Crossed Fields: Discovery of the Electron — Question 9

Question 9

Deflection controls identify the sign of cathode-ray charge without assuming its mass

A beam moves along +x. A downward electric field alone deflects it upward. An into-page magnetic field alone deflects it downward. With both fields applied it is straight. The electric field is E and magnetic magnitude B.

1. Determine the charge sign independently from each control observation.

2. Find the beam speed and predict both deflection directions if the charge sign were reversed without changing the velocity.

Question 9 - Solution

Deflection controls identify the sign of cathode-ray charge without assuming its mass

(a) Method and calculation

$$F_{Ey}=q(-E)>0\Rightarrow q<0.$$$$\vec v\times\vec B=vB\hat y,\quad F_{By}=qvB<0\Rightarrow q<0.$$

(b) Interpretation and further calculation

$$q(-E+vB)=0\Rightarrow v=E/B.$$$$q\to-q:\quad F_{Ey}<0,\quad F_{By}>0,\quad F_{Ey}+F_{By}=0\text{ at the same speed}.$$

Physical check

The two independent polarity tests agree; the force-balance speed does not require knowing the mass.