Question

The de Broglie wavelength of a proton and $\alpha$-particle are equal. The ratio of their velocities is:

• A. $4:2$
• B. $4:1$
• C. $1:4$
• D. $4:3$

Answer 1

The correct option is B

$4:1$

Step 1: Given data:

Let the de Broglie wavelength of a proton is ${\lambda }_p$

The de Broglie wavelength of a $\alpha$-particle is ${\lambda }_{\alpha }$

${\lambda }_p={\lambda }_{\alpha }$

Step 2: Formula used:

From the de Broglie wavelength

$\lambda =\frac{h}{mv}$ where $h,m,v$ are Planck's constant, mass and velocity respectively

Therefore, $v\alpha \frac{1}{m}$since the wavelength is the same for both the particle

Step 3: Calculating the ratios of their velocities:

Let the mass of a proton is = $m_p$

Mass of $\alpha$ particle is $m_{\alpha }$ = $4m_p$ since mass of alpha particle is four times the mass of proton

The ratio of their velocity can calculate as-

$$\begin{gathered} \Rightarrow v_p/v_{\alpha }=m_{\alpha }/m_p \\ \Rightarrow v_p/v_{\alpha }=4m_p/m_p \\ \Rightarrow v_p/v_{\alpha }=4/1 \end{gathered}$$

Therefore, the ratio of their velocities is $4:1$.

Hence, option B is the correct answer.