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A proton and photon both have same energy

Question :- A proton and photon both have same energy of E = 100 keV. The de-Broglie wavelength of proton and photon be λ₁ and λ₂ then λ₁/λ₂ is proportional to

(A) E^-1/2

(B) E^1/2

(D) E

Solution:-

de Broglie wavelength associated with a proton is given by

$\displaystyle {{\lambda }_{1}}=\frac{h}{\sqrt{2{{m}_{p}}E}}$ …..(1)

And for photon

$\displaystyle E=h\nu =\frac{hc}{{{\lambda }_{2}}}$

$\displaystyle \Rightarrow {{\lambda }_{2}}=\frac{hc}{E}$ …..(2)

Dividing (1) by (2), we get

$\displaystyle \frac{{{\lambda }_{1}}}{{{\lambda }_{2}}}=\frac{h}{\sqrt{2{{m}_{p}}E}}\times \frac{E}{hc}=\frac{1}{c}\sqrt{\frac{E}{2{{m}_{p}}}}$

$\displaystyle \Rightarrow \frac{{{\lambda }_{1}}}{{{\lambda }_{2}}}\propto {{E}^{1/2}}$

Hence option (B) is correct.

A proton and photon both have same energy

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