Force on moving charges and conductors

Magnetic force on charges and wires

Lorentz force: a charge q moving with velocity v in a magnetic field B feels a force F = qvB sin θ, where θ is the angle between v and B. The force is perpendicular to both v and B (F = q v × B). If an electric field is also present, F = q(E + v × B).

Because the magnetic force is always perpendicular to the velocity, it does no work and does not change the speed or kinetic energy; it only changes the direction. A charge moving perpendicular to a uniform B therefore moves in a circle of radius r = mv/(qB), with time period T = 2πm/(qB), which does not depend on speed. This is the basis of the cyclotron.

Force on a current-carrying conductor: F = BIL sin θ, where L is the length in the field. Direction is given by Fleming’s left-hand rule: forefinger = field, middle finger = current, thumb = force (motion).

Worked example: a 0.5 m wire carrying 3 A at right angles to B = 0.4 T: F = 0.4 × 3 × 0.5 = 0.6 N.

Force between two parallel wires carrying currents I₁ and I₂ at separation d: force per unit length f = μ₀I₁I₂/(2πd). Parallel currents attract; antiparallel currents repel.

Definition of the ampere (the classical definition used until the 2019 SI revision, which now fixes the value of e instead): one ampere is the current which, flowing in two infinitely long parallel straight wires 1 m apart in vacuum, produces a force of 2 × 10⁻⁷ N per metre of length between them.

Key points

• F = qvB sin θ; zero when v is parallel to B.
• Magnetic force does no work; speed stays constant.
• r = mv/qB; T = 2πm/qB.
• F = BIL sin θ; f = μ₀I₁I₂/2πd.
• Like (parallel) currents attract.

Exam tips

• Check the angle first: parallel motion gives zero force.
• For parallel-wire problems, μ₀/2π = 2 × 10⁻⁷ makes the arithmetic easy.

Common mistakes

• Thinking like currents repel (they attract, unlike like charges).
• Saying the magnetic force increases the speed of a charge.
• Using the right hand instead of the left for Fleming’s motor rule.