According to de-Broglie hypothesis, the wavelength associated with moving electron of mass 'm' is `'lambda_(e)'`. Using mass energy relation and Planck's quantum theory, the wavelength associated with photon is `'lambda_(p)'`. If the energy (E) of electron and photonm is same, then relation between `lambda_e` and `'lambda_(p)'` is

Options

A. `lambda_(p) prop lambda_(e)`
B. `lambda_(p) prop lambda_(e)^(2)`
C. `lambda_(p) prop sqrt(lambda_(p))`
D. `lambda_(p) prop 1/(lambda_(e))`

Explanation

To find the relationship between the wavelengths associated with a moving electron (\( \lambda_e \)) and a photon (\( \lambda_p \)) when both have the same energy, we can follow these steps:

### Step-by-Step Solution:

1. **Energy of the Photon**:
The energy of a photon can be expressed using Planck's equation:
\[ E_p = \frac{hc}{\lambda_p} \]
where \( h \) is Planck's constant and \( c \) is the speed of light.

2. **Energy of the Electron**:
The energy of a moving electron can be expressed as:
\[ E_e = mc^2 \]
where \( m \) is the mass of the electron.

3. **Setting Energies Equal**:
Since the problem states that the energy of the electron and the photon is the same, we can set the two energy equations equal to each other:
\[ E_e = E_p \]
This gives us:
\[ mc^2 = \frac{hc}{\lambda_p} \]

4. **Rearranging for \( \lambda_p \)**:
Rearranging the equation to solve for \( \lambda_p \):
\[ \lambda_p = \frac{hc}{E_e} \]

5. **Using de Broglie's Hypothesis**:
According to de Broglie's hypothesis, the wavelength associated with a moving electron is given by:
\[ \lambda_e = \frac{h}{p} \]
where \( p \) is the momentum of the electron. For a non-relativistic electron, \( p = mv \), but we can also express energy in terms of momentum:
\[ E_e = pc \implies p = \frac{E_e}{c} \]
Substituting this into the de Broglie wavelength equation gives:
\[ \lambda_e = \frac{h}{p} = \frac{hc}{E_e} \]

6. **Relating \( \lambda_e \) and \( \lambda_p \)**:
From the equations derived for \( \lambda_p \) and \( \lambda_e \):
\[ \lambda_p = \frac{hc}{E_e} \quad \text{and} \quad \lambda_e = \frac{hc}{E_e} \]
This shows that:
\[ \lambda_p = \lambda_e \]

7. **Conclusion**:
Therefore, the relationship between the wavelengths is:
\[ \lambda_p \propto \lambda_e \]

### Final Answer:
The correct relation is \( \lambda_p \) is directly proportional to \( \lambda_e \).