An electron of mass 'm' is moving with initial velocity `v_0hati` in an electric field `vecE=E_0hati` Which of the following is correct de Broglie wavelength at a given time t `(lamda_0` is initial de Broglie wavelength ).

Options

A. `lamda_0/((1+(eE_0)/(mlamda_0)t))`
B. `lamda_0/((1+(eE_0)/(mv_0)t))`
C. `h/((1+(mv_0_0)/(Ev_0)t))`
D. `h/((1+(eE_0)/(mv_0)t))`

Explanation

To find the correct de Broglie wavelength of an electron moving in an electric field, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Parameters:**
- Mass of the electron: \( m \)
- Initial velocity of the electron: \( \vec{v_0} = v_0 \hat{i} \)
- Electric field: \( \vec{E} = E_0 \hat{i} \)
- Charge of the electron: \( e \) (approximately \( 1.6 \times 10^{-19} \) C)

2. **Calculate the Force on the Electron:**
- The force \( \vec{F} \) acting on the electron due to the electric field is given by:
\[ \vec{F} = e \vec{E} = e E_0 \hat{i} \]

3. **Determine the Acceleration of the Electron:**
- Using Newton's second law, \( \vec{F} = m \vec{a} \):
\[ m \vec{a} = e E_0 \hat{i} \implies \vec{a} = \frac{e E_0}{m} \hat{i} \]

4. **Use the Kinematic Equation to Find the Velocity at Time \( t \):**
- The final velocity \( \vec{v} \) after time \( t \) can be calculated using the equation:
\[ \vec{v} = \vec{u} + \vec{a} t \]
- Here, \( \vec{u} = v_0 \hat{i} \) and \( \vec{a} = \frac{e E_0}{m} \hat{i} \):
\[ \vec{v} = v_0 \hat{i} + \left(\frac{e E_0}{m}\right) t \hat{i} = \left(v_0 + \frac{e E_0}{m} t\right) \hat{i} \]

5. **Calculate the De Broglie Wavelength:**
- The de Broglie wavelength \( \lambda \) is given by:
\[ \lambda = \frac{h}{p} \]
- Where \( p \) is the momentum, \( p = mv \):
\[ p = m \left(v_0 + \frac{e E_0}{m} t\right) = mv_0 + e E_0 t \]
- Therefore, the de Broglie wavelength becomes:
\[ \lambda = \frac{h}{mv_0 + e E_0 t} \]

6. **Express the Wavelength in Terms of Initial Wavelength \( \lambda_0 \):**
- The initial de Broglie wavelength \( \lambda_0 \) is:
\[ \lambda_0 = \frac{h}{mv_0} \]
- Thus, we can express \( \lambda \) as:
\[ \lambda = \lambda_0 \cdot \frac{1}{1 + \frac{e E_0 t}{mv_0}} \]

7. **Final Expression for the Wavelength:**
- Rearranging gives:
\[ \lambda = \lambda_0 \left(1 + \frac{e E_0 t}{mv_0}\right) \]

### Conclusion:
The correct de Broglie wavelength at a given time \( t \) is:
\[ \lambda = \lambda_0 \left(1 + \frac{e E_0 t}{mv_0}\right) \]