Unit 17: Magnetic Properties of Materials

Magnetic Field Lines and Flux

Magnetic field lines are a visual tool used to represent the direction and strength of a magnetic field. The direction of the field at any point is tangent to the field line, and the density of lines (number per unit area) indicates the magnitude of the magnetic flux density B. Unlike electric field lines, magnetic field lines form continuous closed loops because there are no magnetic monopoles.

Magnetic Flux (Φ)

Magnetic flux through a surface S is defined as the surface integral of the magnetic flux density:

Φ = ∫S **B**·d**A**

For a uniform field and a flat surface of area A making an angle θ with the field direction, this simplifies to:

Φ = B A cosθ

The unit of magnetic flux is the weber (Wb), where 1 Wb = 1 T·m².

Flux Density B in Materials

Inside a material, the magnetic flux density depends not only on the applied magnetizing field H but also on the material’s magnetization M. The fundamental relation is:

B = μ₀ (H + M)

where μ₀ = 4π × 10⁻⁷ T·m/A is the permeability of free space.

For linear, isotropic materials the magnetization is proportional to the magnetizing field:

M = χₘ H

with χₘ the (dimensionless) magnetic susceptibility. Substituting gives:

B = μ₀ (1 + χₘ) H

Defining the relative permeability μᵣ as the ratio of the material’s permeability to that of free space leads to the compact form:

B = μ₀ μᵣ H

and the important link between susceptibility and relative permeability:

μᵣ = 1 + χₘ

Example 1: Flux Density in a Paramagnetic Rod

An aluminium rod (paramagnetic) has a susceptibility χₘ = +2.2 × 10⁻⁵. It is placed in a solenoid producing a magnetizing field H = 500 A/m. Calculate the flux density.

1. Find relative permeability: μᵣ = 1 + χₘ = 1 + 2.2×10⁻⁵ ≈ 1.000022.
2. Compute B: B = μ₀ μᵣ H = (4π×10⁻⁷) × 1.000022 × 500 ≈ 6.2832×10⁻⁴ T.

The result shows that paramagnetism produces only a tiny increase in B over the vacuum value.

Example 2: Flux Density in a Ferromagnetic Core

A ferromagnetic core has μᵣ = 1200 and is subjected to H = 150 A/m. Determine B.

B = μ₀ μᵣ H = (4π×10⁻⁷) × 1200 × 150 ≈ 0.226 T

Such a high B value illustrates why ferromagnets are used to concentrate magnetic flux in transformers and inductors.

Flux Density, Permeability and Susceptibility

This section consolidates the definitions and interrelationships among the key magnetic quantities.

Key Definitions

• Magnetizing field (H): The magnetic field that would exist in a vacuum produced by free currents; measured in amperes per metre (A/m).
• Magnetic flux density (B): Also called magnetic induction; represents the total magnetic field inside a material; measured in tesla (T).
• Magnetization (M): Magnetic dipole moment per unit volume; measures how strongly the material responds to H; unit A/m.
• Relative permeability (μᵣ): Ratio of the material’s permeability to that of free space; dimensionless.
• Magnetic susceptibility (χₘ): Ratio of magnetization to magnetizing field; dimensionless; indicates ease of magnetization.

Mathematical Relationships

Typical Values

Example 3: Determining Susceptibility from Experimental Data

A sample of unknown material is placed in a magnetizing field H = 800 A/m. The measured flux density is B = 1.256 × 10⁻³ T. Find χₘ.

1. Compute the vacuum flux density for the same H: B₀ = μ₀ H = (4π×10.4π×10⁻⁶) × 800 = 1.0053×10⁻³ T.
2. Find relative permeability: μᵣ = B / B₀ = 1.256×10⁻³ / 1.0053×10⁻³ ≈ 1.249.
3. Obtain susceptibility: χₘ = μᵣ − 1 ≈ 0.249.

The positive χₘ of order 10⁻¹ indicates a strongly paramagnetic (or possibly weak ferromagnetic) response.

Hysteresis

When a ferromagnetic material is subjected to a cyclically varying magnetizing field H, the magnetic flux density B does not follow the same path on increasing and decreasing H. This lag of magnetization behind the magnetizing field is called hysteresis. The resulting B–H curve forms a closed loop known as the hysteresis loop.

