Unit 12: Magnetic Fields and Electromagnetism

1. Introduction to Magnetic Fields

Magnetic fields are fundamental forces in nature that arise from moving charges and intrinsic magnetic moments of particles.

Key Properties:
  • Magnetic fields are vector quantities
  • Field lines form closed loops
  • No magnetic monopoles exist
  • Fields can be created by moving charges or permanent magnets
B = magnetic field strength (tesla, T)
1 T = 1 N/(A·m) = 10⁴ gauss

2. Magnetic Force on Moving Charges

Charged particles moving through magnetic fields experience a force perpendicular to both their velocity and the magnetic field.

F = qv × B
F = qvB sin θ
where:
F = magnetic force
q = charge
v = velocity
B = magnetic field
θ = angle between v and B
Right-Hand Rules:
  • RHR-1: Point fingers in direction of v
  • Curl fingers toward B
  • Thumb points in direction of F for positive charge

3. Motion of Charged Particles in Magnetic Fields

Charged particles in magnetic fields follow circular or helical paths depending on their initial velocity.

r = mv/(qB) (radius of circular motion)
T = 2πm/(qB) (period of motion)
ω = qB/m (angular velocity)
Example: An electron enters a 0.5T magnetic field with velocity 2×10⁶ m/s perpendicular to the field. Calculate:
a) Radius of circular path
b) Period of motion

4. Magnetic Force on Current-Carrying Conductors

Current-carrying wires in magnetic fields experience forces that form the basis for electric motors.

F = ILB sin θ
where:
I = current
L = length of conductor
B = magnetic field
θ = angle between current and field
Applications:
  • DC motors
  • Loudspeakers
  • Galvanometers
  • Mass spectrometers

5. Biot-Savart Law

The Biot-Savart Law describes the magnetic field created by a current element.

dB = (μ₀/4π)(IdL × r̂/r²)
where:
μ₀ = 4π × 10⁻⁷ T·m/A (permeability of free space)
dL = current element vector
r = distance to observation point
Example: Calculate the magnetic field at the center of a circular current loop.
B = μ₀I/(2R)

6. Ampère's Law

Ampère's Law relates the magnetic field around a closed loop to the current enclosed by that loop.

∮B·dL = μ₀Ienclosed
Applications:
  • Long straight wires: B = μ₀I/(2πr)
  • Solenoids: B = μ₀nI
  • Toroids: B = μ₀NI/(2πr)

7. Magnetic Materials

Different materials respond differently to magnetic fields based on their magnetic properties.

Types of Magnetic Materials:
  • Ferromagnetic (iron, nickel, cobalt)
  • Paramagnetic (aluminum, platinum)
  • Diamagnetic (copper, silver, gold)
  • Antiferromagnetic (chromium, manganese)

8. Electromagnetic Induction

Changing magnetic fields induce electric fields and voltages in conductors.

ε = -dΦ/dt (Faraday's Law)
Φ = BA cos θ (magnetic flux)
where:
ε = induced EMF
Φ = magnetic flux
Lenz's Law: Induced current produces magnetic field that opposes the change in flux that caused it.

9. Inductance

Inductance is the property of a conductor to oppose changes in current through electromagnetic induction.

V = L(dI/dt)
L = N²μ₀A/l (solenoid inductance)
Energy stored = ½LI²
Example: A 0.5H inductor carries a current changing at 100A/s. Calculate:
a) Induced EMF
b) Energy stored at 2A

10. Applications and Devices

Electromagnetic principles are used in many practical devices and technologies.

Common Applications:
  • Electric generators
  • Transformers
  • Electric motors
  • Magnetic resonance imaging (MRI)
  • Electromagnetic brakes
  • Magnetic levitation

11. Practice Problems

Problem 1: A proton moves at 5×10⁶ m/s perpendicular to a 0.2T magnetic field. Calculate:
a) Magnetic force
b) Radius of circular path
c) Period of motion
Problem 2: A solenoid with 500 turns/meter carries 2A of current. Calculate:
a) Magnetic field inside
b) Magnetic flux through a 10cm² cross-section
c) Inductance per meter
Problem-Solving Tips:
  • Draw clear diagrams
  • Use right-hand rules correctly
  • Check units and reasonable values
  • Consider symmetry in field problems
  • Apply conservation laws