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 ) =- E_(0) hat (j)` `(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. `lamda_(0)`
B. `lamda_(0)sqrt(1+(e^(2)E_(0)^(2)t^(2))/(m^(2)v_(0)^(2)))`
C. `lamda_(0)/(sqrt(1+(e^(2)E_(0)^(2)t^(2))/(m^(2)v_(0)^(2))))`
D. `lamda_(0)/((1+(e^(2)E_(0)^(2)t^(2))/(m^(2)v_(0)^(2))))`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Initial Conditions
We have an electron with mass \( m \) and initial velocity \( \vec{v} = v_0 \hat{i} \) (where \( v_0 > 0 \)). The electron enters an electric field \( \vec{E} = -E_0 \hat{j} \) (where \( E_0 \) is a constant and \( E_0 > 0 \)) at time \( t = 0 \).

### Step 2: Calculate the Initial de Broglie Wavelength
The de Broglie wavelength \( \lambda_0 \) of the electron is given by the formula:
\[ \lambda_0 = \frac{h}{mv_0} \]
where \( h \) is Planck's constant.

### Step 3: Determine the Force and Acceleration
The force \( \vec{F} \) on the electron due to the electric field is given by:
\[ \vec{F} = q \vec{E} \]
For an electron, the charge \( q = -e \) (where \( e \) is the elementary charge). Thus,
\[ \vec{F} = -e (-E_0 \hat{j}) = eE_0 \hat{j} \]
The acceleration \( \vec{a} \) can be calculated using Newton's second law:
\[ \vec{a} = \frac{\vec{F}}{m} = \frac{eE_0}{m} \hat{j} \]

### Step 4: Calculate the Velocity After Time \( t \)
The velocity in the \( \hat{j} \) direction after time \( t \) can be calculated using the kinematic equation:
\[ v_y = u_y + a_y t \]
Since the initial velocity in the \( \hat{j} \) direction \( u_y = 0 \):
\[ v_y = 0 + \left(\frac{eE_0}{m}\right)t = \frac{eE_0 t}{m} \]
The velocity in the \( \hat{i} \) direction remains unchanged:
\[ v_x = v_0 \]

### Step 5: Calculate the Resultant Velocity
The resultant velocity \( \vec{v} \) can be found using the Pythagorean theorem:
\[ v = \sqrt{v_x^2 + v_y^2} = \sqrt{v_0^2 + \left(\frac{eE_0 t}{m}\right)^2} \]

### Step 6: Calculate the de Broglie Wavelength After Time \( t \)
The de Broglie wavelength at time \( t \) is given by:
\[ \lambda_t = \frac{h}{mv} \]
Substituting the expression for \( v \):
\[ \lambda_t = \frac{h}{m \sqrt{v_0^2 + \left(\frac{eE_0 t}{m}\right)^2}} \]

### Step 7: Relate \( \lambda_t \) to \( \lambda_0 \)
We can express \( \lambda_t \) in terms of \( \lambda_0 \):
\[ \lambda_t = \lambda_0 \cdot \frac{1}{\sqrt{1 + \frac{e^2 E_0^2 t^2}{m^2 v_0^2}}} \]

### Final Result
Thus, the de Broglie wavelength of the electron at time \( t \) is:
\[ \lambda_t = \lambda_0 \cdot \frac{1}{\sqrt{1 + \frac{e^2 E_0^2 t^2}{m^2 v_0^2}}} \]

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