Physics › Magnetic fields › Force on a moving charge

Force on a moving charge

Strip the wire away and the rule survives for a single flying charge: F = BQv, always at right angles to the motion. A force that can never do work can only steer, and steering at constant speed draws circles, and the cyclotron is built on precisely that.

Year 13AQA 3.7.5.2CIE 20.3OCR A 6.3.2

Builds on Magnetic flux density and the force on a wire and Circular motion.

IN THIS TOPIC

  • Use F = BQv for a charge moving perpendicular to the field, with the correct direction for either sign.
  • Explain why the path is a circle and derive r = mv/BQ.
  • Describe the cyclotron: magnetic steering, electric acceleration, a widening spiral.

WHAT YOU PROBABLY THINK

Magnetic fields can speed particles up.

One charge, same rule

A current is charge in motion, so the wire's force law has a single-particle version. A charge Q moving at speed v at right angles to a field of flux density B feels

F = BQvON YOUR DATA SHEET
A moving charge feels the same rule, F = BQv, and the direction flips with the sign of the chargepositive chargenegative chargesame v, same B: the force flips with the charge's sign
FIG. 1Same field, same velocity: the force on a negative charge is exactly opposite to the force on a positive one.

with the direction from Fleming's left hand once more, remembering that the seCond finger follows conventional current: a negative charge moving right counts as conventional current moving left, so electrons deflect exactly opposite to protons in the same field. A stationary charge, with v = 0, feels nothing at all.

Circles, because no work is done

constant speed is not constant velocitywatch the arrows:velocity: length pinnedacceleration: always inwardso a force must aim at the centre:gravity, for the satellite;the magnetic force, for the charge
FIG. 2The dot goes round at constant speed, and the cyan velocity arrow never changes length; but it never stops changing direction, so the velocity is changing and there must be an acceleration. The coral arrow shows it: aimed at the centre at every instant. Restyled, the same motion is a satellite held by gravity or a charge held by a magnetic field.

Because the force is always perpendicular to the velocity, it can never do work on the charge: the speed never changes, only the direction, which disposes of the opening misconception outright. A constant-magnitude force forever at right angles to the motion is precisely the condition for circular motion, with F = BQv in the centripetal seat. Setting the two faces equal, BQv = mv2/r, gives the radius:

r = mvBQNOT ON THE DATA SHEET — LEARN IT

so faster or heavier particles sweep wider circles, while stronger fields and bigger charges bend tighter. This one relation is how bubble-chamber photographs are read: the curvature of a track hands over the particle's momentum, and the direction of curl hands over its sign.

The cyclotron

the cyclotron's secret: a fixed half-turn timefaster, so widerthe gap kicks in perfect rhythm: the half-turn time never changes
FIG. 3Each time the particle crosses the gap the alternating voltage kicks it, so every half-circle is wider and faster than the last. But watch the gap flashes: perfectly even. Radius grows in step with speed, so the half-turn time never changes, and that is the machine's whole trick: one fixed frequency can keep kicking a particle that is forever speeding up.

The spec names the cyclotron as the application. Two hollow D-shaped electrodes sit in a uniform magnetic field, with an alternating pd across the gap between them. Inside each dee the magnetic field steers the particle round a half-circle; at each gap crossing the electric field does the accelerating, timed to push whichever way the particle is crossing. Every crossing raises v, and by r = mv/BQ every half-circle is wider than the last: the path is an outward spiral, and the particle exits at the rim with the energy of many small kicks. The design rests on that division of labour: magnetic fields steer for free, electric fields do the work.

THE EXAM BIT

  • F = BQv needs the perpendicular condition stated, and gives zero for a charge moving along the field or standing still.
  • Sign handling first: convert the particle's motion to conventional current before applying the left hand. Electrons curl opposite to protons.
  • The no-work argument earns marks verbatim: the force is perpendicular to the velocity, so no work is done and the speed is constant; only the direction changes.
  • Derive r = mv/BQ by equating BQv to mv2/r; the derivation is quick and frequently asked.
  • For the cyclotron, split the jobs cleanly: the magnetic field provides the circular steering, the alternating electric field between the dees provides the energy. Mixing them up loses the explanation marks.

CHECK YOURSELF

An electron moves at 2.0 × 107 m s−1 at right angles to a field of 0.50 mT. Find the force on it and the radius of its circular path.

Show a hint

BQv for the force; then let it be the centripetal force.

Show the answer

F = BQv = 5.0 × 10−4 × 1.60 × 10−19 × 2.0 × 107 = 1.6 × 10−15 N.

r = mv/BQ = (9.11 × 10−31 × 2.0 × 107) / (5.0 × 10−4 × 1.60 × 10−19) = 0.23 m.

The speed stays 2.0 × 107 m s−1 the whole way round: the field steered the electron without giving it a single joule.

F = BQv steers and never works: circles at constant speed.

Radius mv/BQ: momentum written as curvature.

No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.