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.

### Step 1: Determine the de Broglie wavelength of the proton (\( \lambda_1 \))

The de Broglie wavelength of a particle is given by the formula:

\[ \lambda_1 = \frac{h}{p} \]

where \( h \) is Planck's constant and \( p \) is the momentum of the proton.

### Step 2: Express the momentum of the proton in terms of kinetic energy

The momentum \( p \) of the proton can be expressed in terms of its kinetic energy \( E \):

\[ p = \sqrt{2mE} \]

where \( m \) is the mass of the proton and \( E \) is its kinetic energy. Since the energy of the photon is equal to the kinetic energy of the proton, we can use \( E \) for both.

### Step 3: Substitute the momentum into the de Broglie wavelength formula

Now substituting the expression for momentum into the de Broglie wavelength formula:

\[ \lambda_1 = \frac{h}{\sqrt{2mE}} \]

### Step 4: Determine the wavelength of the photon (\( \lambda_2 \))

The wavelength of a photon is related to its energy by the equation:

\[ \lambda_2 = \frac{hc}{E} \]

where \( c \) is the speed of light.

### Step 5: Find the ratio \( \frac{\lambda_1}{\lambda_2} \)

Now we can find the ratio of the two wavelengths:

\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{h}{\sqrt{2mE}}}{\frac{hc}{E}} = \frac{h \cdot E}{\sqrt{2mE} \cdot hc} \]

This simplifies to:

\[ \frac{\lambda_1}{\lambda_2} = \frac{E}{\sqrt{2mE} \cdot c} \]

### Step 6: Simplify the expression

We can further simplify this expression:

\[ \frac{\lambda_1}{\lambda_2} = \frac{E^{1/2}}{\sqrt{2m} \cdot c} \]

### Conclusion: Determine the proportionality

From the final expression, we can see that the ratio \( \frac{\lambda_1}{\lambda_2} \) is proportional to \( E^{1/2} \).

Thus, the correct answer is:

**(b) \( E^{1/2} \)**