n YDSE, an electron beam is used to obtain interference pattern. If speed of electron is increased then

Options

A. (A) no interference pattern will be observed
B. (B) distance between two consecutive fringes will increase
C. (C) distance between two consecutive fringes will decrease.
D. (D) distance between two consecutive fringes remains same

Answer

To solve the problem regarding the interference pattern in Young's Double Slit Experiment (YDSE) when the speed of an electron beam is increased, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Wavelength and Velocity**:
The wavelength (λ) of a particle is given by the de Broglie wavelength formula:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( v \) is its velocity.

2. **Analyze the Effect of Increasing Velocity**:
If the speed (velocity) of the electron is increased, according to the formula, the wavelength (λ) will be affected. Specifically, since λ is inversely proportional to \( v \):
\[ \lambda \propto \frac{1}{v} \]
This means that as \( v \) increases, \( λ \) decreases.

3. **Determine the Fringe Width in YDSE**:
The fringe width (β) in YDSE is given by the formula:
\[ \beta = \frac{\lambda D}{d} \]
where \( D \) is the distance from the slits to the screen, and \( d \) is the distance between the slits.

4. **Analyze the Effect on Fringe Width**:
Since \( \beta \) is directly proportional to \( λ \), if \( λ \) decreases (due to an increase in \( v \)), then the fringe width \( β \) will also decrease:
\[ \beta \propto \lambda \quad \Rightarrow \quad \text{If } \lambda \text{ decreases, then } \beta \text{ decreases.} \]

5. **Conclusion**:
Therefore, if the speed of the electron is increased, the distance between two consecutive fringes (the fringe width) will decrease.

### Final Answer:
The distance between two consecutive fringes will decrease.

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