Force on a Moving Charge

A current is moving charge, so the motor effect applies to individual charged particles too:

F = B Q v sin θ

with the same angle rule: maximum force when the velocity is perpendicular to the field, and zero force when the particle moves along the field.

A stationary charge feels no magnetic force at all — v = 0 gives F = 0. This is a genuine difference from an electric field, which acts on a charge whether it is moving or not.

Circular motion in a magnetic field — the force is always perpendicular to the velocity, so it changes direction but never speed. That is precisely the condition for circular motion. Setting the magnetic force equal to the centripetal force:

B Q v = m v² / r

which rearranges to

r = m v / (B Q)

What the radius tells you — faster or heavier particles curve less; stronger fields and larger charges curve them more. The radius therefore depends on the momentum-to-charge ratio, which is what makes a magnetic field into a measuring instrument.

No work is done by a magnetic force — because it acts perpendicular to the velocity at all times, it can never change the particle's kinetic energy. A magnetic field can steer a particle but cannot speed it up. Speeding particles up is the job of the electric field, which is why accelerators use both.

Example: this pairing is the basis of the cyclotron. An electric field across a gap accelerates the particle, and a magnetic field bends it back round for another pass. As its momentum rises, so does the radius, and the particle spirals outwards, gaining energy on every crossing.

Directions for negative charges — Fleming's left-hand rule uses conventional current, so for an electron the current direction is opposite to the velocity. This sign flip is the most common error in the topic; state the current direction explicitly before applying the rule.

The velocity selector — crossed electric and magnetic fields exert opposing forces on a moving charge. Only particles for which EQ = BQv, that is v = E/B, pass through undeflected, whatever their charge or mass. This filters a beam to a single speed before it enters a mass spectrometer.