Question

An electron of mass ‘m’ and charged ‘e’ is accelerated from rest through a potential difference ‘V’ in vacuum. Its final velocity will be
(A) ${\displaystyle \frac{eV}{2m}}$
(B)${\displaystyle \frac{eV}{m}}$
(C)$\sqrt{{\displaystyle \frac{2eV}{m}}}$
(D)$\sqrt{{\displaystyle \frac{eV}{m}}}$

Answer 1

Hint:-In this question can easily be solved by applying the law of transformation of energy. In this the electric potential energy of the electron has been changed into kinetic energy. In order to determine the velocity we have equated the potential energy and kinetic energy of the electron. In solving the equation for velocity we can easily determine the velocity of an electron.

Complete step-by-step solution:-
The potential difference applied in vacuum provides energy to electrons which are at rest and it starts moving with velocity. Let’s say the velocity of an electron is $v$.
So, the energy of applied potential is converted into kinetic energy of electron i.e. conversion of energy. As there is no other energy acting on electrons in vacuum, it is safe to say the above statement.
We know that kinetic energy of electron is given by
$K.E.={\displaystyle \frac{1}{2}}m{v}^2$
Where,
m is the mass of electron
v is the velocity of electron
So, the work done on electron by potential is charge x Voltage i.e. $qV\Rightarrow eV$ , and this work produces the kinetic energy of electron,
So according to given condition we have
Potential energy = kinetic energy
$eV={\displaystyle \frac{1}{2}}m{v}^2$
On solving for velocity, v we get
$$\begin{gathered} v^2={\displaystyle \frac{2eV}{m}} \\ v=\sqrt{{\displaystyle \frac{2eV}{m}}} \end{gathered}$$
Hence its final velocity will be $v=\sqrt{{\displaystyle \frac{2eV}{m}}}$ .

Hence the correct option is C.

Additional Information:-
Energy transfer refers to the movement of energy from one place to another. Think of the electricity that flows from your wall socket, then moves through a charger and into a battery. The energy is being transferred from the wall socket to the battery.
One type of energy can change into another type of energy. Energy transformation means the changing of energy from one type to another, e.g. from kinetic energy to electrical energy, or from potential energy to kinetic energy. The above given question is the simplest example of it.

Note:- Here we assumed that this is a non-relativistic particle i.e. the electron doesn’t approach the velocity of light. If it would have been the case of relativistic particles, the solution would have been different. Here the voltage could have ripped out electrons from the atom if it was high enough but we assume it did not for simplicity. If that would have happened then, the electron had produced wavelength corresponding to atomic level.