Question

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

• A. $1/16$ times of mass of electron
• B. $1/6$ times the mass of electron
• C. $1/8$ times the mass of electron
• D. $8$ times the mass of electron

Answer 1

The correct option is C

$1/8$ times the mass of electron

Step 1: Given data

The ratio of the de-Broglie wavelength, $\frac{{\lambda }_p}{{\lambda }_e}=\frac{2}{1}$ [ ${\lambda }_p,{\lambda }_e$ is wavelength of the particle and the electron respectively]

Velocity of particle, $v_p=4v_e$ [$v_p,v_e$ is velocity of the particle and the electron respectively.]

Step 2: Find mass of particle, $m$

The wavelength is given by

$$\begin{gathered} \lambda =\frac{h}{p} \\ \lambda =\frac{h}{mv} \\ \frac{{\lambda }_p}{{\lambda }_e}=\frac{\frac{h}{m_p{v}_p}}{\frac{h}{m_e{v}_e}} \\ \frac{{\lambda }_p}{{\lambda }_e}=\frac{m_e{v}_e}{m_p{v}_p} \\ \frac{2}{1}=\frac{m_e}{m_p}\times \frac{v_e}{4v_e} \\ {m}_p=\frac{m_e}{8} \end{gathered}$$ [Where, $p=mv$ is momentum, $h$ is Planck's constant, $m_p,m_e$ is the mass of the particle and the electron respectively. ]

Hence, option$\left(C\right)$ is correct.