A proton has kinetic energy which is equal to that of a photon. The wavelength of the photon is and that of the proton is . The ratio is proportional to [2004]

Question

A proton has kinetic energy $E = 100 keV$ which is equal to that of a photon. The wavelength of the photon is $λ_{2}$ and that of the proton is $λ_{1}$ . The ratio $\frac{λ_{2}}{λ_{1}}$ is proportional to [2004]

Smart Achievers · Accepted Answer

(4)

For photon, $λ_{2} = \frac{h c}{E}$ $\dots (i)$

For proton, $λ_{1} = \frac{h}{p} = \frac{h}{\sqrt{2 m E}}$ $∵$ $p = \sqrt{2 m E}$ $\dots (i i)$

$∴$ $\frac{λ_{2}}{λ_{1}} = \frac{h c}{E \times \frac{h}{\sqrt{2 m E}}} \propto E^{- 1 / 2}$

Solution

Q.
A proton has kinetic energy $E = 100 keV$ which is equal to that of a photon. The wavelength of the photon is $λ_{2}$ and that of the proton is $λ_{1}$ . The ratio $\frac{λ_{2}}{λ_{1}}$ is proportional to [2004]

1 $E^{2}$

2 $E^{1 / 2}$

3 $E^{- 1}$

4 $E^{- 1 / 2}$

Ans.
(4)

For photon,   $λ_{2} = \frac{h c}{E}$    $\dots (i)$

For proton,   $λ_{1} = \frac{h}{p} = \frac{h}{\sqrt{2 m E}}$    $∵$    $p = \sqrt{2 m E}$    $\dots (i i)$

$∴$    $\frac{λ_{2}}{λ_{1}} = \frac{h c}{E \times \frac{h}{\sqrt{2 m E}}} \propto E^{- 1 / 2}$

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Solution

Q. A proton has kinetic energy E=100 keV which is equal to that of a photon. The wavelength of the photon is λ2 and that of the proton is λ1. The ratio λ2λ1 is proportional to [2004]

1 E2

2 E1/2

3 E-1

4 E-1/2

Ans. (4) For photon, λ2=hcE ⋯(i) For proton, λ1=hp=h2mE ∵ p=2mE ⋯(ii) ∴ λ2λ1=hcE×h2mE∝E-1/2