The energy of a photon is equal to the kinetic energy of a
proton. The energy of the photon is E. Let `lambda_1`be the de-Broglie wavelength of the
proton and `lambda_2` be the wavelength of the photon. The ratio `(lambda_1)/(lambda_2)` is proportional to
(a)`E^0` (b) `E^(1//2)` (c ) `E^(-1)` (d)`E^(-2)`

Explanation

To solve the problem, we need to find the ratio of the de Broglie wavelength of a proton (\( \lambda_1 \)) to the wavelength of a photon (\( \lambda_2 \)) when the energy of the photon is equal to the kinetic energy of the proton. Let's break it down step by step.

### Step 1: Understand the given information
The energy of the photon (\( E \)) is equal to the kinetic energy of the proton. Therefore, we can write:
\[ KE = E \]

### Step 2: Write the formula for the de Broglie wavelength of the proton
The de Broglie wavelength (\( \lambda_1 \)) of a particle is given by the formula:
\[ \lambda_1 = \frac{h}{p} \]
where \( p \) is the momentum of the proton. The momentum can be expressed in terms of kinetic energy:
\[ p = \sqrt{2m \cdot KE} = \sqrt{2m \cdot E} \]
Thus, we can rewrite the de Broglie wavelength as:
\[ \lambda_1 = \frac{h}{\sqrt{2mE}} \]

### Step 3: Write the formula for the wavelength of the photon
The wavelength (\( \lambda_2 \)) of a photon is given by:
\[ \lambda_2 = \frac{hc}{E} \]

### Step 4: Find the ratio \( \frac{\lambda_1}{\lambda_2} \)
Now, we need to find the ratio of \( \lambda_1 \) to \( \lambda_2 \):
\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{h}{\sqrt{2mE}}}{\frac{hc}{E}} = \frac{h \cdot E}{hc \cdot \sqrt{2mE}} = \frac{E}{c \cdot \sqrt{2mE}} \]

### Step 5: Simplify the ratio
We can simplify this expression:
\[ \frac{\lambda_1}{\lambda_2} = \frac{E}{c \cdot \sqrt{2mE}} = \frac{1}{c \sqrt{2m}} \cdot E^{1/2} \]
This shows that the ratio \( \frac{\lambda_1}{\lambda_2} \) is proportional to \( E^{1/2} \).

### Conclusion
Thus, the correct answer is:
\[ \frac{\lambda_1}{\lambda_2} \propto E^{1/2} \]
The correct option is (b) \( E^{1/2} \).