Analyse the circular motion of a charged particle in a uniform magnetic field, relating radius to mass, charge, speed and field strength.

What this dot point is asking

You need to analyse the circular motion of a charged particle in a uniform magnetic field and relate the radius to the particle's mass, charge, speed and the field.

Why the path is a circle

The magnetic force $F = qvB$ on a charge moving perpendicular to the field is always at right angles to the velocity. A force that is always perpendicular to motion changes only the direction, not the speed - exactly the condition for uniform circular motion. So the particle traces a circle at constant speed.

The radius of the path

Rearranging the balance of forces:

So the radius:

• increases with mass and speed (more momentum is harder to bend),
• decreases with charge and field strength (a stronger push bends it tighter).

The period of the circular motion, $T = \dfrac{2\pi m}{qB}$, is independent of speed: faster particles travel larger circles in the same time. This is the principle that lets a cyclotron use a fixed-frequency accelerating voltage.

Applications: the mass spectrometer

The radius equation is the basis of the mass spectrometer. Ions of known charge and speed (often selected by a velocity selector) enter a uniform field. Since $r = \dfrac{mv}{qB}$, ions of different mass-to-charge ratio land at different radii, separating them so their masses can be measured. This is how isotopes are identified and their abundances measured. On a graph of radius against speed for ions of the same charge in the same field, the gradient is $\dfrac{m}{qB}$, so a steeper line means a larger mass.

If a charge enters the field at an angle (not perpendicular), the component of velocity along the field is unaffected and the perpendicular component drives circular motion, so the overall path is a helix.

The cyclotron

A cyclotron accelerates charged particles to high energy using the speed-independent period. The particle spirals outward through two D-shaped electrodes, gaining energy from an alternating voltage applied across the gap each time it crosses. Because $T = \dfrac{2\pi m}{qB}$ does not depend on the speed or radius, the accelerating voltage can alternate at a single fixed frequency and stay synchronised with the particle no matter how fast it is moving. The radius grows as the speed grows, but the time for each half-circle stays the same, which is exactly why the design works.

How SACE assesses this

SACE Stage 2 questions on this dot point typically give a particle's mass, charge, speed and the field, then ask for the radius using $r = \dfrac{mv}{qB}$, or rearrange it to find the speed or field. A second common style supplies a graph of radius against speed and asks you to interpret the gradient $\dfrac{m}{qB}$ to compare masses or charges. "Show that" parts asking you to verify a cyclotron period use $T = \dfrac{2\pi m}{qB}$ directly. Quote the formula, substitute in consistent SI units (convert any radius given in centimetres to metres), and rearrange before substituting numbers rather than after.