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

To solve the problem, we need to find the ratio of the wavelengths of a photon and a proton given that the kinetic energy of the proton is equal to the energy of the photon.

### Step-by-Step Solution:

1. **Identify the Energy of the Photon:**
The energy of the photon \( E \) is given as \( 100 \, \text{keV} \). We can express this in joules:
\[ E = 100 \, \text{keV} = 100 \times 1.6 \times 10^{-19} \, \text{J} = 1.6 \times 10^{-14} \, \text{J} \]

2. **Calculate the Wavelength of the Photon (\( \lambda_2 \)):**
The wavelength of the photon can be calculated using the formula:
\[ \lambda_2 = \frac{hc}{E} \]
where \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{Js} \)) and \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)).
Substituting the values:
\[ \lambda_2 = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{1.6 \times 10^{-14}} \approx 1.24 \times 10^{-6} \, \text{m} \]

3. **Calculate the Wavelength of the Proton (\( \lambda_1 \)):**
The momentum \( p \) of the proton can be related to its kinetic energy \( E \) using:
\[ E = \frac{p^2}{2m} \]
Rearranging gives:
\[ p = \sqrt{2mE} \]
The wavelength of the proton is given by:
\[ \lambda_1 = \frac{h}{p} = \frac{h}{\sqrt{2mE}} \]
Substituting the values:
\[ \lambda_1 = \frac{h}{\sqrt{2m \cdot 1.6 \times 10^{-14}}} \]
Here, the mass of the proton \( m \approx 1.67 \times 10^{-27} \, \text{kg} \).

4. **Calculate the Ratio of Wavelengths (\( \frac{\lambda_2}{\lambda_1} \)):**
Now we can find the ratio:
\[ \frac{\lambda_2}{\lambda_1} = \frac{\frac{hc}{E}}{\frac{h}{\sqrt{2mE}}} = \frac{hc \sqrt{2mE}}{E} \]
Simplifying gives:
\[ \frac{\lambda_2}{\lambda_1} = \frac{hc \sqrt{2m}}{E^{1/2}} \]
Since \( h \) and \( c \) are constants, we can express the ratio as:
\[ \frac{\lambda_2}{\lambda_1} \propto \sqrt{2m} E^{-1/2} \]

5. **Final Result:**
Therefore, the ratio \( \frac{\lambda_2}{\lambda_1} \) is proportional to \( E^{-1/2} \).