An electron is moving initially with velocity `v_o hati+v_ohatj` in uniform electric field `vec E=-E_0 hatk`. If initial wavelength of electron is `lambda_0` and mass of electron is m. Find wavelength of electron as a function of time.

Options

A. Option1 `lambda_0/sqrt(1+(e^2E^2)/(2m^2v_0^2)t^2)`
B. Option2 `lambda_0/sqrt(2+(e^2E^2)/(m^2v_0^2)t^2)`
C. Option3 `(lambda_0mv_0)/(eE_0t)`
D. Option4 `(2lambda_0mv_0)/(eE_0t)`

Explanation

To solve the problem, we need to determine the wavelength of the electron as a function of time when it is moving in a uniform electric field. We will use the de Broglie wavelength formula and the kinematics of the electron under the influence of the electric field.

### Step-by-Step Solution:

1. **Identify Initial Conditions**:
- The initial velocity of the electron is given as \( \vec{v_0} = v_0 \hat{i} + v_0 \hat{j} \).
- The electric field is \( \vec{E} = -E_0 \hat{k} \).
- The mass of the electron is \( m \).
- The initial de Broglie wavelength is \( \lambda_0 \).

2. **Calculate the Initial Wavelength**:
- The de Broglie wavelength \( \lambda \) is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( p \) is the momentum of the electron. The initial momentum \( p_0 \) can be calculated as:
\[ p_0 = m |\vec{v_0}| = m \sqrt{(v_0)^2 + (v_0)^2} = m \sqrt{2}v_0 \]
- Therefore, the initial wavelength \( \lambda_0 \) is:
\[ \lambda_0 = \frac{h}{m \sqrt{2} v_0} \]

3. **Determine the Acceleration of the Electron**:
- The force on the electron due to the electric field is:
\[ F = qE = -eE_0 \]
- The acceleration \( a \) of the electron is given by:
\[ a = \frac{F}{m} = \frac{-eE_0}{m} \]

4. **Calculate the Velocity as a Function of Time**:
- The velocity of the electron in the \( \hat{k} \) direction (z-direction) changes due to the acceleration. Initially, the velocity in the z-direction is zero.
- The velocity in the z-direction at time \( t \) is:
\[ v_z(t) = 0 + a t = \frac{-eE_0}{m} t \]
- Therefore, the total velocity vector becomes:
\[ \vec{v}(t) = v_0 \hat{i} + v_0 \hat{j} + \left(\frac{-eE_0}{m} t\right) \hat{k} \]

5. **Calculate the Magnitude of the Velocity**:
- The magnitude of the total velocity \( v(t) \) is:
\[ v(t) = \sqrt{(v_0)^2 + (v_0)^2 + \left(\frac{-eE_0}{m} t\right)^2} = \sqrt{2v_0^2 + \left(\frac{eE_0}{m} t\right)^2} \]

6. **Determine the Wavelength as a Function of Time**:
- The de Broglie wavelength at time \( t \) is:
\[ \lambda(t) = \frac{h}{mv(t)} \]
- Substituting for \( v(t) \):
\[ \lambda(t) = \frac{h}{m \sqrt{2v_0^2 + \left(\frac{eE_0}{m} t\right)^2}} \]

7. **Express Wavelength in Terms of \( \lambda_0 \)**:
- We can relate this to the initial wavelength \( \lambda_0 \):
\[ \lambda(t) = \lambda_0 \cdot \frac{1}{\sqrt{1 + \frac{e^2E_0^2 t^2}{2m^2v_0^2}}} \]

### Final Result:
The wavelength of the electron as a function of time \( t \) is:
\[ \lambda(t) = \lambda_0 \cdot \frac{1}{\sqrt{1 + \frac{e^2E_0^2 t^2}{2m^2v_0^2}}} \]