The Magnetic Force on a Moving Charge

An electric field pushes a charge along the field line, the same way gravity pulls straight down. A magnetic field does something stranger: it pushes a moving charge sideways, perpendicular to both the charge's velocity and the field itself. That sideways push comes from the formula F = qvB sin(theta), where theta is the angle between velocity and field. No motion means no force at all, and motion parallel to the field also means no force, since sin(0) = 0. The force is strongest, F = qvB, when velocity and field are perpendicular.

Direction comes from the right hand rule. Point your fingers along the velocity, curl them toward the field direction, and your thumb gives the force direction for a positive charge. For a negative charge, flip the result. Because the force is always perpendicular to velocity, it never points along the direction of motion, which is exactly why it only steers the charge instead of speeding it up or slowing it down. A force doing zero work can't change kinetic energy, so speed stays fixed while direction constantly turns, the same logic as a ball on a string moving in a circle.

This sets up the next idea directly: a constant sideways force acting on a constant speed is the recipe for uniform circular motion. That's why a charged particle crossing a uniform magnetic field doesn't fly off in a straight line or spiral in or out, it curves into a perfect circle, with the magnetic force playing the role of centripetal force.