Path of a Charged Particle in a Magnetic Field: MDCAT Physics notes
Path of a Charged Particle in a Magnetic Field for MDCAT: circular motion, r = mv/qB, time period and cyclotron frequency, straight and helical paths.
Three possible paths
The magnetic force $F = qvB\sin\theta$ depends on the angle $\theta$ between the velocity and a uniform field. This decides the path.
| Direction of entry | Force | Path |
|---|---|---|
| Parallel or antiparallel to $B$ | Zero | Straight line, speed unchanged |
| Perpendicular to $B$ | $qvB$, always perpendicular to $v$ | Circle |
| At some other angle | From the perpendicular component | Helix (spiral) about the field lines |
Inside a long current-carrying solenoid the field is along the axis. A proton fired along the axis moves parallel to $B$, so it continues in a straight line with the same velocity: it is neither accelerated nor deflected.
Circular motion
When $\vec{v}\perp\vec{B}$ the magnetic force is always at right angles to the velocity, so it provides the centripetal force:
$$qvB = \frac{mv^2}{r} \quad\Rightarrow\quad r = \frac{mv}{qB} = \frac{p}{qB}$$
- A heavier particle or a faster one moves in a larger circle.
- A stronger field or a larger charge gives a smaller circle.
- The speed and kinetic energy stay constant, because the force does no work.
Time period and frequency
$$T = \frac{2\pi r}{v} = \frac{2\pi m}{qB} \qquad f = \frac{qB}{2\pi m}$$
$f$ is the cyclotron frequency. It is independent of speed and radius: faster particles move in bigger circles but take the same time per revolution. Doubling the mass halves the frequency; doubling $B$ doubles it.
Helical path
If $\vec{v}$ makes an angle $\theta$ with $\vec{B}$, split it into $v\cos\theta$ along the field (unaffected, uniform motion) and $v\sin\theta$ across it (circular motion of radius $mv\sin\theta/qB$). The combination is a helix.
Worked examples
1. A proton ($m = 1.67\times10^{-27}$ kg, $q = 1.6\times10^{-19}$ C) moves at $2\times10^6$ m s$^{-1}$ perpendicular to a 0.5 T field. $r = \dfrac{1.67\times10^{-27}\times2\times10^6}{1.6\times10^{-19}\times0.5} \approx 0.042$ m.
2. If the proton's speed doubles, $r$ doubles but the period is unchanged.
Applications
The radius of the circular path lets us measure $e/m$ of the electron ($e/m = 2V/B^2r^2$) and separate ions of different mass in a mass spectrometer. The fixed cyclotron frequency is the principle of the cyclotron accelerator.
Key formulas
- $r = mv/qB$
- $T = 2\pi m/qB$
- $f = qB/2\pi m$
Common MDCAT traps
- Entering parallel to the field gives a straight path, not a circle.
- The path in a uniform perpendicular field is circular, not parabolic; a parabola is the path in a uniform electric field.
- Cyclotron frequency does not depend on speed.
- $f \propto 1/m$: doubling mass halves the frequency.
Quick revision
- Perpendicular entry gives a circle.
- Oblique entry gives a helix.
- Radius is proportional to momentum.
- Kinetic energy is constant in a magnetic field.