A perpendicular force makes a circle
Suppose a charge moves at right angles to a uniform field. The force is perpendicular to \vec{v}, so it acts exactly like a string on a whirling ball: a constant-magnitude force that always points sideways to the motion. That is the recipe for uniform circular motion. Setting the magnetic force equal to the required centripetal force lets us solve for the radius.
The radius of the circular path. Faster or heavier particles trace bigger circles; a stronger field winds them tighter.
This closed circular path is called cyclotron motion. It is the reason charged particles in the Sun's atmosphere, in fusion reactors, and in accelerators travel in curved arcs rather than straight lines.
The cyclotron frequency
How long does one loop take? Divide the circumference 2\pi r by the speed v and — remarkably — the v cancels. The time for one revolution, and hence the frequency, does not depend on how fast the particle goes or how big its circle is.
The cyclotron frequency depends only on charge, mass, and field — not on speed. This is what lets a cyclotron accelerator keep its timing as the particles speed up.
That speed-independence is the beautiful secret behind the cyclotron frequency: an oscillating voltage tuned to f_c can kick the particle every half-turn, no matter how fast it has become. If the velocity also has a component along \vec{B}, that part is unaffected (no force along the field), so the circle stretches into a helix — exactly the spiral that charged particles follow down Earth's field lines to paint the aurora.
Where circular motion is put to work
Because r = mv/(qB), particles of different mass (for the same charge and speed) curve to different radii — this is how a mass spectrometer separates isotopes and identifies molecules. A velocity selector crosses electric and magnetic fields so that only charges with v = E/B pass straight through (the forces cancel). In the Hall effect, magnetic deflection of the current carriers builds up a sideways voltage that reveals their sign and density — the basis of countless field sensors.
The force on a current-carrying wire
A current is just a parade of moving charges, and each one feels the Lorentz force. Add them up over a length L of wire and the whole conductor is pushed. This is the force on a current-carrying wire — the effect that makes loudspeakers move and motors turn.
Force on a straight wire of length L carrying current I in field B; θ is the angle between the wire and the field. The same right-hand rule gives the direction.
Worked example: a proton in a field
- Problem. A proton (q = 1.6\times10^{-19} C, m = 1.67\times10^{-27} kg) enters a B = 0.20 T field at v = 1.0\times10^{6}\,\text{m/s}, perpendicular to the field. Find its orbit radius and cyclotron frequency.
- Radius. r = \dfrac{mv}{qB} = \dfrac{(1.67\times10^{-27})(1.0\times10^{6})}{(1.6\times10^{-19})(0.20)} = \dfrac{1.67\times10^{-21}}{3.2\times10^{-20}} \approx 0.052\,\text{m}, about 5.2 cm.
- Frequency. f_c = \dfrac{qB}{2\pi m} = \dfrac{(1.6\times10^{-19})(0.20)}{2\pi(1.67\times10^{-27})} \approx 3.0\times10^{6}\,\text{Hz}, roughly 3.0 MHz.
- Check the physics. Double the speed and the radius doubles, but the frequency stays 3.0 MHz — the speed-independence that makes cyclotrons work. A faster proton simply runs a bigger loop in the same time.