An electron (mass m) with an initial velocity `vecv=v_(0)hati` is in an electric field `vecE=E_(0)hatj`. If `lambda_(0)h//mv_(0)`. It's de-broglie wavelength at time t is given by

Options

A. `lambda_(0)`
B. `lambda_(0)sqrt(1+(e^(2)E_(0)^(2)t^(2))/(m^(2)v_(0)^(2)))`
C. `(lambda_(0))/(1+(e^(2)E_(0)^(2)t^(2))/(m^(2)v_(0)^(2)))`
D. `(lambda_(0))/(sqrt(1+((e^(2)E_(0)^(2)t^(2))/(m^(2)v_(0)^(2))))`

Explanation

To solve the problem, we will follow these steps:

1. **Identify the Given Information**:
- Mass of the electron: \( m \)
- Initial velocity: \( \vec{v} = v_0 \hat{i} \)
- Electric field: \( \vec{E} = E_0 \hat{j} \)
- Initial de Broglie wavelength: \( \lambda_0 = \frac{h}{mv_0} \)

2. **Calculate the Force on the Electron**:
The force \( \vec{F} \) acting on the electron due to the electric field is given by:
\[ \vec{F} = q \vec{E} \]
where \( q \) is the charge of the electron (denoted as \( -e \)). Thus,
\[ \vec{F} = -e E_0 \hat{j} \]

3. **Determine the Acceleration**:
Using Newton's second law, the acceleration \( \vec{a} \) can be calculated as:
\[ \vec{a} = \frac{\vec{F}}{m} = \frac{-e E_0 \hat{j}}{m} \]

4. **Calculate the Final Velocity**:
The final velocity \( \vec{v}_2 \) at time \( t \) can be found using the equation:
\[ \vec{v}_2 = \vec{v}_1 + \vec{a} t \]
where \( \vec{v}_1 = v_0 \hat{i} \). Therefore,
\[ \vec{v}_2 = v_0 \hat{i} + \left(\frac{-e E_0}{m} \hat{j}\right) t = v_0 \hat{i} - \frac{e E_0 t}{m} \hat{j} \]

5. **Calculate the Magnitude of the Final Velocity**:
The magnitude of the final velocity \( |\vec{v}_2| \) is:
\[ |\vec{v}_2| = \sqrt{(v_0)^2 + \left(-\frac{e E_0 t}{m}\right)^2} \]
Simplifying this gives:
\[ |\vec{v}_2| = \sqrt{v_0^2 + \frac{(e E_0 t)^2}{m^2}} \]

6. **Determine the de Broglie Wavelength at Time \( t \)**:
The de Broglie wavelength \( \lambda \) is given by:
\[ \lambda = \frac{h}{m |\vec{v}_2|} \]
Substituting the expression for \( |\vec{v}_2| \):
\[ \lambda = \frac{h}{m \sqrt{v_0^2 + \frac{(e E_0 t)^2}{m^2}}} \]

7. **Express in Terms of \( \lambda_0 \)**:
Recall that \( \lambda_0 = \frac{h}{mv_0} \). We can rewrite \( \lambda \) as:
\[ \lambda = \lambda_0 \cdot \frac{1}{\sqrt{1 + \frac{(e E_0 t)^2}{m^2 v_0^2}}} \]

Thus, the final expression for the de Broglie wavelength at time \( t \) is:
\[ \lambda = \lambda_0 \cdot \frac{1}{\sqrt{1 + \frac{(e E_0 t)^2}{m^2 v_0^2}}} \]