An electron of mass `m` and charge `e` initially at rest gets accelerated by a constant electric field `E`. The rate of change of de-Broglie wavelength of this electron at time `t` ignoring relativistic effects is

Options

A. `(-h)/(eEt^(2))`
B. `(-eEt)/E`
C. `(-mh)/(eEt^(2))`
D. `(-h)/(e.E)`

Explanation

To solve the problem of finding the rate of change of the de-Broglie wavelength of an electron accelerated by a constant electric field, we can 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:
\[ F = eE \]
where \( e \) is the charge of the electron.

**Hint:** Remember that the force on a charged particle in an electric field is given by the product of charge and electric field strength.

### Step 2: Calculate the Acceleration of the Electron
Using Newton's second law, the acceleration \( a \) of the electron can be calculated as:
\[ a = \frac{F}{m} = \frac{eE}{m} \]
where \( m \) is the mass of the electron.

**Hint:** Recall that acceleration is the force divided by mass.

### Step 3: Find the Velocity of the Electron at Time \( t \)
Since the electron starts from rest, we can use the equation of motion:
\[ v = u + at \]
where \( u = 0 \) (initial velocity), so:
\[ v = 0 + \left(\frac{eE}{m}\right)t = \frac{eEt}{m} \]

**Hint:** Use the basic kinematic equation for velocity when starting from rest.

### Step 4: Calculate the de-Broglie Wavelength
The de-Broglie wavelength \( \lambda \) is given by:
\[ \lambda = \frac{h}{p} \]
where \( p \) is the momentum of the electron. The momentum \( p \) can be expressed as:
\[ p = mv = m \left(\frac{eEt}{m}\right) = eEt \]
Thus, the de-Broglie wavelength becomes:
\[ \lambda = \frac{h}{eEt} \]

**Hint:** Remember that momentum is mass times velocity.

### Step 5: Find the Rate of Change of the de-Broglie Wavelength
To find the rate of change of the de-Broglie wavelength with respect to time, we differentiate \( \lambda \) with respect to \( t \):
\[ \frac{d\lambda}{dt} = \frac{d}{dt}\left(\frac{h}{eEt}\right) \]
Using the quotient rule:
\[ \frac{d\lambda}{dt} = -\frac{h}{eE} \cdot \frac{1}{t^2} \]
Thus, we have:
\[ \frac{d\lambda}{dt} = -\frac{h}{eEt^2} \]

**Hint:** Use differentiation rules to handle the time dependence in the expression for wavelength.

### Final Answer
The rate of change of the de-Broglie wavelength of the electron at time \( t \) is:
\[ \frac{d\lambda}{dt} = -\frac{h}{eEt^2} \]