Magnetic Materials: Para, Dia, and Ferromagnetism

Movie Hook: Magneto’s Metal Manipulation

In X-Men, Magneto controls all metals with magnetism. But in reality, not all materials respond to magnetic fields the same way! Iron is strongly magnetic (ferromagnetic), aluminum is weakly attracted (paramagnetic), and copper actually repels fields slightly (diamagnetic). Understanding magnetic materials is crucial for designing transformers, hard drives, MRI machines, and yes, even separating metals at recycling centers!

The Big Picture

What’s the Deal?

• All materials respond to magnetic fields, but differently
• Response depends on atomic structure and electron configurations
• Three main types: diamagnetic, paramagnetic, ferromagnetic
• Applications: electromagnets, transformers, data storage, medical imaging

JEE Perspective:

• JEE Main: Classification, magnetic susceptibility, relative permeability
• JEE Advanced: Curie law, hysteresis, domain theory, quantitative problems

Core Concepts: Magnetization and Magnetic Intensity

Magnetic Intensity (H)

Definition: Magnetic field strength independent of the medium

For solenoid:

where n = turns per unit length, I = current

Unit: A/m (ampere per meter)

Physical meaning: H depends only on currents, not on material properties

Magnetization (M)

Definition: Magnetic dipole moment per unit volume

Unit: A/m (same as H)

Relation to atomic dipoles: If N atoms per volume, each with moment m:

Magnetic Permeability (μ)

Relation between B and H:

where μ = permeability of the material

Relative permeability:

For vacuum: μ_r = 1

Magnetic Susceptibility (χ)

Definition: Measure of how much a material magnetizes in response to field

Relation to permeability:

Key equation:

Memory Trick: “CHI makes Material respond to H!”

Three Types of Magnetic Materials

1. Diamagnetic Materials

Properties:

• Weakly repelled by magnetic fields
• χ is negative (small magnitude: ~10⁻⁵)
• μ_r < 1 (slightly less than 1)
• No permanent magnetic moment
• Effect is temperature independent

Mechanism:

• External field induces currents in electron orbits (Lenz’s law)
• Induced magnetic moment opposes applied field
• Present in ALL materials, but usually masked by stronger effects

Examples:

• Bismuth (strongest diamagnet)
• Copper, Gold, Silver
• Water, Wood
• Superconductors (perfect diamagnetism, χ = -1)

Behavior in non-uniform field:

• Move from stronger to weaker field region
• Repelled by magnets

JEE Key Values:

• χ ≈ -10⁻⁵ to -10⁻⁶
• μ_r ≈ 0.99999

2. Paramagnetic Materials

Properties:

• Weakly attracted by magnetic fields
• χ is positive (small magnitude: ~10⁻⁵ to 10⁻³)
• μ_r > 1 (slightly greater than 1)
• Have permanent atomic dipole moments
• Effect decreases with temperature (thermal agitation)

Mechanism:

• Atoms have unpaired electrons → permanent magnetic moments
• Without field: random orientation → no net magnetization
• With field: partial alignment → net magnetization
• Thermal motion opposes alignment

Examples:

• Aluminum, Platinum
• Oxygen (liquid O₂ is paramagnetic!)
• Manganese, Chromium
• CuSO₄, FeSO₄

Behavior in non-uniform field:

• Move from weaker to stronger field region
• Attracted by magnets

Curie Law:

where C is Curie constant, T is absolute temperature

Key insight: Susceptibility inversely proportional to temperature!

JEE Key Values:

• χ ≈ 10⁻⁵ to 10⁻³
• μ_r ≈ 1.00001 to 1.001

3. Ferromagnetic Materials

Properties:

• Strongly attracted by magnetic fields
• χ is large positive (~10² to 10⁵)
• μ_r » 1 (can be 1000 to 100,000!)
• Retain magnetization even after field removed (permanent magnets)
• Effect decreases with temperature

Mechanism:

• Strong interaction between atomic dipoles
• Spontaneous alignment in small regions (domains)
• External field aligns domains → strong magnetization

Examples:

• Iron, Cobalt, Nickel
• Alnico (Al-Ni-Co alloy)
• Ferrites (iron oxides)
• Rare earth magnets (Nd-Fe-B)

Curie-Weiss Law:

where T_C is Curie temperature

Above Curie temperature: Ferromagnetic → Paramagnetic!

Curie Temperatures:

• Iron: 1043 K (770°C)
• Cobalt: 1394 K (1121°C)
• Nickel: 631 K (358°C)

JEE Key Values:

• χ ≈ 10² to 10⁵
• μ_r ≈ 1000 to 100,000

Comparison Table: The Big Three

Memory Trick: “DPF - Diamagnetic Pulls Faintly away, Paramagnetic Pulls Faintly toward, Ferromagnetic Pulls Forcefully!”

