Magnetic Materials & Field Response (Advanced)

Every electron in an atom carries two contributions to its magnetic dipole moment: its orbital angular momentum and its intrinsic spin. The magnetic dipole moment due to spin is $\mu_s = -g_s \mu_B \mathbf{S}/\hbar$, where $\mu_B = e\hbar/2m_e = 9.274 \times 10^{-24}$ J/T is the Bohr magneton and $g_s \approx 2.002$ is the electron spin g-factor. In most filled electron shells, spins cancel in pairs (Pauli exclusion), leaving no net moment—this is why most materials are only weakly magnetic. Atoms with partially filled d- or f-shells (transition metals, rare earths) retain a net moment.

Magnetization $\mathbf{M}$ (units: A/m) is the net dipole moment per unit volume: $\mathbf{M} = n\langle\boldsymbol{\mu}\rangle$, where $n$ is the number density of dipoles and $\langle\boldsymbol{\mu}\rangle$ is the thermally averaged moment. The applied field $\mathbf{H}$ (A/m) is generated by free currents (wires, coils) and is independent of the medium. Inside a linear isotropic material, $\mathbf{M} = \chi_m \mathbf{H}$, giving $\mathbf{B} = \mu_0(1 + \chi_m)\mathbf{H} = \mu_r \mu_0 \mathbf{H} = \mu \mathbf{H}$. The relative permeability $\mu_r = 1 + \chi_m$ is the key material parameter: $\mu_r < 1$ for diamagnets ($\chi_m < 0$, e.g., bismuth: $\chi_m \approx -1.7 \times 10^{-4}$), $\mu_r$ slightly above 1 for paramagnets ($\chi_m > 0$, e.g., aluminum: $\chi_m \approx 2.2 \times 10^{-5}$), and $\mu_r \gg 1$ for ferromagnets.

Diamagnetism arises in all materials but is only observable when no permanent dipoles are present. When an external field is applied, it induces a change in the orbital velocity of electrons—by Faraday's law of induction applied at the atomic scale—creating a small opposing moment. The induced moment per atom is $\Delta\mu = -\frac{e^2 B}{6m_e}\langle r^2 \rangle$, where $\langle r^2 \rangle$ is the mean-square orbital radius. Since $\Delta\mu \propto -B$, the susceptibility $\chi_m$ is small and negative. Diamagnetic materials are repelled by magnetic fields—spectacularly demonstrated by the levitation of a frog in a 16 T field.

In paramagnetic materials, atoms possess permanent dipoles but thermal agitation keeps them randomly oriented, giving $\mathbf{M} = 0$ with no applied field. An external field $\mathbf{H}$ biases the orientations; the resulting magnetization follows the Curie Law: $\chi_m = C/T$, where $C$ is the material-dependent Curie constant and $T$ is temperature in Kelvin. At room temperature, the alignment energy $\mu_B B \approx 10^{-23}$ J is much less than thermal energy $k_BT \approx 4 \times 10^{-21}$ J, so alignment is only partial. This weak, temperature-dependent response distinguishes paramagnets from ferromagnets.

Ferromagnetism cannot be explained by classical physics. A quantum mechanical effect called the exchange interaction causes neighboring electron spins to align parallel, lowering the total energy. In iron, this produces spontaneous magnetization within microscopic regions called Weiss domains (typically 1–100 μm). Each domain is fully magnetized; the directions vary randomly in an unmagnetized sample. Applying $\mathbf{H}$ causes domain walls to move and favorable domains to grow, producing the nonlinear, history-dependent $B$–$H$ curve known as the hysteresis loop. The Curie–Weiss law describes the paramagnetic phase above the Curie temperature $T_C$ (1043 K for iron): $\chi_m = C/(T - T_C)$.

The hysteresis loop is central to magnetic material applications. Starting from an unmagnetized sample, increasing $H$ traces the initial magnetization curve up to saturation magnetization $M_s$. Reducing $H$ to zero leaves residual magnetization called remanence $M_r$ (or $B_r$). To demagnetize the sample, a reverse field equal to the coercive field $H_c$ must be applied. The area enclosed by the loop equals the energy dissipated per cycle per unit volume: $W = \oint B \, dH$ (J/m³). Soft magnetic materials (small $H_c$, low losses) are used in transformer cores; hard magnetic materials (large $H_c$, high remanence) make permanent magnets.

$H = 1591$ A/m, $B \approx 8.0$ T, $\Phi = 1.2 \times 10^{-3}$ Wb

• Magnetic Flux and Gauss's Law for Magnetism — Understanding $\Phi = \int \mathbf{B} \cdot d\mathbf{A}$ is prerequisite for calculating flux through cores with permeability $\mu_r$, as done in the worked example above.
• Ampère's Law and Solenoids — Ampère's law $\oint \mathbf{H} \cdot d\mathbf{l} = NI_{free}$ directly gives the $H$ field in magnetic circuits, the starting point for all permeability calculations.
• Electromagnetic Induction and Faraday's Law — Faraday's law explains both the origin of diamagnetism (induced opposing moments) and the EMF generated when flux through a high-$\mu_r$ core changes in transformers.
• Electron Spin and Quantum Numbers — The Bohr magneton and spin quantum number $m_s = \pm 1/2$ are the microscopic origin of magnetic dipole moments that underpin all magnetic material behavior.