An electron (mass m) with an initial velocity `v=v_(0)hat(i)(v_(0)gt0)` is in an electric field `E=-E_(0)hat(l)(E_(0)="constant"gt0)`. Its de-Broglie wavelength at time t is given by

Options

A. `(lamda_(0))/((1-(eE_(0)t)/(mv_(0))))`
B. `lamda_(0)(1+(eE_(0)t)/(mv_(0)))`
C. `lamda_(0)`
D. `lamda_(0)t`

Explanation

To solve the problem of finding the de-Broglie wavelength of an electron in an electric field after a time \( t \), we can follow these steps:

### Step 1: Identify the Given Information
- Initial velocity of the electron:
\[ \mathbf{v} = v_0 \hat{i} \]
- Electric field:
\[ \mathbf{E} = -E_0 \hat{i} \]
- Mass of the electron:
\[ m \]
- Charge of the electron:
\[ q = -e \]

### Step 2: Calculate the Force Acting on the Electron
The force \( \mathbf{F} \) acting on the electron due to the electric field is given by:
\[ \mathbf{F} = q \mathbf{E} \]
Substituting the values:
\[ \mathbf{F} = (-e)(-E_0 \hat{i}) = eE_0 \hat{i} \]

### Step 3: Calculate the Acceleration of the Electron
Using Newton's second law, \( \mathbf{F} = m \mathbf{a} \), we can find the acceleration \( \mathbf{a} \):
\[ \mathbf{a} = \frac{\mathbf{F}}{m} = \frac{eE_0}{m} \hat{i} \]

### Step 4: Calculate the Final Velocity After Time \( t \)
Using the first equation of motion:
\[ \mathbf{v} = \mathbf{u} + \mathbf{a} t \]
where \( \mathbf{u} = v_0 \hat{i} \):
\[ \mathbf{v} = v_0 \hat{i} + \left(\frac{eE_0}{m} \hat{i}\right) t \]
This simplifies to:
\[ \mathbf{v} = \left(v_0 + \frac{eE_0}{m} t\right) \hat{i} \]

### Step 5: Calculate the Momentum of the Electron
The momentum \( p \) of the electron is given by:
\[ p = m \mathbf{v} \]
Substituting the expression for \( \mathbf{v} \):
\[ p = m \left(v_0 + \frac{eE_0}{m} t\right) \hat{i} = m v_0 \hat{i} + eE_0 t \hat{i} \]

### Step 6: Calculate the de-Broglie Wavelength
The de-Broglie wavelength \( \lambda \) is given by:
\[ \lambda = \frac{h}{p} \]
Substituting the expression for \( p \):
\[ \lambda = \frac{h}{m v_0 + eE_0 t} \]

### Step 7: Express the Wavelength in Terms of Initial Wavelength
If we denote the initial de-Broglie wavelength \( \lambda_0 \) as:
\[ \lambda_0 = \frac{h}{m v_0} \]
Then we can express \( \lambda \) as:
\[ \lambda = \frac{\lambda_0}{1 + \frac{eE_0}{m v_0} t} \]

### Final Answer
The de-Broglie wavelength of the electron at time \( t \) is:
\[ \lambda = \frac{\lambda_0}{1 + \frac{eE_0}{m v_0} t} \]

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