Question

An electron having charge $e$ and mass $m$ at rest starts from the lower plate of two metallic plates separated by a distance $d$. If the potential difference between the plates is $V$, the time taken by the electron to reach the upper plate is

• A. $\sqrt{\frac{2m{d}^2}{eV}}$
• B. $\sqrt{\frac{m{d}^2}{eV}}$
• C. $\sqrt{\frac{m{d}^2}{2eV}}$
• D. $\sqrt{\frac{4m{d}^2}{eV}}$

Answer 1

The correct option is A $\sqrt{\frac{2m{d}^2}{eV}}$

Electric field between the plates is towards lower plate of magnitude,

$E=\frac{V}{d}$

Force experience by the electron due to electric field $E$ is towards upper plate of magnitude,

$F=eE=\frac{eV}{d}$

From Newton's second law of motion, the acceleration of the electron,

$a=\frac{F}{m}=\frac{eV}{md}$

From equation of motion,
$d=ut+\frac{1}{2}a{t}^2$

Since the initial velocity of electron is zero.i.e. $u=0$ and the time taken by electron to reach upper plate is $t$, so,
$d=\frac{1}{2}a{t}^2$

$\Rightarrow t=\sqrt{\frac{2d}{a}}$

Substituting the value of $a$ in the above equation, we get

$t=\sqrt{\frac{2dmd}{eV}}$

$\therefore t=\sqrt{\frac{2m{d}^2}{eV}}$

Hence, option $\left(\text{A}\right)$ is the correct answer.