Interactive Demo: Material Response to Magnetic Fields

See how different materials respond to magnetic fields based on their atomic structure.

Magnetic Domains: The Secret of Ferromagnetism

Domain Theory

Key Concept: Ferromagnetic material consists of small regions (~10⁻¹⁰ m³) called domains

Within each domain:

• All atomic dipoles are perfectly aligned
• Domain has large magnetic moment

Without external field:

• Domains point in random directions
• Net magnetization = 0

With external field:

• Domains aligned with field grow (domain wall movement)
• Domains not aligned shrink or rotate
• Net magnetization increases

At saturation:

• All domains aligned
• Maximum possible magnetization
• Further increase in H doesn’t increase M

Hysteresis Loop

Hysteresis: Lagging of magnetization behind magnetizing field

Key Points on Loop:

1. O → A: Initial magnetization (virgin curve)

   • Domains start aligning
   • M increases with H

2. A: Saturation

   • All domains aligned
   • M = M_s (saturation magnetization)

3. A → B: Reducing H to zero

   • Some domains misalign
   • M = M_r at H = 0 (retentivity or remanence)

4. B → C: Reversing field

   • Domains flip to opposite direction
   • M = 0 at H = -H_c (coercivity)

5. C → D: Saturation in opposite direction

   • M = -M_s

6. D → E → A: Completing the loop

Area of loop: Energy dissipated per cycle per unit volume (as heat)

Memory Trick: “ReCo - REtentivity at H=0, COercivity at M=0”

Hard vs Soft Magnetic Materials

JEE Gold:

• Soft: Easy to magnetize/demagnetize (good for AC applications)
• Hard: Retains magnetization (good for permanent magnets)

Interactive Demo: Susceptibility vs Temperature

import numpy as np
import matplotlib.pyplot as plt

def curie_law_paramagnetic(T, C):
    """Paramagnetic susceptibility: χ = C/T"""
    return C / T

def curie_weiss_ferromagnetic(T, C, T_c):
    """Ferromagnetic susceptibility: χ = C/(T - T_c)"""
    # Only defined for T > T_c
    return np.where(T > T_c, C / (T - T_c), np.nan)

# Temperature range
T_para = np.linspace(50, 500, 100)  # Kelvin
T_ferro = np.linspace(650, 1500, 100)  # Above Curie temp

# Constants
C_para = 0.5
C_ferro = 100
T_curie_Fe = 1043  # Iron

# Calculate susceptibilities
chi_para = curie_law_paramagnetic(T_para, C_para)
chi_ferro = curie_weiss_ferromagnetic(T_ferro, C_ferro, T_curie_Fe)

# Plotting
fig, axes = plt.subplots(1, 3, figsize=(16, 5))

# Plot 1: Paramagnetic
axes[0].plot(T_para, chi_para * 1e3, linewidth=2.5, color='blue')
axes[0].set_xlabel('Temperature (K)', fontsize=12)
axes[0].set_ylabel('Susceptibility χ (×10⁻³)', fontsize=12)
axes[0].set_title('Paramagnetic: Curie Law (χ ∝ 1/T)', fontsize=13, fontweight='bold')
axes[0].grid(True, alpha=0.3)
axes[0].annotate('χ decreases\nwith T', xy=(300, chi_para[150]*1e3),
                xytext=(350, chi_para[50]*1e3),
                arrowprops=dict(arrowstyle='->', color='red'),
                fontsize=11, bbox=dict(boxstyle='round', facecolor='yellow', alpha=0.7))

# Plot 2: Ferromagnetic (above T_c)
axes[1].plot(T_ferro, chi_ferro, linewidth=2.5, color='red')
axes[1].axvline(T_curie_Fe, color='green', linestyle='--', linewidth=2, label=f'T_C = {T_curie_Fe} K')
axes[1].set_xlabel('Temperature (K)', fontsize=12)
axes[1].set_ylabel('Susceptibility χ', fontsize=12)
axes[1].set_title('Ferromagnetic: Curie-Weiss Law', fontsize=13, fontweight='bold')
axes[1].grid(True, alpha=0.3)
axes[1].legend(fontsize=11)
axes[1].set_ylim(0, 50)

# Plot 3: Comparison at room temperature effect
materials = ['Diamagnetic\n(Cu)', 'Paramagnetic\n(Al)', 'Ferromagnetic\n(Fe)']
chi_values = [-1e-5, 2e-5, 1000]  # Representative values
colors_mat = ['cyan', 'orange', 'darkred']

axes[2].bar(materials, chi_values, color=colors_mat, alpha=0.7, edgecolor='black', linewidth=2)
axes[2].set_ylabel('Susceptibility χ (log scale)', fontsize=12)
axes[2].set_title('Susceptibility Comparison', fontsize=13, fontweight='bold')
axes[2].set_yscale('symlog', linthresh=1e-4)
axes[2].axhline(y=0, color='black', linewidth=0.8)
axes[2].grid(True, alpha=0.3, axis='y')

