Force on a moving charge: the Lorentz force
A charge q at a point with electric field E and magnetic field B, moving with velocity v, feels
F = q (E + v × B)
This is the Lorentz force. The magnetic part has size F = qvB sinθ, where θ is the angle between v and B.
- If the charge is at rest (v = 0) or moves along B (θ = 0), the magnetic force is zero.
- The magnetic force is always perpendicular to v. So it does no work and cannot change the speed, only the direction.
- For a negative charge (like an electron) the force is opposite.
1 tesla is the field in which a charge of 1 C moving at 1 m/s at right angles to B feels a force of 1 N.
Motion of a charge in a uniform magnetic field
If v is perpendicular to B, the force qvB is always towards one centre. It acts as the centripetal force:
qvB = mv²/r ⇒ r = mv / qB
The time for one round is T = 2πm / qB, and the frequency is f = qB / 2πm. These do not depend on speed! A faster charge makes a bigger circle but takes the same time.
If v makes an angle with B, the part along B carries on unchanged while the part across B goes round: the path becomes a helix (a spring shape). Pitch = v cosθ × T.
Crossed E and B fields: velocity selector
Let E and B be at right angles to each other and to v. The electric force qE and the magnetic force qvB point opposite ways. They cancel when
qE = qvB ⇒ v = E / B
Only charges with exactly this speed go straight through a slit; slower ones bend towards the E-force side, faster ones towards the B-force side. This works for any charge and mass, so it is used to pick out a beam of one speed.
Force on a current-carrying wire
A current is many charges drifting along the wire. Add the Lorentz force on all of them in a straight length L:
F = I L × B, size F = BIL sinθ
θ is the angle between the wire and B. Largest when the wire is across the field (θ = 90°), zero when it lies along the field. Direction: Fleming's left-hand rule or the cross product.
Force between two parallel currents; the ampere
Wire 1 makes a field B₁ = μ₀I₁ / 2πd at wire 2. Wire 2 feels F = B₁I₂L, so the force per metre is
F / L = μ₀ I₁ I₂ / 2πd
- Currents in the same direction attract.
- Currents in opposite directions repel.
- The forces on the two wires are equal and opposite (Newton's third law).
The ampere
1 ampere is the current which, flowing in two very long, thin, straight, parallel wires 1 m apart in vacuum, makes each wire feel a force of 2 × 10⁻⁷ N per metre of length. (Since 2019 the SI fixes the ampere through the charge of the electron, but this definition gives the same size and is what the syllabus asks.)
Torque on a current loop in a uniform field
Take a rectangular loop (sides a and b, N turns, current I) in a uniform field B. The net force on it is zero, because opposite sides get opposite forces. But the two forces on the sides of length a are not in one line, so they form a couple that turns the loop.
τ = N I A B sinθ, A = ab
θ is the angle between B and the normal to the loop. Maximum torque when the plane of the loop is along B (θ = 90°); zero when the plane is across B (θ = 0), which is the stable position. This works for a loop of any shape.
Current loop as a magnetic dipole
Define the magnetic moment of the loop: m = N I A, a vector along the normal given by the right-hand rule (fingers along current, thumb gives m). Unit: A m².
Then the torque is τ = m × B, exactly like the torque on a bar magnet. The loop's field far away also looks like a bar magnet's (it falls as 1/r³). So a current loop is a magnetic dipole: one face acts as N, the other as S.
An electron going round a nucleus is also a tiny current loop, with a tiny magnetic moment. This links electricity and magnetism in matter.
Moving coil galvanometer
A galvanometer detects and measures small currents. Parts: a coil of many turns on a light frame, pivoted between concave magnet poles; a soft iron cylinder in the middle; a spring; a pointer on a scale.
- The concave poles and the iron core make the field radial: the plane of the coil is always along B, so sinθ = 1 at every position.
- Torque by current: τ = NIAB. The spring gives a restoring torque kφ (k = torsion constant).
- At balance: NIAB = kφ ⇒ φ = (NAB / k) I. The deflection is proportional to current, so the scale is linear.
Sensitivity
Current sensitivity = φ / I = NAB / k (radians or divisions per ampere).
Voltage sensitivity = φ / V = NAB / (kG), where G is the coil resistance.
