CBSE 2022 · Region 2 · Set 1 · Q10 · 3 marks
Find the ratio of the de Broglie wavelengths associated with an alpha particle and a proton, if both(a)have the same speeds,(b)have the same kinetic energy,(c)are accelerated through the same potential difference.
Find the ratio of the de Broglie wavelengths associated with an alpha particle and a proton, if both
(a)
have the same speeds,
(b)
have the same kinetic energy,
(c)
are accelerated through the same potential difference.
Marking-scheme solution
(a)
$\displaystyle \lambda = \dfrac{h}{p}$
\[\frac{\lambda_\alpha}{\lambda_p} = \frac{h}{m_\alpha v_\alpha} \times \frac{m_p v_p}{h} = \frac{1}{4}\]
(b)
$\displaystyle p = \sqrt{2m(K.E)}$
\[\frac{\lambda_\alpha}{\lambda_p} = \frac{h}{\sqrt{2m_\alpha (\mathrm{K.E.})_\alpha}} \times \frac{\sqrt{2m_p (\mathrm{K.E.})_p}}{h} = \sqrt{\frac{m_p}{m_\alpha}} = \frac{1}{2}\]
(c)
$\displaystyle v = \sqrt{\dfrac{2qV}{m}}$
\[\frac{\lambda_\alpha}{\lambda_p} = \frac{h}{m_\alpha v_\alpha} \times \frac{m_p v_p}{h} = \frac{m_p}{m_\alpha}\sqrt{\frac{2q_p V}{m_p}} \times \sqrt{\frac{m_\alpha}{2q_\alpha V}}\]
\[= \frac{m_p}{m_\alpha} \times \sqrt{\frac{m_\alpha}{m_p}} \times \sqrt{\frac{q_p}{q_\alpha}} = \frac{1}{2\sqrt{2}}\]
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CBSE Class 12 Physics past-paper question from the 2022board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.