A proton has kinetic energy E = 100 keV which is equal to that of a photon. The wavelength of photon is `lamda_(2)` and that of proton is `lamda_(1)`. The ratio of `lamda_(2)//lamda_(1)` is proportional to

Options

A. `E^2`
B. `E^(1//2)`
C. `E^(-1)`
D. `E^(-1//2)`

Answer

D

Explanation

For photon , `E = (hc)/(lambda_2) or lambda_(2) = (hc)/(E)......... (i)`
For proton kinetic energy `K= (1)/(2) m_p v_(p)^(2)`
or `2m_p K- p^2`
or `2m_p K = ((h)/(lambda_1))^2` , by de Broglie equation
K.E = energy E.
or `lambda_1 = (h)/(sqrt(2 m_p K)) = (h)/(sqrt(2m_p E))`
From (i) and (ii), or `(lambda_2)/(lambda) = (hc)/(E) xx sqrt(2m_pE)/h`
or `(lambda_2)/(lambda_1) = (c xxsqrt(2m_P))/(sqrt(E))= c sqrt(2m_p) xx E^(-1//2) or (lambda_2)/(lambda_1)prop E^(-1//2)`