# Add values on bars
for i, (mat, chi, color) in enumerate(zip(materials, chi_values, colors_mat)):
    if chi < 0:
        axes[2].text(i, chi/2, f'{chi:.0e}', ha='center', fontsize=10, fontweight='bold')
    else:
        axes[2].text(i, chi*2 if chi > 1 else chi+1e-4, f'{chi:.0e}', ha='center', fontsize=10, fontweight='bold')

plt.tight_layout()
plt.show()

# Print comparison table
print("\n" + "=" * 70)
print("MAGNETIC SUSCEPTIBILITY AT ROOM TEMPERATURE (300 K)")
print("=" * 70)
print(f"{'Material':<20} {'Type':<15} {'χ (approx)':<20} {'μ_r':<15}")
print("-" * 70)
print(f"{'Bismuth':<20} {'Diamagnetic':<15} {'-1.7×10⁻⁵':<20} {'0.999983':<15}")
print(f"{'Copper':<20} {'Diamagnetic':<15} {'-1.0×10⁻⁵':<20} {'0.99999':<15}")
print(f"{'Aluminum':<20} {'Paramagnetic':<15} {'+2.2×10⁻⁵':<20} {'1.000022':<15}")
print(f"{'Platinum':<20} {'Paramagnetic':<15} {'+2.8×10⁻⁴':<20} {'1.00028':<15}")
print(f"{'Iron (soft)':<20} {'Ferromagnetic':<15} {'+5000':<20} {'5000':<15}")
print(f"{'Permalloy':<20} {'Ferromagnetic':<15} {'+100,000':<20} {'100,000':<15}")
print("=" * 70)

Observe:

• Paramagnetic χ decreases with temperature (thermal disorder)
• Ferromagnetic becomes paramagnetic above Curie temperature
• Huge difference in magnitude between types!

Common JEE Mistakes (Avoid These!)

Mistake 1: Sign of Diamagnetic Susceptibility

Wrong: Diamagnetic χ is positive Right: Diamagnetic χ is NEGATIVE (repulsion!)

Mistake 2: Permeability Relations

Wrong: μ_r = 1 - χ Right: μ_r = 1 + χ (plus sign!)

Mistake 3: Temperature Effect Direction

Wrong: Paramagnetic χ increases with temperature Right: Paramagnetic χ DECREASES with temperature (χ ∝ 1/T)

Mistake 4: Hysteresis Terminology

Wrong: Retentivity when M = 0 Right: Retentivity when H = 0, Coercivity when M = 0

Mistake 5: Domain Misconception

Wrong: Domains exist only when external field applied Right: Domains exist ALWAYS; field just aligns them!

Problem-Solving Strategy: The MCHR Method

Material type identification (dia/para/ferro) Chi and mu_r calculation Hysteresis if ferromagnetic Relation between B, H, M

Practice Problems

Level 1: JEE Main Foundations

Problem 1.1: A material has relative permeability μ_r = 1.00005. Classify the material and find its susceptibility.

Problem 1.2: The magnetic susceptibility of a material is -4.2 × 10⁻⁶. Is it diamagnetic or paramagnetic? Find its relative permeability.

Problem 1.3: A paramagnetic material has susceptibility 3 × 10⁻⁴ at 300 K. Find its susceptibility at 400 K (assuming Curie law).

Level 2: JEE Advanced Application

Problem 2.1: A solenoid with 1000 turns/m carrying 5 A has a core of relative permeability 500. Find: (a) magnetic intensity H, (b) magnetic field B, (c) magnetization M.

Problem 2.2: The retentivity of a ferromagnetic material is 1.5 T and coercivity is 6000 A/m. Sketch the hysteresis loop qualitatively and find the approximate energy dissipated per cycle if the loop can be approximated as a rectangle.

     B (T)
      ↑
      |    +1.5 ──────┐
      |              │
      |              │
      |              ↓
──────┼──────────────── → H (A/m)
-6000 |         +6000
      |      ↑
      |      │
      | ─────┘
      |   -1.5

Problem 2.3: A magnetic material has susceptibility that varies with temperature as χ = 0.6/(T - 400) in SI units. (a) What type of material is it? (b) What is its Curie temperature? (c) Find χ at 500 K.

