Crossed Fields: Discovery of the Electron — Question 8

Question 8

A selector isolates velocity while an accelerating voltage couples velocity to charge-to-mass ratio

Two positive ion species have masses m and 4m, and charges q and 2q respectively. Both start from rest and must pass the same ideal selector with speed E/B.

1. Find the ratio of accelerating potential magnitudes needed to transmit the two species.

2. If both instead receive the same accelerating potential, find their speed ratio and explain whether both can pass the same force-balanced selector.

Question 8 - Solution

A selector isolates velocity while an accelerating voltage couples velocity to charge-to-mass ratio

(a) Method and calculation

$$U_1=\frac m{2q}\left(\frac EB\right)^2,\quad U_2=\frac{4m}{2(2q)}\left(\frac EB\right)^2.$$$$U_2/U_1=2.$$

(b) Interpretation and further calculation

$$v_1=\sqrt{2qU/m},\quad v_2=\sqrt{2(2q)U/(4m)}=v_1/\sqrt2.$$$$v_1\ne v_2\Rightarrow\text{both cannot satisfy }v=E/B\text{ at the same setting}.$$

Physical check

The selector alone does not measure mass or charge; the accelerating stage supplies the additional relation.