Magnetic Susceptibility & Permeability

When an external magnetic field of strength $H$ (measured in A m⁻¹) is applied to a material, the material develops a magnetisation $M$ (also in A m⁻¹), defined as the magnetic dipole moment per unit volume. For linear magnetic materials, the relationship is simply proportional: $M = \chi_m H$. Here $\chi_m$ is the magnetic susceptibility — dimensionless and characteristic of the material. Paramagnetic materials have small positive values (e.g. $\chi_m \approx 10^{-5}$ for aluminium), diamagnetic materials have small negative values (e.g. $\chi_m \approx -9 \times 10^{-6}$ for water), and ferromagnetic materials like iron have $\chi_m$ values that can exceed $10^5$, though they are not truly linear.

The total magnetic flux density $B$ inside a material accounts for both the applied field and the material's own magnetisation response. Using the SI relation $B = \mu_0 (H + M)$, where $\mu_0 = 4\pi \times 10^{-7}$ T m A⁻¹ is the permeability of free space, and substituting $M = \chi_m H$, we get $B = \mu_0 (H + \chi_m H) = \mu_0 (1 + \chi_m) H$. The factor $(1 + \chi_m)$ is called the relative permeability, $\mu_r$. So $B = \mu_0 \mu_r H = \mu H$, where $\mu = \mu_0 \mu_r$ is the absolute permeability of the material.

The relative permeability $\mu_r = 1 + \chi_m$ is a dimensionless number that compares how well a material supports a magnetic field compared to a vacuum. For vacuum (and approximately for air), $\mu_r = 1$. For paramagnets, $\mu_r$ is just slightly greater than 1. For diamagnets, $\mu_r$ is just slightly less than 1. For soft iron, $\mu_r$ can be around 5000–200,000, meaning field lines are concentrated powerfully inside the iron — exactly what you want in a transformer core. The absolute permeability $\mu$ (in T m A⁻¹ or equivalently H m⁻¹) is what appears directly in Ampère's law for materials: $\nabla \times H = J_f$, with $B = \mu H$.

For paramagnetic materials, susceptibility is not constant — it decreases as temperature rises because thermal agitation disrupts the alignment of magnetic dipoles. Pierre Curie found empirically that $\chi_m = \frac{C}{T}$, where $T$ is the absolute temperature in kelvin and $C$ is the Curie constant, specific to each material. This is the Curie Law. At very low temperatures, susceptibility is large (easy alignment); at high temperatures, it is small (thermal randomness wins). Ferromagnetic materials lose their ferromagnetism above the Curie temperature $T_C$, above which they become paramagnetic and the Curie–Weiss law $\chi_m = C/(T - T_C)$ applies.

$B \approx 2.51\text{ T},\quad \chi_m = 499,\quad M \approx 2.00 \times 10^6\text{ A m}^{-1}$

• Types of Magnetic Materials — Understanding diamagnetic, paramagnetic, and ferromagnetic behaviour is the prerequisite for making sense of why susceptibility values differ so dramatically between material classes.
• Magnetic Flux and Flux Density — Magnetic permeability directly determines the flux density B inside a material, so a firm grasp of flux and flux density (B and Φ) is essential to applying the equation B = μ₀μᵣH correctly.
• Hysteresis and the B–H Curve — Ferromagnetic materials have a susceptibility that varies with field strength and history, leading to hysteresis loops — a direct extension of the permeability concept to nonlinear materials.
• Magnetic Domains — The microscopic origin of high permeability and susceptibility in ferromagnets lies in domain structure — understanding domains explains why μᵣ can reach tens of thousands in soft iron.
• Electromagnetic Induction — Transformers and inductors rely on high-permeability cores to maximise magnetic flux linkage, directly applying the concept of μᵣ to real electrical engineering devices.