Crossed Fields: Discovery of the Electron — Question 10

Question 10

The plate aperture limits an electric-deflection measurement before the screen does

A particle of charge magnitude q and mass m enters midway between parallel plates separated by d, with horizontal speed v and no transverse velocity. A uniform transverse electric field E acts over length L. Neglect fringing and relativity.

1. Derive the largest E for which the particle does not strike a plate before exiting.

2. For a screen distance D beyond the plates, find the largest screen displacement compatible with that aperture constraint.

Question 10 - Solution

The plate aperture limits an electric-deflection measurement before the screen does

(a) Method and calculation

$$|y(x)|=\frac{qE}{2mv^2}x^2,\quad|y(L)|\le d/2.$$$$E\le E_{\max}=\frac{mv^2d}{qL^2}.$$

(b) Interpretation and further calculation

$$|Y|=\frac{qE}{mv^2}L(D+L/2).$$$$|Y|_{\max}=d\left(\frac DL+\frac12\right).$$

Physical check

The transverse displacement grows monotonically inside the plates, so the exit gives the limiting clearance; equality is a grazing limit.