Magnetism Lecture Notes

Basic Principles of Magnetism

• Bar magnets have North and South poles.
• Like poles repel, while opposite poles attract.
• Each bar magnet generates a magnetic field that emanates from the North pole to the South pole.
• Magnetic fields cancel out in regions of repulsion and add up in regions of attraction.

Creation of Magnetic Fields

• Moving electric charges create magnetic fields.

• A current-carrying wire generates a circular magnetic field around it.

• The right-hand rule helps determine the direction of the magnetic field around a wire.

• Formula to calculate magnetic field strength around a wire:

  [ B = \frac{μ₀ I}{2 π r} ]

  • B: Magnetic field strength (Tesla)
  • μ₀: Permeability of free space (4π imes 10^{-7}Tm/A)
  • I: Current (Amps)
  • r: Distance from the wire (meters)

Magnetic Force on a Current-Carrying Wire

• Magnetic force (F): A magnetic field exerts a force on a moving charge or current-carrying wire.

• Formula for magnetic force on a wire:

  [ F = ILB , \sin \theta ]

  • I: Current (Amps)
  • L: Length of the wire in the magnetic field (meters)
  • B: Magnetic field strength (Tesla)
  • θ: Angle between current direction and magnetic field

• Right-hand rule is used to determine the direction of the magnetic force.

Example Problems

Example 1: Calculating Magnetic Field 2 cm from Wire Carrying 45 Amps

• Given:

  • I = 45 Amps
  • r = 2 cm = 0.02 meters

• Calculate:

  [ B = \frac{4π imes 10^{-7} imes 45}{2π imes 0.02} = 4.5 imes 10^{-4} T ]

Example 2: Distance to Produce Magnetic Field of 8 × 10⁻⁴ Tesla with 10 Amps

• Given:

  • B = 8 × 10⁻⁴ T
  • I = 10 Amps

• Calculate:

  [ r = \frac{4π imes 10^{-7} imes 10}{2π imes 8 imes 10^{-4}} = 2.5 imes 10^{-3} m = 2.5 mm ]

Magnetic Force on a Moving Charge

• Formula: For a moving charge in a magnetic field, the magnetic force is given by:

  [ F = Bqv , \sin \theta ]

  • B: Magnetic field strength (Tesla)
  • q: Charge of the particle (Coulombs)
  • v: Velocity of the particle (m/s)
  • θ: Angle between velocity and magnetic field

• Example: Proton moving east at 4 × 10⁶ m/s in a 2 × 10⁻⁴ T magnetic field

  • Calculate magnetic force:

  [ F = 2 imes 10^{-4} imes 1.6 imes 10^{-19} imes 4 imes 10^{6} = 1.28 imes 10^{-16} N ]

Magnetic Force in a Circular Path

• A charged particle in a perpendicular magnetic field will move in a circular path.

• The radius of such a path can be determined using:

  [ r = \frac{mv}{Bq} ]

• Example: Proton with speed 5 × 10⁶ m/s in a 2.5 T magnetic field

  • Calculate radius:

  [ r = \frac{1.673 imes 10^{-27} imes 5 imes 10^{6}}{2.5 imes 1.6 imes 10^{-19}} = 2.09 imes 10^{-2} m = 2.09 cm ]

Interaction Between Parallel Wires

• Attraction: Parallel wires carrying current in the same direction attract each other.

• Repulsion: Parallel wires carrying current in opposite directions repel each other.

• Magnetic field created by one wire affects the other.

• Formula to calculate force per unit length between two wires:

  [ F/L = \frac{μ₀ I₁ I₂}{2π d} ]

Example: Force Between Two Wires

• Given:

  • I₁ = I₂ = 50 Amps
  • L = 30 meters
  • d = 2 cm = 0.02 meters

• Calculate:

  [ F = \frac{4π imes 10^{-7} imes 50 imes 50 imes 30}{2π imes 0.02} = 0.75 N ]

Ampere's Law

• Ampere's Law relates the integrated magnetic field around a closed loop to the electric current passing through the loop:

  [ \oint \mathbf{B} , \cdot d \mathbf{l} = μ₀ Iₑ ]

• Helps derive the magnetic field for different geometries like a solenoid.

• Solenoid: A coil of wire with many loops; magnetic field strength inside is given by:

  [ B = μ₀ n I ]

  • n: Number of turns per meter
  • I: Current (Amps)

  Example: Magnetic Field in Solenoid

  • Given:
    • Length = 15 cm = 0.15 m
    • Turns = 800
    • Current = 5 Amps
  • Calculate: Turns per meter

  [ n = \frac{800}{0.15} = 5333 , \text{turns/meter} ]

  • Calculate magnetic field:

  [ B = 4π imes 10^{-7} imes 5333 imes 5 = 0.0335 T ]

Torque on a Current-Carrying Loop

• A current-carrying loop in a magnetic field experiences a torque:

  [ \tau = n I A B , \sin \theta ]

  • τ: Torque
  • n: Number of loops
  • I: Current
  • A: Area of the loop (length × width for rectangular)
  • B: Magnetic field strength
  • θ: Angle between normal to the loop and magnetic field

• Maximum torque occurs when angle θ = 90°.

Example: Maximum Torque on a Circular Coil

• Given:

  • Radius = 30 cm = 0.3 m
  • Loops = 50
  • Current = 8 Amps
  • Magnetic Field = 5 Tesla

• Area:

  [ A = πr^2 = π imes 0.3^2 = 0.2827 m^2 ]

• Calculate torque:

  [ \tau = 50 imes 8 imes 0.2827 imes 5 = 565.5 , \text{N.m} ]

Conclusion

• Recap of key formulas and concepts:
  • Magnetic field around wire: ( B = \frac{μ₀ I}{2πr} )
  • Magnetic force on wire: ( F = ILB , \sin \theta )
  • Magnetic force on charge: ( F = Bqv , \sin \theta )
  • Radius of particle path: ( r = \frac{mv}{Bq} )
  • Force between parallel wires: ( F/L = \frac{μ₀ I₁ I₂}{2π d} )
  • Magnetic field inside solenoid: ( B = μ₀ n I )
  • Torque on loop: ( \tau = n I A B , \sin \theta )

Always remember to use the right-hand rule to determine the direction of magnetic fields and forces.