Magnetic Materials

All materials respond to an applied magnetic field, but the nature and magnitude of this response varies widely depending on the electronic structure of the material. Classifying materials into diamagnetic, paramagnetic, and ferromagnetic categories based on their relative permeability is essential for selecting core materials in transformers, inductors, and permanent magnets. This classification also appears regularly in GATE questions on electromagnetics.

Magnetic Susceptibility and Permeability

The response of a material to an applied magnetic field H is characterized by the magnetization M, defined as the magnetic dipole moment per unit volume. The relationship M = chi_m * H defines the magnetic susceptibility chi_m. The total flux density inside the material is B = mu_0 (H + M) = mu_0 (1 + chi_m) H = mu_0 * mu_r * H, where the relative permeability mu_r = 1 + chi_m.

For diamagnetic materials, chi_m is a very small negative number (around -10^-5), so mu_r is slightly less than 1. The induced magnetization opposes the applied field. Examples include copper, bismuth, and silicon. This effect is present in all materials but is usually overwhelmed by the other effects.

For paramagnetic materials, chi_m is a small positive number (around 10^-5 to 10^-3), so mu_r is slightly greater than 1. Permanent atomic or molecular magnetic dipoles exist but are randomly oriented. An applied field partially aligns these dipoles, causing weak magnetization in the same direction as the field. Examples include aluminum, platinum, and oxygen.

Ferromagnetic Materials and Magnetic Domains

In ferromagnetic materials such as iron, nickel, and cobalt, exchange interactions between neighboring atoms cause large regions called magnetic domains to form where all atomic dipoles are spontaneously aligned. In an unmagnetized sample, these domains point in random directions so the net magnetization is zero. When an external H is applied, domain walls move and domains aligned with H grow at the expense of others, resulting in a very large net magnetization.

The relationship between B and H in a ferromagnetic material is nonlinear, described by the B-H curve (also called the magnetization curve or hysteresis loop). Key parameters include saturation flux density Bs (maximum B achievable), remanent flux density Br (B remaining after H is removed), and coercive force Hc (H needed to reduce B back to zero). Hard magnetic materials have large Hc and are used for permanent magnets; soft magnetic materials have small Hc and are used for transformer cores.

Practical Understanding

The choice of core material in electromagnetic devices directly depends on these classifications. Transformer cores use soft ferromagnetic materials like silicon steel (mu_r around 5000) to minimize hysteresis losses and maximize inductance. Permanent magnets use hard ferromagnetic or ferrimagnetic materials with large remanence. The effective permeability determines how much flux a core can carry for a given magnetomotive force.

Given:
A ferromagnetic core with mu_r = 4000, cross-section A = 2 cm2 = 2e-4 m2.
Applied H = 500 A/m inside the core.

Why this formula applies:
In a linear approximation (below saturation) B = mu_0 * mu_r * H.
Magnetization M = chi_m * H where chi_m = mu_r - 1.

Formula:
B = mu_r * mu_0 * H
M = (mu_r - 1) * H

Substitution:
B = 4000 * 4*pi*1e-7 * 500
M = (4000 - 1) * 500

Calculation:
mu_0 = 1.2566e-6 H/m
B = 4000 * 1.2566e-6 * 500 = 2.513 T
M = 3999 * 500 = 1.9995e6 A/m

Final Answer:
B = 2.513 T  (approaching saturation for silicon steel)
M = 1.999 x 10^6 A/m

Exam Tip: Remember mu_r = 1 + chi_m. For diamagnetic: chi_m is negative, mu_r less than 1. For paramagnetic: chi_m is small positive, mu_r slightly greater than 1. For ferromagnetic: chi_m is very large positive, mu_r much greater than 1. Ferromagnetics have nonlinear B-H curves; other two are approximately linear.

• Diamagnetic: chi_m is small negative, mu_r less than 1; opposes applied field weakly.
• Paramagnetic: chi_m is small positive, mu_r slightly greater than 1; weak alignment of existing dipoles.
• Ferromagnetic: chi_m is very large positive, mu_r up to 10^5 to 10^6; domain alignment mechanism.
• B-H curve is nonlinear for ferromagnetics; hysteresis loop area represents energy loss per cycle per unit volume.
• Soft magnetic materials (low Hc): used in transformer cores to minimize hysteresis loss.
• Hard magnetic materials (high Hc and high Br): used for permanent magnets.

Quick Revision

• mu_r = 1 + chi_m; B = mu_0 * mu_r * H; M = chi_m * H.
• Diamagnetic: chi_m is negative, repels field slightly (copper, bismuth, silicon).
• Paramagnetic: chi_m is small positive, attracts weakly (aluminum, platinum).
• Ferromagnetic: chi_m is very large positive, domain-based strong attraction (iron, nickel, cobalt).
• Ferromagnetic B-H is nonlinear; characterized by Bs, Br, and Hc.
• Trap: mu_r is not constant for ferromagnetics; it depends on operating point on the B-H curve.
• Hysteresis loss per cycle = area enclosed by B-H loop (in J/m3 per cycle).