An electron (mass m ) with initival velocity `vecv = v_(0) hati + v_(0) hatj` is the an electric field `vecE = - E_(0)hatk` . It `lambda_(0)` is initial de - Broglie wavelength of electron, its de-Broglie wave length at time t is given by :

Options

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

Answer

To solve the problem of finding the de Broglie wavelength of an electron in an electric field, we will follow these steps:

### Step 1: Understand the initial conditions
The electron has an initial velocity given by:
\[ \vec{v}_0 = v_0 \hat{i} + v_0 \hat{j} \]
This means that the electron is moving in the XY plane with equal components in both the x and y directions.

### Step 2: Determine the force acting on the electron
The electron is subjected to an electric field given by:
\[ \vec{E} = -E_0 \hat{k} \]
The force 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, the force becomes:
\[ \vec{F} = -e (-E_0 \hat{k}) = e E_0 \hat{k} \]

### Step 3: Calculate the acceleration of the electron
Using Newton's second law, the acceleration \(\vec{a}\) can be calculated as:
\[ \vec{a} = \frac{\vec{F}}{m} = \frac{e E_0}{m} \hat{k} \]
where \(m\) is the mass of the electron.

### Step 4: Determine the velocity of the electron at time \(t\)
The velocity of the electron at time \(t\) can be expressed as:
\[ \vec{v}(t) = \vec{v}_0 + \vec{a} t \]
Substituting the values:
\[ \vec{v}(t) = (v_0 \hat{i} + v_0 \hat{j}) + \left(\frac{e E_0}{m} \hat{k}\right) t \]
Thus, the velocity vector becomes:
\[ \vec{v}(t) = v_0 \hat{i} + v_0 \hat{j} + \frac{e E_0 t}{m} \hat{k} \]

### Step 5: Calculate the magnitude of the velocity
The magnitude of the velocity \(|\vec{v}(t)|\) is given by:
\[ |\vec{v}(t)| = \sqrt{(v_0)^2 + (v_0)^2 + \left(\frac{e E_0 t}{m}\right)^2} \]
This simplifies to:
\[ |\vec{v}(t)| = \sqrt{2 v_0^2 + \left(\frac{e E_0 t}{m}\right)^2} \]

### Step 6: Relate the de Broglie wavelength to the velocity
The de Broglie wavelength \(\lambda\) is given by:
\[ \lambda = \frac{h}{mv} \]
where \(h\) is Planck's constant. Thus, the de Broglie wavelength at time \(t\) becomes:
\[ \lambda(t) = \frac{h}{m |\vec{v}(t)|} \]

### Step 7: Substitute the magnitude of velocity into the wavelength equation
Substituting the expression for \(|\vec{v}(t)|\):
\[ \lambda(t) = \frac{h}{m \sqrt{2 v_0^2 + \left(\frac{e E_0 t}{m}\right)^2}} \]

### Step 8: Express the new wavelength in terms of the initial wavelength
The initial de Broglie wavelength \(\lambda_0\) is:
\[ \lambda_0 = \frac{h}{mv_0} \]
Thus, we can express \(\lambda(t)\) in terms of \(\lambda_0\):
\[ \lambda(t) = \frac{\lambda_0}{\sqrt{1 + \frac{e^2 E_0^2 t^2}{2 m^2 v_0^2}}} \]

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