but also on the material’s magnetic history, graphically represented by the hysteresis loop. This energy loss mechanism is fundamental to understanding practical applications such as iron-core inductors and transformers, where B and H are frequently related by a permeability factor (mu) for small field ranges. The chapter delves into the origin of this strong effect through spontaneous magnetization, utilizing the Weiss mean-field theory to model the tendency of atomic magnets to align against thermal agitation (k times T). This theory yields a formula for magnetization that uses the hyperbolic tangent function, and successfully predicts the existence of a Curie temperature (T sub C), below which the material remains spontaneously magnetized even without an external field. However, the theory implies that the true cause of ferromagnetism lies in powerful, nonmagnetic quantum mechanical interactions between neighboring atoms, rather than classical magnetic forces. Finally, the chapter formalizes the analogy between the equations of electrostatics and static ferromagnetism, which clarifies the roles of B, H, and M in various geometric situations.