An electron `(-|e|, m)` is released in Electric field E from rest. rate of change of de-Broglie wavelength with time will be.

Options

A. `-h/(2|e|)`
B. `-h/(2|e|t)`
C. `-h/(|e|Et^2)`
D. `-(2ht^2)/(|e|E)`

Answer

To solve the problem of finding the rate of change of the de Broglie wavelength of an electron released in an electric field \( E \), we will follow these steps:

### Step 1: Determine the force on the electron
The force \( F \) acting on the electron due to the electric field \( E \) is given by:
\[ F = qE \]
where \( q = -|e| \) is the charge of the electron. Therefore, the force is:
\[ F = -|e|E \]

### Step 2: Calculate the acceleration of the electron
Using Newton's second law, the acceleration \( a \) of the electron can be calculated as:
\[ a = \frac{F}{m} = \frac{-|e|E}{m} \]

### Step 3: Find the velocity of the electron as a function of time
Since the electron starts from rest, its velocity \( v \) at time \( t \) can be expressed as:
\[ v = at = \left(\frac{-|e|E}{m}\right)t \]

### Step 4: Calculate the momentum of the electron
The momentum \( p \) of the electron is given by:
\[ p = mv = m\left(\frac{-|e|E}{m}\right)t = -|e|Et \]

### Step 5: Write the expression for the de Broglie wavelength
The de Broglie wavelength \( \lambda \) is given by:
\[ \lambda = \frac{h}{p} \]
Substituting the expression for momentum, we get:
\[ \lambda = \frac{h}{-|e|Et} \]

### Step 6: Differentiate the de Broglie wavelength with respect to time
To find the rate of change of the de Broglie wavelength with respect to time, we differentiate \( \lambda \) with respect to \( t \):
\[ \frac{d\lambda}{dt} = \frac{d}{dt}\left(\frac{h}{-|e|Et}\right) \]
Using the power rule for differentiation:
\[ \frac{d\lambda}{dt} = -\frac{h}{-|e|E} \cdot \frac{d}{dt}(t^{-1}) = -\frac{h}{-|e|E} \cdot (-t^{-2}) = \frac{h}{|e|E} \cdot \frac{1}{t^2} \]

### Final Result
Thus, the rate of change of the de Broglie wavelength with respect to time is:
\[ \frac{d\lambda}{dt} = \frac{h}{|e|E} \cdot \frac{1}{t^2} \]