Crossed Fields: Discovery of the Electron — Question 2

Question 2

Acceleration voltage and a force-balance speed determine the charge-to-mass ratio

A cathode-ray particle starts essentially at rest and gains kinetic energy eU through an accelerating potential magnitude U=180 V. It then passes undeflected through perpendicular fields E=30.0 kilovolt per metre and B=5.00 millitesla. Treat its motion as nonrelativistic.

1. Derive e/m from U, E and B and evaluate it.

2. Predict the selected speed and the new balancing magnetic field if U is quadrupled while E is unchanged.

Question 2 - Solution

Acceleration voltage and a force-balance speed determine the charge-to-mass ratio

(a) Method and calculation

$$eU=\tfrac12mv^2,\quad v=E/B\Rightarrow\frac em=\frac{E^2}{2UB^2}.$$$$v=6.00\times10^6\,\mathrm{m/s},\quad e/m=1.00\times10^{11}\,\mathrm{C/kg}.$$

(b) Interpretation and further calculation

$$U\prime=4U\Rightarrow v\prime=\sqrt{4}\,v=1.20\times10^7\,\mathrm{m/s}.$$$$B\prime=E/v\prime=B/2=2.50\,\mathrm{mT}.$$

Physical check

The inferred ratio is a property of the measured beam under the model; the supplied example is not a claim of an exact physical electron constant.