Question

A beam of cathode rays after gaining kinetic energy once it passes throught a potential difference of $V$ is subjected to crossed electric ($E$) and magnetic fields ($B$). The fields are adjusted such that the beam is not deflected. The specific charge of the cathode rays is given by

• A. $\frac{B^2}{2V{E}^2}$
• B. $\frac{2V{B}^2}{E^2}$
• C. $\frac{2V{E}^2}{B^2}$
• D. $\frac{E^2}{2V{B}^2}$

Answer 1

The correct option is D $\frac{E^2}{2V{B}^2}$

When a beam of cathode rays (or electrons) are subjected to crossed electric ($E$) and magnetic ($B$) fields and the beam is not deflected, we can say that the magnitude of the forces due to the electric and magnetic fields are equal.
$i.eBeV=eE--\left(1\right)or,v=\frac{E}{B}$
If $V$ is the potential differnce between the anode and cathode, then,
$\frac{1}{2}m{v}^2=eVor,\frac{e}{m}=\frac{v^2}{2V}--\left(2\right)$
Substituting the value of $v$ from equation (1) in equation (2), we get
$\frac{e}{m}=\frac{E^2}{2V{B}^2}$
$\therefore$ Specific charge of the cathode rays$\frac{e}{m}=\frac{E^2}{2V{B}^2}$