Cyclotron Frequency

A charged particle moving perpendicular to a uniform magnetic field experiences a sideways magnetic force, qvB, which acts as the centripetal force needed to keep it moving in a circle. Setting qvB = mv^2/r gives the circular motion condition, and eliminating r shows the angular speed is qB/m, so the rotation frequency is f = (qB)/(2\pi m). The result is independent of the particle’s speed and radius as long as the motion is non-relativistic and the field is uniform.

Use this when analyzing the orbit of ions or electrons in a uniform magnetic field, such as in cyclotrons, mass spectrometers, or plasma physics. It is also useful whenever you need the rotation rate, period, or charge-to-mass ratio from magnetic circular motion.

• Using the particle speed in the formula instead of the charge, field strength, and mass
• Forgetting that the charge should be used as a magnitude, so the frequency is positive
• Mixing up frequency and angular frequency, or omitting the factor of 2\pi