A) \[2\sqrt{2}:1\]
B) \[1:2\sqrt{2}\]
C) \[2:1\]
D) none of these
Correct Answer: A
Solution :
The relation for energy is given by \[E=\frac{1}{2}m\,{{V}^{2}}\] or \[\sqrt{2mE}=m\,V\] and relation for wavelength is given by \[\lambda =\frac{h}{m\upsilon }=\frac{h}{\sqrt{2m\,E}}\] Now for proton \[{{\lambda }_{p}}=\frac{h}{\sqrt{2\,P\,E}}\] ?..(i) and for a particle, \[{{\lambda }_{\alpha }}=\frac{h}{\sqrt{2\times 4m\times 2E}}\] ?.(ii) (mass of \[\alpha \] - particle is 4 times to that of proton) From equation (i) and equation (ii), we get \[\frac{{{\lambda }_{p}}}{{{\lambda }_{\alpha }}}=\frac{h}{\sqrt{2m\,E}}\times \frac{\sqrt{16m\,E}}{h}\] \[\Rightarrow \] \[\frac{{{\lambda }_{p}}}{{{\lambda }_{\alpha }}}=\frac{2\sqrt{2}}{1}\] Hence \[{{\lambda }_{p}}:{{\lambda }_{\alpha }}=2\sqrt{2}:1\]
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