Force on a moving charge

Physics A2 · Magnetic fields. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A charge moving across a magnetic field feels a force at right angles to both its velocity and the field. Use Fleming's left-hand rule with the current finger along the motion of a positive charge. For an electron, point it opposite to the motion.

Because the force is always perpendicular to the velocity, it does no work. The speed stays constant and the particle moves in a circle, with the magnetic force as the centripetal force.

In a conductor carrying a current across a field, the moving charge carriers are pushed to one side. This builds up the Hall voltage across the slice. A velocity selector uses crossed electric and magnetic fields so that only one speed passes straight through.

Rules to remember

force = flux density × charge × speed × sin θ (F = BQv sin θ), θ is the angle between v and B
Circular path: BQv = mv²/r, so r = mv ÷ (BQ)
Time for one revolution T = 2πm ÷ (BQ), which does not depend on the speed
Hall voltage V_H = BI ÷ (ntq): n = number density of charge carriers, t = thickness of the slice, q = charge of each carrier
Hall probe: V_H is proportional to B for a constant current; the slice must be perpendicular to the field
Semiconductors have a much smaller n than metals, so they give a larger Hall voltage
Velocity selector: qE = Bqv, so v = E ÷ B, with E = V ÷ d between the plates

Common mistake

For electrons and other negative particles the force is opposite to the direction the left-hand rule gives for the motion. Point the current finger backwards along the electron's path.

Worked example

A proton (mass 1.67 × 10⁻²⁷ kg, charge 1.60 × 10⁻¹⁹ C) moves at 4.0 × 10⁶ m s⁻¹ at right angles to a uniform magnetic field of flux density 0.50 T. What is the radius of its path?

The magnetic force is the centripetal force: BQv = mv²/r, so r = mv ÷ (BQ).
mv = 1.67 × 10⁻²⁷ × 4.0 × 10⁶ = 6.68 × 10⁻²¹ kg m s⁻¹.
BQ = 0.50 × 1.60 × 10⁻¹⁹ = 8.0 × 10⁻²⁰.
r = 6.68 × 10⁻²¹ ÷ 8.0 × 10⁻²⁰ = 0.084 m.

Answer: 0.084 m (8.4 cm)

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