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Why does a charged particle move in a circular path when its velocity is perpendicular to a uniform magnetic field?

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Diagram showing a circular loop of wire with an electric current, surrounded by curved blue lines representing magnetic field lines that loop around the wire and pass through its center. — magnetism Some lines of the magnetic field B (blue) for an electric current I (gold) in a loop.

When a charged particle moves through a uniform magnetic field with a velocity perpendicular to the field lines, it experiences a Lorentz force that is always directed perpendicular to both its velocity and the magnetic field.12

Because this magnetic force acts at a right angle to the particle's direction of motion, it does not change the particle's speed; instead, it acts as a centripetal force that continuously changes the direction of the velocity. This constant, perpendicular force causes the particle to follow a circular trajectory. The radius of this circular path, denoted as r, is determined by the particle's mass m, velocity v, charge q, and the magnetic field strength B, expressed by the formula:12

$r = \frac{m v}{q B}$

This principle is fundamental to the operation of particle accelerators such as cyclotrons, which utilize these circular orbits to manipulate and accelerate charged particles for scientific and medical applications.12