Study Notes on Electricity and Magnetism

1.0 Electricity and Magnetism

1.10 Magnetic Properties of Materials and Their Uses

• The discussion begins by defining various terms related to magnetic materials.
• Purpose of Investigation:
    - Understanding magnetic properties helps determine material suitability for:
      - Permanent magnets: Used in devices like loudspeakers.
      - Temporary magnets: Used in transformers as cores.
• Source of Magnetic Properties:
    - Attributed to the movement of electrons within atoms.

1.11 Flux Density in the Magnetic Material

Magnetic Field Intensity (B)

• Definition of Magnetic Field:
    - The space around a magnet or a current-carrying conductor where a magnetic field exists.
• Field Vector (H):
    - Represents force due to the magnetic field.
    - Direction indicates the force direction for a north-seeking pole.
• Magnetic Flux Density (B):
    - Also known as magnetic field induction.
    - Visualized as lines of induction, with tangent lines indicating direction.
    - Density of lines per unit area represents the magnitude of B.
    - Close lines → High B; Far lines → Low B.

Force Equation and Units

• A positive charge $q_0$ moving with velocity $v$ experiences a magnetic force $F$, given by:
  $F = q_0 v B an heta$
• Key Points:
    - $F$ is perpendicular to both $v$ and $B$.
    - $heta$ is the angle between $v$ and $B$.
• Units of B:
    - Derived units: $N/(m/s)$ (newton per meter per second) = tesla (T).
    - Also defined as force per unit current length at right angles to the magnetic field.

Relationship of B and H

• Generally, $B$ is proportional to $H$, with the constant of proportionality denoted as $<br /> u$ (permeability):
  $B = <br /> u H$
• Permeability in Different Mediums:
    - In a vacuum: $<br /> u = <br /> u_0$ (permeability of free space).
• Relative Permeability ($u_r$):
    - Ratio of permeability of a material to that of free space:
  u_r = rac{
  u}{
  u_0}

Magnetic Flux (φ)

• Definition:
    - The product of intensity $B$ normal to an area $A$, given by:
  $φ = B A ext{cos} heta$
    - Where $heta$ is the angle between field lines and the area.
• Units of Flux:
    - SI unit is Weber (Wb).

Magnetic Moment (µ)

• Definition:
    - Vector pointing from south to north of the magnet, indicating orientation.
• For a coil:
    - Specified as the vector oldsymbol{
  u} along the axis related to current direction by the right-hand rule.
• Magnitude of Magnetic Moment of a Coil:
    - Given by:
    oldsymbol{
  u} = N A I
    - Where:
      - $N$ = number of turns
      - $A$ = area of the coil
      - $I$ = current through each turn.
• The direction of oldsymbol{
  u} is perpendicular to the coil plane, as determined by the right-hand rule.
• In a uniform magnetic field, a magnet experiences a couple leading to angular acceleration, ultimately aligning with the magnetic field.
• The orientation at a point indicates the tendency of a magnetic moment to align with the magnetic field.

1.12 Magnetization

• Definition of Magnetization:
    - The term describing the state of a material's magnetism, denoted as the magnetic moment per unit volume, $M$:
    M = rac{ ext{Total magnetic dipole moment}}{ ext{Volume}}
• Key Question:
    - What are the units of $M$?

Evaluating B and M Relationship in a Magnetic Field

• Set-Up with a Toroid:
    - Length $L$, total turns $N$, mean radius $r$, circumference $L$.
• Factors Affecting Flux Density (B):
    - Magnetic properties depend on:
      1. Current through wire $I$.
      2. Magnetization of the material.
• Total flux density is:
  $B = B_0 + B_m$
    - Where:
      - $B_0$ = flux density due to current $I$
      - $B_m$ = flux density due to magnetization.
• Typically, B_0 >> B_m which can be further described by:
  $B_m = n I <br /> u$
• Where:
    - $n = ext{turns per unit length}$.

