An electron of mass `m` with an initial velocity
`vec(v) = v_(0) hat`(i) `(v_(0) gt 0)` enters an electric field
`vec(E ) = v_(0) hat (i) `(E_(0) = constant gt 0)` at `t = 0` . If `lambda_(0)` is its de - Broglie wavelength initially, then its de - Broglie wavelength at time `t` is

Options

A. `lambda_(0)`
B. `(lambda_(0))/(( 1 + (e E_(0))/(mv_(0)) t)`
C. `lambda_(0) t`
D. `lambda_(0) (1 + ( e E_(0))/(mv_(0)) t)`

Explanation

To solve the problem, we need to find the de Broglie wavelength of an electron at time \( t \) after it has entered a constant electric field. Here's a step-by-step breakdown of the solution:

### Step 1: Initial De Broglie Wavelength
The initial de Broglie wavelength \( \lambda_0 \) of the electron can be calculated using the formula:
\[ \lambda_0 = \frac{h}{p_0} \]
where \( p_0 \) is the initial momentum of the electron, given by:
\[ p_0 = mv_0 \]
Thus, we can express the initial wavelength as:
\[ \lambda_0 = \frac{h}{mv_0} \]

### Step 2: Force on the Electron
When the electron enters the electric field \( \vec{E} \), it experiences a force \( \vec{F} \) given by:
\[ \vec{F} = q\vec{E} \]
For an electron, the charge \( q \) is \( -e \) (where \( e \) is the elementary charge). Therefore, the force acting on the electron is:
\[ \vec{F} = -e\vec{E} = -eE_0 \hat{i} \]

### Step 3: Acceleration of the Electron
The acceleration \( a \) of the electron can be calculated using Newton's second law:
\[ F = ma \implies a = \frac{F}{m} = \frac{-eE_0}{m} \]

### Step 4: Velocity of the Electron at Time \( t \)
The velocity \( v(t) \) of the electron at time \( t \) can be found using the equation of motion:
\[ v(t) = v_0 + at = v_0 - \frac{eE_0}{m}t \]

### Step 5: Final Momentum
The final momentum \( p \) of the electron at time \( t \) is:
\[ p = mv(t) = m\left(v_0 - \frac{eE_0}{m}t\right) = mv_0 - eE_0t \]

### Step 6: Final De Broglie Wavelength
The final de Broglie wavelength \( \lambda(t) \) at time \( t \) can be calculated using the updated momentum:
\[ \lambda(t) = \frac{h}{p} = \frac{h}{mv_0 - eE_0t} \]

### Step 7: Relating Final Wavelength to Initial Wavelength
Now we can express the final wavelength in terms of the initial wavelength:
\[ \lambda(t) = \frac{h}{mv_0 - eE_0t} = \frac{h}{mv_0} \cdot \frac{1}{1 - \frac{eE_0t}{mv_0}} = \lambda_0 \cdot \frac{1}{1 - \frac{eE_0t}{mv_0}} \]

### Final Answer
Thus, the de Broglie wavelength of the electron at time \( t \) is:
\[ \lambda(t) = \lambda_0 \cdot \frac{1}{1 - \frac{eE_0t}{mv_0}} \]

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