Magnetized Matter, Bound Currents and H

The Idea: Magnetization as a Current Density

Imagine a block of material where every atom acts like a tiny bar magnet. If these dipoles are randomly oriented, their effects cancel. If they align, the material becomes magnetized. We define the magnetization \(\vec{M}\) as the net magnetic dipole moment per unit volume. While we cannot see these individual atomic currents, their collective effect is mathematically identical to a continuous flow of charge. This "bound" current is not a flow of charge carriers moving through the material like in a wire, but rather the result of circulating charges within atoms. By treating magnetization as a source of current, we can apply the standard tools of magnetostatics, such as Ampère's law, to matter itself.

Bound Volume and Surface Currents

Consider a volume of magnetized material. The derivation D-056 shows that the vector potential generated by a distribution of dipoles is identical to that generated by a specific current distribution. This current splits into two parts: a volume current and a surface current. The volume current density is given by the curl of the magnetization:

This term is non-zero only where the magnetization changes direction or magnitude within the material. For a uniformly magnetized object, \(\nabla \times \vec{M} = 0\), meaning there is no volume current. However, at the boundary of the material, the magnetization drops abruptly to zero. This discontinuity creates a surface current density:

Here, \(\hat{n}\) is the outward normal to the surface. This surface current is the dominant source of the external magnetic field for uniformly magnetized bodies. It is crucial to note that \(\vec{J}_b\) and \(\vec{K}_b\) are not independent sources; they are entirely determined by \(\vec{M}\). They are "bound" because they cannot be separated from the material itself.

Generalizing Ampère's Law

In vacuum, Ampère's law states that the curl of \(\vec{B}\) is proportional to the current density. In matter, the total current density is the sum of free currents \(\vec{J}_f\) (conduction currents we control) and bound currents \(\vec{J}_b\). Thus, the fundamental equation becomes:

Substituting \(\vec{J}_b = \nabla \times \vec{M}\), we get:

Rearranging terms to group the curls together:

This structure suggests defining a new field quantity, the magnetic field strength \(\vec{H}\), such that:

The beauty of this definition is that the curl of \(\vec{H}\) depends only on the free current:

Integrating over a surface \(S\) bounded by a loop \(C\), we obtain the macroscopic Ampère's law:

This is the primary reason \(\vec{H}\) is useful in engineering and macroscopic physics: it allows us to calculate fields in complex materials by considering only the currents we explicitly drive, ignoring the intricate atomic details.

Boundary Conditions for H

Just as \(\vec{B}\) has boundary conditions derived from Gauss's and Ampère's laws, \(\vec{H}\) has its own conditions. Applying the integral form of \(\nabla \times \vec{H} = \vec{J}_f\) to a rectangular loop straddling the interface between two media (with negligible free surface current), we find that the tangential component of \(\vec{H}\) is continuous:

where \(\vec{K}_f\) is the free surface current density. If there is no free current on the surface, \(\vec{H}_{1t} = \vec{H}_{2t}\). This contrasts with \(\vec{B}\), whose tangential component is discontinuous by \(\mu_0 \vec{K}_{\text{total}}\). The normal component of \(\vec{B}\) is always continuous (\(\nabla \cdot \vec{B} = 0\)), but the normal component of \(\vec{H}\) is discontinuous if the magnetization changes across the boundary.