Circular Motion in a Magnetic Field

The magnetic force on a moving charge is F = qvB sin(theta), and its direction is always perpendicular to the velocity. A force perpendicular to velocity cannot change speed, since it does no work on the charge; it can only change direction. When the charge moves perpendicular to the field, the force stays perpendicular to v at every instant, constantly steering the particle without ever speeding it up. That steady sideways push is exactly what a centripetal force does: it bends a path into a circle instead of letting it fly straight.

Because the magnetic force is the centripetal force here, you can set them equal: qvB = mv²/r. Solve for r and you get r = mv/(qB). Everything you need, mass, charge, speed, field strength, plugs directly into this one equation. Notice the speed appears once in the numerator after cancellation, so faster particles sweep bigger circles, and stronger fields or larger charges pull the radius tighter.

If the particle's velocity has a component along the field direction, that component feels no force at all, since sin(theta) = 0 for motion parallel to B. That piece of the motion just coasts along at constant speed while the perpendicular piece keeps circling, and the two combine into a helix, a corkscrew path spiraling along the field line. This is the same centripetal reasoning from circular motion in mechanics, where gravity or tension or friction supplied F = mv²/r; here the magnetic force simply takes over that role.