Magnetic force on moving charges

A magnetic field pushes on a charge only while the charge is moving. The force is:

F = q(v × B), of size qvB sin θ, where θ is the angle between v and B.

Its direction is perpendicular to both v and B (use the right-hand rule for the cross product; flip it for a negative charge). A charge moving along B feels no force.

Example: a proton at 10⁶ m/s, at right angles to B = 0.1 T: F = 1.6 × 10⁻¹⁹ × 10⁶ × 0.1 = 1.6 × 10⁻¹⁴ N.

The force is always sideways to the motion, so it does no work. It changes the direction of a charge, never its speed or kinetic energy.

Circular motion: a charge moving at right angles to a uniform B goes round a circle. The magnetic force supplies the centripetal force, qvB = mv²/r, so:

r = mv / (qB) and the frequency ν = qB / (2πm).

For the proton above: r = 1.67 × 10⁻²⁷ × 10⁶ / (1.6 × 10⁻¹⁹ × 0.1) ≈ 10 cm.

A faster particle moves in a bigger circle but takes the same time per round, because ν does not depend on v. The cyclotron uses this to speed up particles.

If v is at an angle to B, the part along B is unchanged and the part across B goes round: the path is a helix, a spring shape.

Lorentz force: with both fields present, F = q(E + v × B). If they balance, qE = qvB, only particles with v = E/B go straight through: a velocity selector.

Force on a wire: a current is many moving charges, so a wire of length l feels F = I l × B, size IlB sin θ. Example: 2 m of wire with 3 A, at right angles to 0.5 T: F = 3 N.