A beam of electron is used `YDSE` experiment . The slit width is d when the velocity of electron is increased ,then

Options

A. no interference is observed
B. fringe width increases
C. fringe width decreases
D. fringe width remains same

Explanation

To solve the problem regarding the effect of increasing the velocity of electrons in a Young's Double Slit Experiment (YDSE), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Fringe Width**:
The fringe width (β) in a Young's Double Slit Experiment is given by the formula:
\[ \beta = \frac{\lambda D}{d} \]
where:
- \( \lambda \) is the wavelength of the electrons,
- \( D \) is the distance from the slits to the screen,
- \( d \) is the slit width.

2. **Determine the Wavelength of Electrons**:
According to de Broglie's hypothesis, the wavelength (λ) associated with a moving particle (like an electron) is given by:
\[ \lambda = \frac{h}{p} \]
where \( p \) is the linear momentum of the electron. The momentum \( p \) can be expressed as:
\[ p = mv \]
where \( m \) is the mass of the electron and \( v \) is its velocity.

3. **Substituting the Wavelength into the Fringe Width Formula**:
By substituting the expression for wavelength into the fringe width formula, we have:
\[ \beta = \frac{h D}{mv d} \]

4. **Analyzing the Effect of Increasing Velocity**:
From the formula \( \beta = \frac{h D}{mv d} \), we can see that if \( h \), \( D \), and \( d \) are constants, then fringe width (β) is inversely proportional to the velocity (v) of the electron:
\[ \beta \propto \frac{1}{v} \]
This indicates that as the velocity \( v \) increases, the fringe width \( \beta \) decreases.

5. **Conclusion**:
Therefore, when the velocity of the electron is increased, the fringe width decreases. The correct answer is that the fringe width decreases.

### Final Answer:
As the velocity of the electron is increased, the fringe width decreases.

---