Question

A particle is travelling $4$ times as fast as an electron. Assuming the ratio of de-Broglie wavelength of a particle to that of electron is $2:1$, the mass of the particle is:

• A. $\frac{1}{16}$ times of mass of $e^{-}$
  
• B. $16$ times of mass of $e^{-}$
  
• C. $\frac{1}{8}$ times of mass of $e^{-}$
  
• D. $8$ times of mass of $e^{-}$
  

Answer 1

The correct option is C $\frac{1}{8}$ times of mass of $e^{-}$


De-broglie wavelength, $\lambda =\frac{h}{p}=\frac{h}{mv}$
Let ${\lambda }_p$ and ${\lambda }_e$ be the wavelength of particle and electron respectively.

$\Rightarrow \frac{{\lambda }_p}{{\lambda }_e}=\frac{\frac{h}{m_p{v}_p}}{\frac{h}{m_e{v}_e}}=\frac{m_e{v}_e}{m_p{v}_p}$

$\Rightarrow \frac{2}{1}=\frac{m_e}{m_p}\times \frac{v_e}{4v_e}$

$\Rightarrow m_p=\frac{m_e}{8}$
So mass of the particle is $\frac{1}{8}$ times the mass of electron.