Magnetic Properties Of Matter

Paramagnetic materials:

• Microscopic: Atoms have permanent magnetic moments that partially align with external field
• Susceptibility: Small positive values (10⁻³ to 10⁻⁵)
• Response: Weakly attracted to magnets; field lines become slightly more dense
• Temperature effect: Susceptibility ∝ 1/T (Curie's Law)

Ferromagnetic materials:

• Microscopic: Strong alignment of atomic moments via exchange coupling; form domains
• Susceptibility: Very large positive values (thousands)
• Response: Strongly attracted to magnets; greatly increase field strength
• Temperature effect: Become paramagnetic above Curie temperature

Diamagnetic materials:

• Microscopic: No permanent moments; induced moments oppose external field
• Susceptibility: Small negative values
• Response: Weakly repelled by magnets; field lines become less dense
• Temperature effect: Minimal variation with temperature

Relationship between $\vec{B}$, $\vec{H}$, and $\vec{I}$:

• Starting definition: $\vec{H} = \frac{\vec{B}}{\mu_0} - \vec{I}$
• Rearranging: $\vec{B} = \mu_0(\vec{H} + \vec{I})$
• With $\vec{I} = \chi\vec{H}$ (for para/diamagnetic materials): $\vec{B} = \mu_0(1+\chi)\vec{H}$

Permeability relationships:

• Permeability: $\mu = \mu_0(1+\chi)$, giving $\vec{B} = \mu\vec{H}$
• Relative permeability: $\mu_r = \frac{\mu}{\mu_0} = 1+\chi$
• Physical significance: $\mu_r$ is the factor by which magnetic field increases when material is present
• Example: For a solenoid with a core, $\frac{B}{B_0} = \mu_r$

Magnetic hysteresis: The phenomenon where magnetization (I) depends not only on current magnetic intensity (H) but also on the material's magnetic history.

Hysteresis loop process:

1. Initial magnetization: O→A (virgin curve, increasing H from zero to maximum H₀)
2. Decreasing field: A→C (H decreases to zero, but I remains positive)
3. Reverse magnetization: C→D→E (H becomes negative, I decreases to zero at D, then becomes negative)
4. Return path: E→F→G→A (completing the cycle)

Key points:

• Retentivity: Value of I at point C (magnetization remaining when H=0)
• Coercive force: Value of H at point D (reverse field needed to demagnetize material)
• Area of loop: Proportional to energy dissipated as heat per cycle

Applications significance:

• Materials with large loop area (steel): Good for permanent magnets
• Materials with small loop area (soft iron): Good for electromagnets, transformers

Magnetic susceptibility ($\chi$):

• Defined as the proportionality constant between magnetization and magnetic intensity: $\vec{I} = \chi\vec{H}$
• Dimensionless quantity that characterizes a material's magnetic response

Temperature dependencies:

1. Paramagnetic materials:

   • Follows Curie's Law: $\chi = \frac{C}{T}$ where C is Curie constant
   • Physical explanation: Thermal agitation disrupts alignment of dipoles
   • Higher temperature increases randomization, decreasing susceptibility

2. Ferromagnetic materials:

   • Below Curie temperature: Complex, non-linear relationship
   • Above Curie temperature: Follows Curie-Weiss Law: $\chi = \frac{C'}{T-T_c}$
   • Physical explanation: Exchange coupling weakens with temperature

3. Diamagnetic materials:

   • Minimal temperature dependence
   • Physical explanation: Induced moments not significantly affected by thermal motion

Magnetic intensity $\vec{H}$:

• Definition: $\vec{H} = \frac{\vec{B}}{\mu_0} - \vec{I}$ where $\vec{I}$ is magnetization
• Units: ampere per meter (A·m⁻¹)
• Physical significance: Represents the contribution to magnetic field from external currents only

Differences from magnetic field $\vec{B}$:

• $\vec{B}$ includes contributions from both external currents and material magnetization
• $\vec{B}$ has units of tesla (T)
• $\vec{B}$ determines force on moving charges via $\vec{F} = q\vec{v} \times \vec{B}$

Relationship in vacuum:

