Question

Calculate the de-Broglie wavelength of an electron beam acceleration through a potential difference of 60 V.
A.1.58 A
B.2.58 A
C.3.58 A
D.4.58 A

Answer 1

Hint: We will make use of the de-Broglie wave equation. Firstly, we will calculate the energy of the particle. Then by substituting the values of Planck's constant, mass of electron and energy in the de-Broglie wavelength formula we will get the required answer.

Formula used: $\lambda =\dfrac{h}{\sqrt{2mE}}$
$E=q\Delta V\text{ }$

Complete step by step solution:
We know the equation derived by the scientist Louis de-Broglie which describes the wave nature of any particle. Therefore, the wavelength of any moving body/object is derived by a formula:
$\lambda =\dfrac{h}{\sqrt{2mE}}\text{ }............\text{(1)}$
Where $\lambda $ denotes the wavelength of a moving body, h is the Planck's constant, m is the mass of the particle in kg and E is the energy of the particle.
In the question we have given the potential difference 60 V, that is $\Delta V=60V$
Now, Energy of the particle is given by:
$E=q\Delta V\text{ }..........\text{(2)}$
We know that charge on one electron is equal to $1.6\times {{10}^{-19}}C$, so substituting this in equation(1)
we get,
$\begin{align}
  & E=1.6\times {{10}^{-19}}\times 60 \\
 & \Rightarrow E=9.6\times {{10}^{-18}}J\text{ }.............\text{(3)} \\
\end{align}$
And momentum, p=mv=$\sqrt{2mE}\text{ }............\text{(4)}$
We know that mass of an electron is $m=9.1\times {{10}^{-31}}kg$,
$E=9.6\times {{10}^{-18}}Joules$and
\[h=6.62\times {{10}^{-34}}\text{ Joule second}\]
Substituting the above values in equation (1) we get,
$\begin{align}
  & \lambda =\dfrac{6.62\times {{10}^{-34}}}{\sqrt{2\times 9.1\times {{10}^{-31}}\times 9.6\times {{10}^{-18}}}} \\
 & \Rightarrow \lambda =1.58\times {{10}^{-10}}m \\
 & \therefore \lambda =1.58\text{ Angstrom} \\
\end{align}$
So, the de-Broglie wavelength of an electron beam when accelerated through a potential difference of 60 V will be 1.58 Angstrom.

Hence, Option(A) is correct.

Additional information:
De Broglie equation states that a matter can act as waves just like light which behaves as waves and particles. The equation also explains that a beam of electrons can be diffracted similar to a beam of light. Thus, the de Broglie equation explains the idea of matter having a wavelength. Therefore, according to de Broglie every moving particle whether it is microscopic or macroscopic it will have a wavelength.

Note:
One must remember the value of Planck's constant ‘h’, mass of electron ‘m’ and charge on electron ‘q’, to solve such kinds of questions.
\[h=6.62\times {{10}^{-34}}\text{ Joule second}\]
$m=9.1\times {{10}^{-31}}kg$
$q=1.6\times {{10}^{-19}}C$