Question

If the de-Broglie wavelengths for a proton and alpha particle is same, then the ratio of their velocities is:
A. 1:2
B. 2:1
C. 1:4
D. 4:1

Answer 1

Hint: The de Broglie wavelength is basically the wavelength associated with a moving particle/matter. An alpha particle is basically He²⁺ ion, i.e consisting of two neutrons and two protons.

Formula Used: $\lambda=\dfrac{h}{mv}$

Complete step by step answer: As we all know that the de-broglie equation states that matter can act as waves not much like light and some radiation will behave like both particles and waves. De-broglie equation helps us to imagine that matter is also having wavelength.
We have studied that the de-broglie equation is given by:
$\lambda = \dfrac{h}{{mv}}$ …… (I)
Here $\lambda $ is the De-broglie wavelength, $h$ is the Planck's constant, $m$ is the mass of the particle and $v$ is the velocity of the particle.
So, If the de-broglie wavelength for both the proton and alpha particle is same then the equation (I) can be written as,
$ \Rightarrow {\lambda _1} = {\lambda _2}$
$ \Rightarrow {\left( {\dfrac{h}{{mv}}} \right)_1} = {\left( {\dfrac{h}{{mv}}} \right)_2}$ …… (II)
Here the subscripts 1 and 2 represent the proton and the alpha particle. $m$ represents the mass and $v$ represents the velocity. Also it is given to us in the question that,
${m_2} = 4{m_1}$
Here ${m_1}$ is the mass of the proton and ${m_2}$ is the mass of the alpha particle. Now the equation (II) can be written by substituting ${m_2} = 4{m_1}$ as,
\[
  {m_1}{v_1} = {m_2}{v_2} \\
   \Rightarrow {m_1}{v_1} = 4{m_1}{v_2} \\
   \Rightarrow {v_1} = 4{v_2} \\
  \therefore \dfrac{{{v_1}}}{{{v_2}}} = \dfrac{4}{1} \\
 \]
Therefore, the velocity of the proton is four times the mass of the alpha particle. Therefore, the correct option is (D).

Note: As we all know that the concept regarding wave particle duality is well explained by matter waves. Matter of all kinds can exhibit a wave like behaviour. A best example for it is that a beam of electrons is just diffracted like the beam of light or water wave. But in general the wavelength is very small to impact a great extent on us for our day to day life. We mean that the matter waves related to tennis ball, other objects are not relevant.