A Circulating Charged Particle — Question 9

Question 9

Two helical particles can regain their common transverse phase after different turn counts

Two positive particles have the same charge q and masses m and 3m/2. They start from the same point with the same nonzero perpendicular velocity and the same parallel velocity u in uniform B. Let $T_1=2\pi m/(qB)$.

1. Find the earliest positive time at which both have completed integer numbers of turns and returned to their initial transverse positions.

2. Find their common axial displacement then and compare their radii and pitches.

Question 9 - Solution

Two helical particles can regain their common transverse phase after different turn counts

(a) Method and calculation

$$T_2=\tfrac32T_1,\quad n_1T_1=n_2T_2\Rightarrow2n_1=3n_2.$$$$n_1=3,\quad n_2=2,\quad t_{\mathrm{return}}=3T_1.$$

(b) Interpretation and further calculation

$$\Delta z=ut_{\mathrm{return}}=3uT_1.$$$$r_2/r_1=m_2/m_1=3/2,\quad p_2/p_1=T_2/T_1=3/2.$$

Physical check

Their transverse radii differ, but integer-turn returns remove that spatial difference at the specified instant.