1.13 Intensity of Magnetization

• The magnetic moment due to surface current $I_m$ is given by:
• Total magnetic moment can be quantified per unit volume.
• The magnetic field density (H):
  $B = <br /> u_0 n I$
• Total Flux Density Equations:
• Key Questions:
    1. What are the units for H?
    2. What are the units for $<br /> u$? What is the value for $<br /> u_0$?

1.14 Relative Permeability and Magnetic Susceptibility

• From previous discussions:
  $<br /> u_r = 1 + ext{χ}_M$
• Where $ext{χ}_M$ = magnetic susceptibility defined as:
     ext{χ}_M = rac{M}{H}
• Processes during Magnetization:
    - As a material is magnetized, $M$ increases with increasing field until saturation, where all magnetic domains are aligned.
• Behavior under Increasing H:
    - B can still increase even after saturation.

1.15 Types of Magnetic Materials

• Materials categorized based on response to changes in the magnetic field $H$:
    1. Diamagnetism
    - Induced currents generate a magnetic field opposing the applied magnetic field ($H$).
    - Characteristics:
      - $M$ is opposite to $H$ → Negative susceptibility ($ext{χ}$).
    - Example: Bismuth, with $ext{χ} = -0.000015$.
  
    2. Paramagnetism
    - Atoms in these materials have random thermal motion, leading to no resultant magnetization until an external magnetic field is applied.
    - Characteristics:
      - Small, positive susceptibility ($ext{χ}$).
    - Example: Platinum with $ext{χ} = +0.0001$.
  
    3. Ferromagnetism
    - In these materials, magnetization aligns with the applied field, resulting in a high positive susceptibility ($ext{χ}$).
    - Groups of Ferromagnetic Materials:
      - Soft Magnetic Materials: Easily magnetized and demagnetized.
        - Example: Iron (used in transformers).
      - Hard Magnetic Materials: Difficult to magnetize but retain magnetism.
        - Example: Steel (used for permanent magnets).

1.16 Hysteresis Loop

• Concept: The variation of $B$ with applied field $H$ through a complete cycle for a magnetic specimen leads to specific behavior patterns.
• Stages of Hysteresis:
    - oa to ab: Increasing $B$ with increasing $H$.
    - ab: $H$ reduces to zero → $B$ follows the path ab.
    - bc: $H$ increased in the opposite direction.
• Resulting Effects:
    - At point a, reaching saturation.
    - Residual magnetic field stays at point b when applying $H$ returns to zero.
    - Leftover magnetism creates remanence ($B_r$).
    - The coercive force ($H_c$) measures the difficulty in demagnetizing the material, indicated by the flux density reducing to zero.
• Observation: - The magnetization curve forms a closed loop, known as the hysteresis loop, indicating that $B$ lags behind the applied field $H$ when cycled through a magnetic process.

Homework

1. (a) Discuss the domain theory of magnetization and explain magnetization and demagnetization.
2. (a) Define magnetic hysteresis, sketch a typical hysteresis curve, and explain its implications on the material's magnetic properties.
     (b) Identify desirable magnetic properties for the core of an electromagnet and a permanent magnet.
3. Calculate $B$ and $H$ for a toroid core with:
     - N = 1200 turns
     - Length L = 80 cm
     - Cross-sectional area A = 60 cm²
     - Current I = 1.5 A
     - Assume empty core.
4. Given a cast iron ring with a mean diameter of 0.2 m and cross-section area of $5 imes 10^{-4} m^2$, wound with 2000 turns carrying 2.0 A, and flux in iron at $8 imes 10^{-3} Wb$:
     - Calculate the relative permeability of iron.
5. Determine for a toroid with:
     - Current in windings = 2.0 A
     - Turns = 400 turns
     - Length = 40 cm
     - Magnetic field found = 1.0 T:
     (a) Magnetic intensity
     (b) Magnetization
     (c) Magnetic susceptibility
     (d) Equivalent surface current
     (e) Relative permeability.