Level 3: JEE Advanced Mastery

Problem 3.1: A toroid with 2000 turns, mean radius 10 cm, and cross-sectional area 1 cm² has a ferromagnetic core with μ_r = 1000. When 2 A current flows: (a) Find B inside core, (b) Find magnetization M, (c) If core is removed, what current would produce same B?

Problem 3.2: Two identical samples of a paramagnetic material are placed in regions where magnetic field intensities are H₁ = 10⁴ A/m and H₂ = 2 × 10⁴ A/m. The ratio of magnetizations is found to be M₁/M₂ = 1/3. If the first sample is at 300 K, find temperature of second sample.

Problem 3.3: A bar magnet is cut into two equal pieces perpendicular to its length. Each piece has retentivity B_r. The pieces are placed end-to-end with like poles together (N-N). If they behave like a single soft magnetic material in Earth’s field (B_E = 5 × 10⁻⁵ T), estimate the repulsive force. Assume cross-sectional area A = 1 cm².

Applications: Where Magnetic Materials Matter

1. Transformer Cores

Material: Soft ferromagnetic (silicon steel, ferrites)

Why soft?

• Low hysteresis loss (narrow loop)
• Easy magnetization/demagnetization with AC
• High μ_r (concentrates flux)

Energy loss: Proportional to loop area × frequency

2. Permanent Magnets

Material: Hard ferromagnetic (alnico, NdFeB, SmCo)

Why hard?

• High retentivity (strong residual field)
• High coercivity (resists demagnetization)
• Wide hysteresis loop

Applications: Motors, generators, speakers, MRI, magnetic latches

3. Magnetic Storage

Material: Hard ferromagnetic films

Principle:

• Data stored as magnetization direction (0 or 1)
• High coercivity prevents accidental erasure
• Read head detects field direction

Examples: Hard drives, magnetic tapes, credit card strips

4. MRI Machines

Material: Superconductors (perfect diamagnetism)

Principle:

• χ = -1 (perfect diamagnet)
• Expels all magnetic flux (Meissner effect)
• Creates ultra-strong stable fields (3-7 T)

Connections: The Web of Knowledge

Link to Magnetic Dipole

• Atomic magnetic moments create bulk properties
• M = (dipole moment per atom) × (number density)
• See: Magnetic Dipole

Link to Electromagnetic Induction

• Changing B in magnetic core induces EMF
• Hysteresis loss dissipated as heat
• See: Faraday’s Law

Link to Ampere’s Law

• H = nI from Ampere’s law
• B = μH in material
• See: Ampere’s Circuital Law

Link to Atomic Structure

• Paramagnetic: unpaired electrons
• Diamagnetic: paired electrons
• See: Atomic Structure

Quick Revision: Formula Sheet

Classification:

• Diamagnetic: χ < 0, μ_r < 1, weak repulsion
• Paramagnetic: χ > 0 (small), μ_r > 1 (slightly), weak attraction
• Ferromagnetic: χ » 0, μ_r » 1, strong attraction

Exam Strategy

Pattern Recognition:

• “χ negative” → Diamagnetic
• “χ positive small (~10⁻⁵)” → Paramagnetic
• “χ large (~10³)” → Ferromagnetic
• “Temperature increases, χ decreases” → Para or Ferro
• “Hysteresis loop” → Ferromagnetic

Common Question Types:

1. Classification from χ or μ_r (40%)
2. Temperature dependence (Curie law) (30%)
3. B-H-M calculations (20%)
4. Hysteresis properties (10%)

Time Savers:

• μ_r = 1 + χ (memorize!)
• Curie: χ ∝ 1/T
• Retentivity at H = 0, Coercivity at M = 0

Quick Checks:

• Diamagnetic always has χ < 0
• μ_r close to 1? Dia or Para
• μ_r » 1? Definitely Ferro

Summary: Key Takeaways

1. Three Types: Diamagnetic (χ < 0), Paramagnetic (χ > 0, small), Ferromagnetic (χ » 0, large)

2. Key Relations: χ = μ_r - 1, B = μ₀μ_r H = μ₀(H + M)

3. Temperature: Paramagnetic χ ∝ 1/T (Curie), Ferromagnetic χ ∝ 1/(T - T_C) (Curie-Weiss)

4. Domains: Ferromagnetic has spontaneous magnetization in regions; field aligns domains

5. Hysteresis: Retentivity (B when H = 0), Coercivity (H when B = 0), loop area = energy loss

6. Soft vs Hard: Soft (narrow loop, transformers), Hard (wide loop, permanent magnets)

7. JEE Pattern: 60% classification + χ/μ_r, 25% temperature effects, 15% hysteresis

Last Updated: April 5, 2025 Next Chapter: Electromagnetic Induction Previous Topic: Magnetic Dipole