Charged particle motion: mass spectrometry and cyclotrons

A magnetic force is always perpendicular to a moving charge's velocity, which means it never does any work on the particle — it can redirect the motion but never speed it up or slow it down. Send a charged particle into a uniform field at right angles to it, and this constant-magnitude, ever-sideways force produces exactly what a constant-speed, ever-sideways force always produces: uniform circular motion, with radius r = mv/(qB).

That one relationship runs two very different pieces of technology. A mass spectrometer ionises a sample, accelerates every ion through the same voltage so they enter the field with related speeds, and then simply measures where each one lands — since the radius scales with mass (for the same charge, speed and field), heavier ions swing wider and lighter ones curve tighter, sorting the sample by mass without ever weighing anything directly. A cyclotron runs the logic in reverse: it wants the particle to gain energy, so it lets the particle cross a gap between two D-shaped electrodes each half-turn, gaining speed each time. As the speed climbs, so must the radius (mass, charge and field held fixed), which is why the particle spirals steadily outward rather than looping at a fixed radius.

The two variables worth keeping straight are charge and mass, because they pull the radius in opposite directions. More mass (same speed, charge, field) means a harder particle to turn, so it needs a wider circle to keep curving at all — radius grows with mass. More charge means the magnetic force grabs the particle harder, so it curves into a tighter circle — radius shrinks with charge. Both intuitions can be checked against the formula directly: m sits in the numerator, q in the denominator.