Photon having energy E has the same de Broglie wavelength as that of proton of energy `E_1` . The correct relation between `E and E_1 ` is

Options

A. `E_1 prop sqrtE`
B. `E_1 prop E^2`
C. `E prop sqrtE_1`
D. `E_1 prop 1/E`

Explanation

To solve the problem, we need to find the relationship between the energy of a photon (E) and the energy of a proton (E₁) given that they have the same de Broglie wavelength.

### Step-by-Step Solution:

1. **Understanding the de Broglie Wavelength:**
The de Broglie wavelength (λ) for a photon is given by the formula:
\[ \lambda = \frac{hc}{E} \]
where \( h \) is Planck's constant and \( c \) is the speed of light.

2. **De Broglie Wavelength for a Proton:**
The de Broglie wavelength for a proton can be expressed as:
\[ \lambda_1 = \frac{h}{\sqrt{2mE_1}} \]
where \( m \) is the mass of the proton and \( E_1 \) is the energy of the proton.

3. **Setting the Wavelengths Equal:**
Since the problem states that the de Broglie wavelengths of the photon and the proton are the same, we can set the two equations equal to each other:
\[ \frac{hc}{E} = \frac{h}{\sqrt{2mE_1}} \]

4. **Canceling Planck's Constant:**
We can cancel \( h \) from both sides of the equation:
\[ \frac{c}{E} = \frac{1}{\sqrt{2mE_1}} \]

5. **Cross-Multiplying:**
Cross-multiplying gives us:
\[ c \sqrt{2mE_1} = E \]

6. **Squaring Both Sides:**
To eliminate the square root, we square both sides:
\[ c^2 \cdot 2mE_1 = E^2 \]

7. **Rearranging the Equation:**
Rearranging this equation gives us:
\[ E_1 = \frac{E^2}{2mc^2} \]

8. **Identifying the Relationship:**
From this equation, we can see that \( E_1 \) is proportional to the square of \( E \):
\[ E_1 \propto E^2 \]

### Conclusion:
Thus, the correct relationship between the energies is:
\[ E_1 \propto E^2 \]