Question

An $\alpha -$particle and a proton have their masses in the ratio $4:1$ and charges in the ratio $2:1$.

Find ratio of de-Broglie wavelengths, when both have equal kinetic energy.

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

Answer 1

The correct option is B $1:2$

We know,
De broglie wavelength= $\lambda =\frac{h}{mv}$
or, $\lambda =\frac{h}{\sqrt{2\times m\times KE}}$

At constant $KE$,
$\lambda \propto \frac{1}{\sqrt{m}}$

$\frac{{\lambda }_{\alpha }}{{\lambda }_p}=\sqrt{\frac{m_p}{m_{\alpha }}}$

Mass of $\alpha$ particle is 4 times mass of a proton.
$\frac{m_p}{m_{\alpha }}=\frac{1}{4}$

Hence,
$\frac{{\lambda }_{\alpha }}{{\lambda }_p}=\frac{1}{2}$

Hence option $\text{B}$ is correct.