Circular Motion of Charged Particles

The force on a charge $q$ moving with velocity $\vec{v}$ in a magnetic field $\vec{B}$ is given by the Lorentz force law: $\vec{F} = q\vec{v} \times \vec{B}$. The magnitude is $F = qvB\sin\theta$. When the velocity is perpendicular to the field ($\theta = 90°$), this simplifies to $F = qvB$. This force is always perpendicular to $\vec{v}$, so it cannot do work (recall $W = \vec{F} \cdot \vec{d}$, and a perpendicular force has zero component along displacement). The particle's speed stays constant, but its direction changes continuously.

For any object moving in a circle of radius $r$ at speed $v$, Newton's second law requires a centripetal force $F_c = \frac{mv^2}{r}$ directed toward the center. Since the magnetic force provides exactly this centripetal force, we set them equal: $$qvB = \frac{mv^2}{r}$$. Solving for $r$, we divide both sides by $v$ (since $v \neq 0$) and rearrange to get $r = \frac{mv}{qB}$. This is the fundamental equation for the radius of a charged particle's circular orbit in a magnetic field.

The time for one complete revolution is the period $T$. Since the particle travels a circumference $2\pi r$ at constant speed $v$, we have $T = \frac{2\pi r}{v}$. Substituting $r = \frac{mv}{qB}$ gives $$T = \frac{2\pi m}{qB}$$. Remarkably, the period is independent of the particle's speed! Faster particles travel larger circles but complete orbits in the same time. The cyclotron frequency (number of orbits per second) is $f = \frac{1}{T} = \frac{qB}{2\pi m}$. This speed-independence is the principle behind the cyclotron particle accelerator.

The direction of the circular orbit depends on the sign of the charge and the direction of $\vec{B}$. Using the right-hand rule for $\vec{v} \times \vec{B}$: point fingers in the direction of $\vec{v}$, curl them toward $\vec{B}$, and the thumb points in the direction of the force on a positive charge. A negative charge (like an electron) experiences the opposite force direction and curves the other way. If $\vec{B}$ points into the page and $\vec{v}$ points to the right, a proton curves upward initially and orbits counterclockwise; an electron orbits clockwise.

• r — Orbital radius (metres (m)): The radius of the circular path traced by the particle in the magnetic field.
• m — Mass of particle (kilograms (kg)): The inertial mass of the charged particle; larger mass means larger radius.
• v — Speed of particle (metres per second (m/s)): The constant speed of the particle; the magnetic force cannot change this value.
• q — Charge of particle (coulombs (C)): The magnitude of the particle's electric charge; larger charge means smaller radius.
• B — Magnetic field strength (tesla (T)): The magnitude of the uniform magnetic field; stronger fields produce tighter orbits.

Problem. A proton (mass $m_p = 1.67 \times 10^{-27}$ kg, charge $q = 1.60 \times 10^{-19}$ C) travels at $3.0 \times 10^6$ m/s perpendicular to a uniform magnetic field of strength $B = 0.25$ T. (a) Calculate the radius of its circular orbit. (b) Calculate the period of the orbit.

Answer. The proton orbits with radius approximately 12.5 cm and completes one full orbit in approximately 262 nanoseconds.

• The Lorentz Force Law — The magnetic component of the Lorentz force, $\vec{F} = q\vec{v} \times \vec{B}$, is the fundamental force that causes circular motion in a magnetic field. Understanding vector cross products and the right-hand rule is essential prerequisite knowledge.
• Centripetal Acceleration and Force — The derivation of orbital radius sets the magnetic force equal to the centripetal force $F_c = \frac{mv^2}{r}$. A solid understanding of circular motion mechanics is required to follow and apply this derivation correctly.
• The Mass Spectrometer — Mass spectrometers use circular motion in magnetic fields to separate ions by mass. By measuring orbital radii, scientists determine atomic masses with extreme precision — a direct technological application of $r = \frac{mv}{qB}$.
• The Cyclotron Particle Accelerator — The cyclotron exploits the fact that the orbital period $T = \frac{2\pi m}{qB}$ is independent of speed, allowing particles to be accelerated in synchrony with an alternating electric field as they spiral outward in a magnetic field.