Learning Objectives
- Understand the magnetic force on moving charges and current-carrying conductors
- Study the Biot-Savart law and Ampere's circuital law
- Calculate magnetic fields due to common current configurations
- Understand the motion of charged particles in magnetic fields
- Learn about force between parallel current-carrying conductors
Key Concepts
Magnetic Force on a Moving Charge
Lorentz Force: F = qv × B = qvB sin θ
Direction: perpendicular to both v and B (right-hand rule or Fleming's left-hand rule).
The magnetic force does no work (F ⊥ v), so it changes direction but not speed.
If v ∥ B: F = 0 (no force). If v ⊥ B: F = qvB (maximum force).
Motion of Charged Particle in Magnetic Field
v ⊥ B: Circular motion. Radius: r = mv/(qB). Time period: T = 2πm/(qB). Frequency: f = qB/(2πm) (cyclotron frequency).
v at angle θ to B: Helical motion. Pitch = v cos θ × T = 2πm v cos θ/(qB).
Magnetic Force on a Current-Carrying Conductor
F = IL × B = ILB sin θ (L is the length vector in direction of current).
Direction: Fleming's left-hand rule (First finger = Field, seCond finger = Current, thuMb = Motion/Force).
Biot-Savart Law
dB = (μ₀/4π) × (Idl × r̂)/r²
μ₀ = 4π × 10⁻⁷ T·m/A (permeability of free space).
Magnetic field due to common configurations:
- Straight wire (infinite): B = μ₀I/(2πr) (concentric circles by right-hand rule)
- Centre of circular loop: B = μ₀I/(2R)
- On axis of circular loop: B = μ₀IR²/[2(R² + x²)^(3/2)]
- Solenoid (inside): B = μ₀nI (n = turns per unit length)
- Toroid: B = μ₀NI/(2πr) (inside), B = 0 (outside)
Ampere's Circuital Law
∮B · dl = μ₀I_enclosed
The line integral of magnetic field around a closed loop equals μ₀ times the enclosed current.
Used for symmetric current distributions (long wire, solenoid, toroid).
Force Between Parallel Current-Carrying Conductors
F/L = μ₀I₁I₂/(2πd)
Parallel currents (same direction): attract. Antiparallel currents: repel.
Definition of ampere: 1 A is the current which, when flowing through two parallel conductors 1 m apart, produces a force of 2 × 10⁻⁷ N/m.
Torque on a Current Loop in Magnetic Field
τ = NIAB sin θ = M × B, where M = NIA is the magnetic moment.
This is the principle of the galvanometer and electric motor.
Moving Coil Galvanometer
Current sensitivity: I_s = NAB/k. Voltage sensitivity: V_s = NAB/(kR).
Conversion to ammeter: connect low resistance (shunt) in parallel.
Conversion to voltmeter: connect high resistance in series.
Summary
Moving charges create and experience magnetic fields. The Lorentz force (F = qv × B) causes circular or helical motion. The Biot-Savart law gives the field due to a current element. Ampere's law relates the field around a loop to enclosed current. Parallel currents attract, antiparallel currents repel. A current loop in a magnetic field experiences torque τ = NIAB sin θ.
Important Terms
- Lorentz Force: Total force on a charge in EM field, F = q(E + v × B)
- Cyclotron Frequency: f = qB/(2πm), independent of speed
- Biot-Savart Law: dB = (μ₀/4π)(Idl × r̂)/r²
- Ampere's Law: ∮B · dl = μ₀I_enclosed
- Magnetic Moment: M = NIA (for a coil of N turns)
- Solenoid: Coil of many turns; B = μ₀nI inside
Quick Revision
- F = qvB sin θ (charge); F = BIL sin θ (conductor)
- Circular motion: r = mv/qB; T = 2πm/qB
- Infinite wire: B = μ₀I/2πr; Loop centre: B = μ₀I/2R
- Solenoid: B = μ₀nI; Toroid: B = μ₀NI/2πr
- Force between wires: F/L = μ₀I₁I₂/2πd
- τ = NIAB sin θ; M = NIA