A Circulating Charged Particle — Question 3

Question 3

Helix radius and pitch separately reveal transverse and longitudinal speed

A nonrelativistic particle with $|q|/m=10^7$ C/kg moves helically in B=.100 T. Its radius is .0300 m and its pitch, the axial advance per revolution, is .120 m.

1. Find the perpendicular speed, parallel-speed magnitude and total speed.

2. Find the acute angle between velocity and the field, and the number of turns made during an axial advance of .600 m.

Question 3 - Solution

Helix radius and pitch separately reveal transverse and longitudinal speed

(a) Method and calculation

$$\omega=|q|B/m=10^6\,\mathrm{s^{-1}},\quad v_\perp=\omega r=3.00\times10^4\,\mathrm{m/s}.$$$$|v_\parallel|=p\omega/(2\pi)=1.90986\times10^4\,\mathrm{m/s},\quad v=\sqrt{v_\perp^2+v_\parallel^2}=3.55634\times10^4\,\mathrm{m/s}.$$

(b) Interpretation and further calculation

$$\tan\theta=v_\perp/|v_\parallel|=2\pi r/p\Rightarrow\theta=57.5184^\circ.$$$$N=\Delta z/p=.600/.120=5.00\quad\text{turns}.$$

Physical check

Radius probes only perpendicular momentum; pitch supplies the missing parallel component.