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A proton has kinetic energy E = 100 keV ...

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

A

`E^(2)`

B

`E^(1//2)`

C

`E^(-1)`

D

`E^(-1//2)`

Text Solution

Verified by Experts

The correct Answer is:
D
For photon, `E=(hc)/lambda_(2)" or "lambda_(2)=(hc)/E" …(i)"`
For proton kinetic energy `K=1/2m_(p)v_(p)^(2)`
or `2m_(p)K=m_(p)^(2)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(2m_(p)K)=h/sqrt(2m_(p)E)" ...(ii)"`
From (i) and (ii), `lambda_(2)/lambda_(1)=(hc)/E times sqrt(2m_(p)E)/h`
or `lambda_(2)/lambda_(1)=(c times sqrt(2m_(p)))/sqrt(E)=csqrt(2m_(p)) times E^(-1//2)" or " lambda_(2)/lambda_(1) propto E^(-1//2)`
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