A Circulating Charged Particle — Question 8

Question 8

Orbit timing and rotation sense determine a signed charge-to-mass ratio

A nonrelativistic particle circulates counterclockwise as viewed from +z in a uniform field $+.200\hat z$ T. Its period is 2.00 microseconds. The speed and orbit radius are not measured.

1. Determine the charge sign and signed q/m.

2. Explain why timing is sufficient for this ratio but insufficient to determine kinetic energy.

Question 8 - Solution

Orbit timing and rotation sense determine a signed charge-to-mass ratio

(a) Method and calculation

$$\vec v=v\hat x\Rightarrow\vec v\times\vec B=-vB\hat y;\quad\text{counterclockwise curvature requires }q<0.$$$$\frac qm=-\frac{2\pi}{BT}=-1.57080\times10^7\,\mathrm{C/kg}.$$

(b) Interpretation and further calculation

$$T=\frac{2\pi m}{|q|B}\quad\text{contains no }v.$$$$K=\tfrac12mv^2\quad\text{requires additional information about }m\text{ and }v.$$

Physical check

Rotation sense fixes the sign only after the viewing direction and magnetic-field direction are specified.