Magnetisation and Susceptibility

To describe magnetism inside matter we need more than the bare field $\vec B$: we must separate the contributions from "free" currents we control from the "bound" currents inside the material. This leads to two auxiliary fields, $\vec H$ and $\vec M$, and three material constants $\chi$, $\mu_r$, $\mu$.

Concept

• Magnetisation $\vec M$: dipole moment per unit volume of the material. Units A/m.
• Magnetic intensity $\vec H$: the part of the field due to free (external) currents alone. Units A/m.
• Susceptibility $\chi$ (dimensionless): how strongly a linear material magnetises in response to $\vec H$:

$\vec M = \chi\,\vec H.$

• Relative permeability $\mu_r$ and absolute permeability $\mu$:

$\mu_r = 1 + \chi,\qquad \mu = \mu_0\mu_r.$

The total magnetic field inside a linear material satisfies

$\vec B = \mu_0(\vec H + \vec M) = \mu_0(1+\chi)\vec H = \mu\vec H.$

Derivation

Consider a long solenoid of $n$ turns/m carrying free current $I_f$. Without a core, the field inside is

$B_0 = \mu_0 n I_f \equiv \mu_0 H,$

so $\vec H = n I_f \hat z$ is set entirely by the free current.

Now fill the solenoid with a magnetic material. The material develops a magnetisation $\vec M$ in response. The bound surface current per unit length on the material's cylindrical surface is exactly $M$, so the total effective surface current per unit length is $nI_f + M$. The field inside becomes

$B = \mu_0(n I_f + M) = \mu_0(H + M).$

For a linear material, $M = \chi H$, giving $B = \mu_0(1+\chi)H = \mu_0\mu_r H$.

Identifying $\vec H$: from $\vec B = \mu_0(\vec H + \vec M)$,

$\vec H = \frac{\vec B}{\mu_0} - \vec M.$

Ampere's circuital law for $\vec H$ involves only the free current:

$\oint \vec H\cdot d\vec\ell = I_{\text{free,enc}}.$

This is why $\vec H$ is useful — it isolates the part of magnetism we can directly control.

Worked Example

A solenoid of $n = 2000$ turns/m carries $I = 1.0$ A. Its iron core has $\mu_r = 800$.

$H = nI = 2000\,A/m,$ $B = \mu_0\mu_r H = (4\pi\times 10^{-7})(800)(2000) = 2.0\,T,$ $M = (\mu_r - 1)H = 799 \times 2000 = 1.6\times 10^{6}\,A/m,$ $\chi = \mu_r - 1 = 799.$

Without the iron core $B_0 = \mu_0 H = 2.5\times 10^{-3}\,T$ — three orders of magnitude weaker.

Common Confusions

• $\vec H$ and $\vec M$ have the same units (A/m); $\vec B$ is in Tesla. Don't add $\vec H$ and $\vec B$ directly.
• $\chi$ here is the volume susceptibility (dimensionless). Some textbooks use a mass susceptibility $\chi/\rho$ — check units.
• For nonlinear materials (ferromagnets), $\chi$ depends on $H$ — it is not a constant.
• The relation $\mu_r = 1 + \chi$ holds for linear, isotropic media; for anisotropic crystals $\chi$ becomes a tensor.

Key Takeaways

• $\vec B = \mu_0(\vec H + \vec M)$ — the fundamental relation in matter.
• For linear materials: $\vec M = \chi\vec H$, $\mu_r = 1 + \chi$, $\mu = \mu_0\mu_r$.
• $\vec H$ is set by free currents only and obeys $\oint \vec H\cdot d\vec\ell = I_{\text{free}}$.
• A high-$\mu_r$ core multiplies the field of an air-core coil by $\mu_r$.