Electrons are accelerated through a potential difference V and protons are accelerated through a potential difference 4 V. The de-Broglie wavelengths are $λ_{e} and λ_{p}$ for electrons and protons respectively. The ratio of $\frac{λ_{e}}{λ_{p}}$ is given by (given m_e is mass of electron and m_p is mass of proton).

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$\frac{λ_{e}}{λ_{p}} = \frac{1}{2} \sqrt{\frac{m_{e}}{m_{p}}}$

$\frac{λ_{e}}{λ_{p}} = \sqrt{\frac{m_{p}}{m_{e}}}$

$\frac{λ_{e}}{λ_{p}} = 2 \sqrt{\frac{m_{p}}{m_{e}}}$

$\frac{λ_{e}}{λ_{p}} = \sqrt{\frac{m_{e}}{m_{p}}}$

answer is D.

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Detailed Solution

$λ = \frac{h}{\sqrt{2 mV q}} \Rightarrow λ \propto \frac{1}{\sqrt{mV}}$

$\Rightarrow \frac{λ_{e}}{λ_{p}} = \sqrt{\frac{m_{p} V_{p}}{m_{e} V_{e}}} = \sqrt{\frac{m_{p} (4 V)}{m_{e} (V)}}$

$\Rightarrow \frac{λ_{e}}{λ_{p}} = 2 \sqrt{\frac{m_{p}}{m_{e}}}$

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