Crossed Fields: Discovery of the Electron — Question 3

Question 3

Turning off the magnetic field converts a balanced beam into a charge-to-mass measurement

An electron beam is first balanced at speed $v=10^7$ m/s. The magnetic field is then removed while a downward electric field E=1000 V/m acts over plate length L=.0200 m. A screen is D=.0800 m beyond the plates. The upward screen displacement is Y=3.60 mm. Assume negligible initial transverse speed.

1. Derive the screen displacement including the drift beyond the plates.

2. Infer e/m and check the exit transverse-to-longitudinal speed ratio.

Question 3 - Solution

Turning off the magnetic field converts a balanced beam into a charge-to-mass measurement

(a) Method and calculation

$$t_L=L/v,\quad y_L=\frac{eE L^2}{2mv^2},\quad v_{yL}=\frac{eEL}{mv}.$$$$Y=y_L+v_{yL}D/v=\frac{eE}{mv^2}L(D+L/2).$$

(b) Interpretation and further calculation

$$\frac em=\frac{Yv^2}{EL(D+L/2)}=2.00\times10^{11}\,\mathrm{C/kg}.$$$$v_{yL}/v=\frac{eEL}{mv^2}=.0400.$$

Physical check

Ignoring the field-free drift would systematically overestimate e/m from a given screen displacement.