The Auxiliary Field H

In the previous section, we discovered that magnetized matter contains internal "bound currents" (\(\mathbf{J}_b\)) that generate magnetic fields. However, there is a practical problem: we cannot directly control these microscopic currents.

In a laboratory, we control the current in the copper wires wrapped around a material (the free current, \(\mathbf{J}_f\)). We need a version of Ampère’s law that relies only on \(\mathbf{J}_f\) and the material properties, ignoring the messy details of \(\mathbf{J}_b\). To do this, we introduce the auxiliary field, H.

1. Ampère’s Law in Magnetized Materials

The total current density is the sum of free and bound currents: \(\mathbf{J} = \mathbf{J}_f + \mathbf{J}_b\). From our study of magnetization, we know that bound current is the curl of magnetization: \(\mathbf{J}_b = \nabla \times \mathbf{M}\).

By rearranging the original Ampère’s law, we can group the terms involving the fields (\(\mathbf{B}\) and \(\mathbf{M}\)) on one side and the free current on the other. This gives us the definition of \(\mathbf{H}\):

\mathbf{H} \equiv \frac{1}{\mu_0}\mathbf{B} - \mathbf{M}

Consequently, Ampère's law takes a new, elegant form that depends only on the currents we can control:

\nabla \times \mathbf{H} = \mathbf{J}_f \quad \text{or} \quad \oint \mathbf{H} \cdot d\mathbf{l} = I_{f_{\text{enc}}}

Electric Analogy: This is perfectly analogous to the electric displacement field \(\mathbf{D}\), defined as \(\epsilon_0 \mathbf{E} + \mathbf{P}\), which obeys \(\nabla \cdot \mathbf{D} = \rho_f\). Both \(\mathbf{H}\) and \(\mathbf{D}\) are mathematical constructs designed to handle matter by factoring out bound charges/currents.

Deep Dive: Deriving Ampère’s Law for H

2. A Deceptive Parallel

It is tempting to think that \(\mathbf{H}\) is "the magnetic field due to free current" just as \(\mathbf{B}\) is the field due to total current. This is generally false.

While the curl of \(\mathbf{H}\) is simple (\(\mathbf{J}_f\)), the divergence of \(\mathbf{H}\) is not necessarily zero.

\nabla \cdot \mathbf{H} = - \nabla \cdot \mathbf{M}

Because \(\nabla \cdot \mathbf{H} \neq 0\) (usually), \(\mathbf{H}\) lines act as if they start and stop inside the material (specifically, at the "poles" where magnetization ends). Therefore, you cannot blindly use the Biot-Savart law for \(\mathbf{H}\) using \(\mathbf{J}_f\).

For example, in a permanent magnet with no free current (\(I_f = 0\)), \(\mathbf{H}\) is not zero; it is determined entirely by the divergence of \(\mathbf{M}\).

Deep Dive: The Copper Rod

3. Boundary Conditions

Just as we did for electrostatics, we must determine how fields change when crossing the interface between two materials (or vacuum and matter). We use the integral forms of the divergence and curl laws.

Field Component	Electric Boundary Condition	Magnetic Boundary Condition
Perpendicular	\(D_{1}^{\perp} - D_{2}^{\perp} = \sigma_f\) (Discontinuous by free charge)	\(B_{1}^{\perp} - B_{2}^{\perp} = 0\) (Always Continuous! No magnetic monopoles)
Parallel	\(E_{1}^{\parallel} - E_{2}^{\parallel} = 0\) (Always Continuous)	\(\mathbf{H}_{1}^{\parallel} - \mathbf{H}_{2}^{\parallel} = \mathbf{K}_f \times \mathbf{\hat{n}}\) (Discontinuous by free surface current)

Notice the symmetry breaking here. For electric fields, the fundamental field \(\mathbf{E}\) is continuous parallel to the surface. For magnetic fields, the fundamental field \(\mathbf{B}\) is continuous perpendicular to the surface.

However, notice that \(H^{\perp}\) is discontinuous if there is a discontinuity in Magnetization (\(M^{\perp}\)), acting like a "magnetic surface charge."

Deep Dive: Boundary Conditions & Shielding