• When no material is present (vacuum): $\vec{I} = 0$
• Therefore: $\vec{H} = \frac{\vec{B}}{\mu_0}$
• Simple proportional relationship

Relationship in a material:

• In linear materials: $\vec{B} = \mu_0(1+\chi)\vec{H} = \mu\vec{H}$
• $\vec{H}$ is determined solely by external currents when end effects can be neglected
• Example: In center of a solenoid, $H = ni$ regardless of core material

Curie's Law:

• Mathematical statement: $\chi = \frac{C}{T}$ for paramagnetic materials
• C = Curie constant (material-specific)
• T = absolute temperature (K)
• Valid for paramagnetic materials at all temperatures
• Physical basis: Competition between magnetic alignment and thermal randomization

Curie Temperature ($T_c$):

• Definition: Temperature above which a ferromagnetic material becomes paramagnetic
• Examples: Iron ($T_c = 1043$ K), Cobalt ($T_c = 1394$ K), Nickel ($T_c = 631$ K)

Behavior change at Curie point:

• Below $T_c$: Spontaneous magnetization, domain structure, hysteresis behavior
• At $T_c$: Phase transition occurs, spontaneous magnetization vanishes
• Above $T_c$: Material behaves as paramagnetic substance

Modified law above Curie temperature:

• Curie-Weiss Law: $\chi = \frac{C'}{T-T_c}$ for T > $T_c$
• Accounts for residual interaction effects between atomic moments
• C' = modified Curie constant

This difference arises from the fundamental nature of magnetic and electric sources:

• Magnetic dipoles (current loops):

  • Created by circulating electric current
  • The magnetic field lines form closed loops with no beginning or end
  • The Right-Hand Rule determines that the field at center points in the same direction as μ

• Electric dipoles:

  • Created by separated positive and negative charges
  • Electric field lines begin on positive charges and end on negative charges
  • At the center, the field points from the positive charge toward the negative charge (opposite to p)

This opposite behavior is why paramagnetic materials align with magnetic fields, while polar molecules align against electric fields.

Each material would orient differently based on energy minimization principles:

• Ferromagnetic sample: Aligns parallel to the field lines (long axis along field direction). The strong field concentration minimizes magnetic potential energy when aligned with the field.

• Paramagnetic sample: Also aligns parallel to field lines, but with weaker torque than ferromagnetic materials due to lower susceptibility.

• Diamagnetic sample: Orients perpendicular to field lines (long axis across field direction). This minimizes the volume of material that field lines must pass through, reducing the energetically unfavorable field distortion.

This orientation behavior provides a practical method to identify material types experimentally without measuring susceptibility directly.

The key differences in atomic mechanisms are:

Diamagnetic materials:

• Have no permanent atomic magnetic moments
• Magnetic moments are induced by external fields
• According to Lenz's law, induced moments oppose the applied field
• Repelled from stronger field regions
• Susceptibility (χ) is negative
• Effect is generally weak and temperature-independent

Paramagnetic/Ferromagnetic materials:

• Contain atoms with permanent magnetic moments
• External field tends to align these existing moments
• Thermal motion (in paramagnetics) disrupts alignment
• Domain structures (in ferromagnetics) enhance alignment
• Susceptibility (χ) is positive
• Paramagnetic effect decreases with temperature (Curie's Law: χ ∝ 1/T)
• Ferromagnetic effect is much stronger and exhibits hysteresis

This fundamental difference in atomic structure explains their opposite orientational behaviors in magnetic fields.

Retentivity: The magnetization that remains in a ferromagnetic material after the magnetic intensity (H) is reduced to zero. It represents the material's ability to maintain magnetization without an external field.

Coercive Force: The magnetic intensity (H) required to reduce the magnetization (I) to zero after the material has been saturated. It measures the resistance of a material to demagnetization.

Practical Significance:

• Materials with high retentivity and high coercive force (like steel) make good permanent magnets
• Materials with low retentivity and low coercive force (like soft iron) are ideal for electromagnets and transformer cores
• High coercive force materials resist demagnetization from stray fields
• The values determine energy losses in AC applications (transformer cores, electric motors)