Doubling N doubles the current sensitivity but also doubles G, so the voltage sensitivity does not change. To raise sensitivity, use a strong magnet, a large coil area and a weak spring.
Conversion to an ammeter
A galvanometer gives full-scale deflection at a small current Ig. To measure a bigger current I, connect a small resistance S (the shunt) in parallel. Most current goes through the shunt:
Ig G = (I − Ig) S ⇒ S = IgG / (I − Ig)
An ammeter must have very low resistance, so it does not change the current it measures.
Conversion to a voltmeter
Connect a large resistance R in series. For full scale at voltage V:
V = Ig(G + R) ⇒ R = V / Ig − G
A voltmeter must have very high resistance, so it draws almost no current from the circuit.
Exam corner
Typical board questions: derive r = mv/qB and show T is independent of speed (3 marks); define 1 ampere from parallel currents (2 marks); derive τ = NIAB sinθ (3 marks); galvanometer principle, radial field, sensitivity and conversion to ammeter/voltmeter (3–5 marks); numericals on shunt and series resistance.
Key formulas and definitions
- Lorentz force: F = q(E + v × B)
- Magnetic force: F = qvB sinθ
- Circle radius: r = mv / qB; period T = 2πm / qB
- Velocity selector: v = E / B
- Force on wire: F = BIL sinθ (F = I L × B)
- Parallel wires: F / L = μ₀I₁I₂ / 2πd
- Torque on loop: τ = NIAB sinθ = m × B, m = NIA
- Galvanometer: φ = (NAB/k) I; current sensitivity NAB/k; voltage sensitivity NAB/kG
- Ammeter shunt: S = I_gG / (I − I_g); voltmeter series R = V/I_g − G
Worked examples
1. A proton moves at 2 × 10⁶ m/s at right angles to a 0.5 T field. Find the force on it. (e = 1.6 × 10⁻¹⁹ C)
F = qvB = 1.6 × 10⁻¹⁹ × 2 × 10⁶ × 0.5 = 1.6 × 10⁻¹³ N.
2. Find the radius of the proton's circle in the example above. (m = 1.67 × 10⁻²⁷ kg)
r = mv / qB = 1.67 × 10⁻²⁷ × 2 × 10⁶ / (1.6 × 10⁻¹⁹ × 0.5) ≈ 4.2 × 10⁻² m = 4.2 cm.
3. In a velocity selector E = 3 × 10⁴ V/m and B = 0.1 T. Which speed passes straight through?
v = E / B = 3 × 10⁴ / 0.1 = 3 × 10⁵ m/s. This is true for any charge.
4. A 0.5 m wire carries 4 A in a 0.2 T field at 30° to the field. Find the force.
F = BIL sinθ = 0.2 × 4 × 0.5 × 0.5 = 0.2 N.
5. Two long parallel wires 5 cm apart carry 10 A and 5 A in the same direction. Find the force per metre.
F/L = 2 × 10⁻⁷ × 10 × 5 / 0.05 = 2 × 10⁻⁴ N/m, attractive (same direction).
6. A coil of 100 turns, area 2 × 10⁻³ m², carries 0.5 A in a 0.2 T field with its plane along the field. Find m and the torque.
m = NIA = 100 × 0.5 × 2 × 10⁻³ = 0.1 A m². Plane along B means θ = 90° between m and B, so τ = mB = 0.1 × 0.2 = 0.02 N m.
7. A galvanometer of resistance 20 Ω gives full-scale deflection at 5 mA. Convert it into (a) an ammeter of range 1 A, (b) a voltmeter of range 10 V.
(a) S = I_gG / (I − I_g) = 0.005 × 20 / 0.995 ≈ 0.10 Ω in parallel. (b) R = V/I_g − G = 10 / 0.005 − 20 = 1980 Ω in series.
Common mistakes
- Thinking the magnetic force speeds a charge up. It is always perpendicular to v, so it does no work.
- Using the angle between the loop's plane and B in τ = NIAB sinθ. θ is between B and the normal (m).
- Connecting the shunt in series or the voltmeter resistor in parallel. Shunt: parallel, small. Voltmeter: series, large.
- Saying more turns always make a galvanometer better for voltage. Voltage sensitivity NAB/kG stays the same because G grows with N.