Electrons are accelerated through a potential difference \(V\) and protons are accelerated through a potential difference \(4\, V\). The de-Broglie wavelengths are \(\lambda_e \) and \(\lambda_p \) for electrons and protons respectively. The ratio of \(\frac{{{\lambda _e}}}{{{\lambda _p}}}\) is given by : (given \(m_e\) is mass of electron and \(m_p\) is mass of proton).

\(=\) charge \(\times\) potential diff. in volt

\({{\text{E}}_{{\text{electron }}}} = {q_{\text{e}}}{\text{V}}\) and \({{\text{E}}_{{\text{proton }}}} = {q_{\text{p}}}4{\text{V}}\)

de-Broglie wavelength

\(\lambda=\frac{\mathrm{h}}{\mathrm{P}}=\frac{\mathrm{h}}{\sqrt{2 \mathrm{mE}}}\)

\(\lambda_{\mathrm{c}}=\frac{\mathrm{h}}{\sqrt{2 \mathrm{m}_{\mathrm{e}} \mathrm{e} \mathrm{V}}}\) and \(\lambda_{\mathrm{P}}=\frac{\mathrm{h}}{\sqrt{2 \mathrm{m}_{\mathrm{p}} \mathrm{e} 4 \mathrm{V}}}\)

\({\text{(}}\because {q_c} = {q_p}{\text{)}}\)

\(\therefore \frac{{{\lambda _{\text{c}}}}}{{{\lambda _{\text{P}}}}} = \frac{{\frac{{\text{h}}}{{\sqrt {2{{\text{m}}_{\text{e}}}{\text{eV}}} }}}}{{\frac{{\text{h}}}{{\sqrt {2{{\text{m}}_p}{\text{e4V}}} }}}}\) \( = \sqrt {\frac{{2{{\text{m}}_p}{\text{e4V}}}}{{2{{\text{m}}_e}{\text{eV}}}}} \)

\( = 2\sqrt {\frac{{{{\text{m}}_{\text{p}}}}}{{{{\text{m}}_{\text{e}}}}}} \)