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  4. A proton when accelerated through a potential difference of V volts has a wave length λ associated with it. An α -particle in order to have the same wavelength λ , must be accelerated through a potential difference (in volts)

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Q. A proton when accelerated through a potential difference of $ V $ volts has a wave length $ \lambda $ associated with it. An $ \alpha $ -particle in order to have the same wavelength $ \lambda $ , must be accelerated through a potential difference (in volts)

AMUAMU 2015Dual Nature of Radiation and Matter

Solution:

For a charged particle
$\lambda = \frac{h}{\sqrt{2mqV}}$
Since,$\lambda_p = \lambda_{\alpha}$ we have
$\therefore \frac{h}{\sqrt{2m_pq_p V}} = \frac{h}{\sqrt{2m_{\alpha}q_{\alpha}V_{\alpha}}}$
$V_{\alpha} = \frac{m_p q_p V}{m_{\alpha}q_{\alpha}}$
$m_{p} q_{p} V = m_{\alpha} q_{\alpha} V_{\alpha}$
$ = \frac{m_{p}q_{p}V}{4m_{p} \cdot2q_{p} } = \frac{V}{8}$