Features of the Hysteresis Loop

• Saturation (Bₛ): The maximum flux density attained when all magnetic domains are aligned.
• Retentivity (Bᵣ): The residual flux density when the magnetizing field is reduced to zero (H = 0). It measures the material’s ability to retain magnetism.
• Coercivity (H_c): The reverse magnetizing field required to bring the flux density back to zero (B = 0). It quantifies the resistance to demagnetization.
• Area enclosed by the loop: Represents the energy loss per unit volume per cycle, dissipated as heat (hysteresis loss).

Mathematical Expression for Hysteresis Loss

For a sinusoidal variation of H with peak value Hₘ, the hysteresis loss per unit volume per cycle is often approximated by the Steinmetz equation:

Ph = η Bₘα f

where η is a material constant, α ≈ 1.5–2.5, Bₘ is the maximum flux density, and f is the frequency of magnetization reversal.

Example 4: Estimating Hysteresis Loss in a Transformer Core

Consider a silicon‑steel transformer core with η = 0.02 J/m³·Tα, α = 1.6, peak flux density Bₘ = 1.5 T, and operating frequency f = 50 Hz. Compute the hysteresis loss per unit volume.

Ph = 0.02 × (1.5)1.6 × 50 ≈ 0.02 × 2.05 × 50 ≈ 2.05 J/m³ per cycle

Multiplying by the frequency gives a power loss density of about 2.05 × 50 ≈ 102.5 W/m³. This illustrates why low‑loss (low η) materials are essential for efficient transformers.

Soft vs. Hard Magnetic Materials

Diamagnetic, Paramagnetic and Ferromagnetic Materials

Materials are classified according to the sign and magnitude of their magnetic susceptibility χₘ and their response to an external magnetic field.

Diamagnetic Materials

• χₘ < 0 (negative, typically −10⁻⁵ to −10⁻⁶).
• Weakly repelled by magnetic fields.
• Arises from Lenz’s law: induced magnetic moments oppose the applied field.
• All electrons are paired; no permanent magnetic dipoles.
• Examples: Copper (χₘ ≈ −1.0×10⁻⁵), Silver (−2.6×10⁻⁶), Bismuth (−1.7×10⁻⁴), Water (−9.0×10⁻⁶).
• Applications: Magnetic levitation (e.g., levitating frogs), shielding sensitive equipment.

Paramagnetic Materials

• χₘ > 0 (positive, typically 10⁻⁵ to 10⁻³).
• Weakly attracted to magnetic fields.
• Due to unpaired electron spins that align with the field, but thermal motion disrupts alignment.
• Magnetization disappears when the field is removed (no retentivity).
• Examples: Aluminium (χₘ ≈ +2.2×10⁻⁵), Platinum (+2.6×10⁻⁴), Manganese (+3.0×10⁻³), Oxygen gas (+1.8×10⁻⁶ at STP).
• Applications: Oxygen sensors, magnetic resonance imaging (MRI) contrast agents (gadolinium‑based complexes).

Ferromagnetic Materials

• χₘ ≫ 0 (often 10²–10⁵).
• Strongly attracted to magnetic fields; can retain magnetization after the external field is removed.
• Existence of magnetic domains: regions where atomic magnetic moments are uniformly aligned.
• In an unmagnetized state, domains are randomly oriented, giving zero net magnetization. An applied field causes domain walls to move and domains to grow aligned with the field.
• Above the Curie temperature (TC) thermal energy disrupts domain alignment, and the material becomes paramagnetic.
• Examples: Iron (χₘ ≈ 200), Cobalt (~60), Nickel (~100), alloys such as Permalloy (χₘ ~ 8000).
• Applications: Electromagnets, transformers, magnetic storage (hard disks), motors, generators, magnetic shielding.

Domain Theory of Ferromagnetism

Ferromagnetic materials consist of microscopic regions called domains (typically 0.1–1 mm in size). Within each domain, the magnetic dipole moments of atoms are aligned due to strong exchange interaction, producing a spontaneous magnetization. In the absence of an external field, domains are oriented randomly, resulting in zero net magnetization.

When an external magnetizing field H is applied:

1. Domains favorably aligned with H grow at the expense of unfavorably oriented ones (domain wall motion).
2. With increasing H, domains rotate to align more closely with the field.
3. At saturation, essentially all domains are aligned, giving the maximum flux density Bₛ.
4. Upon reducing H to zero, domain walls may remain pinned, leaving a net magnetization (retentivity).
5. Reversing the field requires overcoming this pinning, which defines the coercivity.