Magnetic fields are produced by moving electric charges (currents). Unlike electric fields that emanate from static charges, magnetic fields arise only when charges are in motion.
Key Concepts
- Magnetic Force on a moving charge (Lorentz Force): F = q(v × B). Magnitude: F = qvB sin(theta), where theta is the angle between v and B. The force is perpendicular to both v and B.
- No work by magnetic force: The magnetic force is always perpendicular to velocity, so it does no work and does not change kinetic energy — it only changes direction.
- Circular motion in magnetic field: When v is perpendicular to B, the particle moves in a circle. Radius r = mv / (qB). Time period T = 2 pi m / (qB) — independent of speed!
- Biot-Savart Law: The magnetic field dB at a point due to a small current element I dl is: dB = (µ0 / 4 pi) × (I dl × rhat) / r2. Here µ0 = 4 pi × 10-7 T·m/A.
- Magnetic field at centre of circular loop: B = µ0 I / (2R).
- Ampere's Circuital Law: The line integral of B around any closed path equals µ0 times the total current enclosed. Integral(B · dl) = µ0 Ienclosed.
- Magnetic field of a long straight wire: B = µ0 I / (2 pi r), directed tangentially (right-hand rule).
- Solenoid: A long cylindrical coil of N turns per unit length. The field inside: B = µ0 n I (n = turns per unit length). Field outside is nearly zero.
- Toroid: A solenoid bent into a closed ring. B inside = µ0 n I; B outside = 0.
- Force on current-carrying conductor in B field: F = IL × B. Magnitude: F = BIL sin(theta).
- Force between two parallel wires: F/L = µ0 I1 I2 / (2 pi d). Parallel currents attract; antiparallel currents repel.
- Torque on a current loop: tau = NIAB sin(theta) = m × B, where m = NIA is the magnetic dipole moment.
- Galvanometer: Uses the torque on a current loop. Converted to ammeter (low shunt in parallel) or voltmeter (high resistance in series).
- Key Formulas
- F = qvB sin(theta); r = mv/(qB)
- Bwire = µ0 I / (2 pi r)
- Bsolenoid = µ0 n I
- Torque tau = NIAB sin(theta)
- m = NIA
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An electron moves at 106 m/s perpendicular to a field B = 0.1 T. Find the radius of its circular path.
r = mv/(qB) = (9.1 × 10-31 × 106) / (1.6 × 10-19 × 0.1) = 9.1 × 10-25 / 1.6 × 10-20 = 5.69 × 10-5 m ≈ 0.057 mm
Find the magnetic field 0.05 m from a long straight wire carrying 5 A.
B = µ0 I / (2 pi r) = (4 pi × 10-7 × 5) / (2 pi × 0.05) = (2 × 10-6) / (0.1) = 2 × 10-5 T
A circular loop of radius 0.1 m carries a current of 2 A. Find B at the centre.
B = µ0 I / (2R) = (4 pi × 10-7 × 2) / (2 × 0.1) = 8 pi × 10-7 / 0.2 = 4 pi × 10-6 ≈ 1.26 × 10-5 T
A solenoid has 1000 turns per metre and carries 2 A. Find B inside.
B = µ0 n I = 4 pi × 10-7 × 1000 × 2 = 8 pi × 10-4 ≈ 2.51 × 10-3 T
A rectangular loop of dimensions 0.2 m × 0.1 m with 50 turns carries 1 A in a field B = 0.5 T. Find the maximum torque.
taumax = NIAB = 50 × 1 × (0.2 × 0.1) × 0.5 = 50 × 0.01 × 0.5 = 0.25 N·m
Two long parallel wires 0.2 m apart carry currents of 3 A and 5 A in the same direction. Find the force per unit length between them.
F/L = µ0 I1 I2 / (2 pi d) = (4 pi × 10-7 × 3 × 5) / (2 pi × 0.2) = (6 × 10-6) / 0.4 = 1.5 × 10-5 N/m (attractive)
A galvanometer with coil resistance 50 Omega and full-scale deflection at 1 mA is to be converted to a 5 A ammeter. Find the shunt resistance.
Shunt S = Ig × G / (I - Ig) = 0.001 × 50 / (5 - 0.001) ≈ 0.05 / 5 = 0.01 Omega
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Common mistakes
- Forgetting that the magnetic force does no work — it cannot change kinetic energy.
- Using sin(theta) vs cos(theta): force F = qvB sin(theta); torque tau = NIAB sin(theta) — both use sine of the angle with B.
- Confusing the direction of force: use the right-hand rule (or the thumb, index, middle finger method) carefully.
Summary
Moving charges create magnetic fields and also experience forces in magnetic fields. The Biot-Savart Law and Ampere's Law help calculate B for various current configurations. The torque on current loops is the basis of galvanometers and motors.