Statement I: An electron beam is used to obtain interference in a simple Young's double-slit experiment arrangement with appropriate distance between the slits. If the speed of electrons in increased, the fringe width decreases.
Statement II: de Broglie wavelength of electron is inversely proportional to the speed of the electrons.

Options

A. Statement I is True, statement II is True, Statement II is a correct explanation for Statement I.
B. Statement I is Ture, Statement II is Ture, Statement II is NOT a correct explanation for Statement I.
C. Statement I is True, Statement II is False.
D. Statement I is False, Statement II is True.

Explanation

To solve the question, we need to analyze both statements and their implications in the context of the Young's double-slit experiment with an electron beam.

### Step-by-Step Solution:

1. **Understanding Fringe Width**:
The fringe width (β) in a Young's double-slit experiment is given by the formula:
\[ \beta = \frac{d \cdot \lambda}{D} \]
where:
- \(d\) = distance between the slits
- \(\lambda\) = wavelength of the light (or in this case, the de Broglie wavelength of the electrons)
- \(D\) = distance from the slits to the screen

2. **Using de Broglie Wavelength**:
For electrons, the wavelength is given by the de Broglie relation:
\[ \lambda_b = \frac{h}{mv} \]
where:
- \(h\) = Planck's constant
- \(m\) = mass of the electron
- \(v\) = speed of the electron

3. **Substituting de Broglie Wavelength into Fringe Width**:
Substituting the de Broglie wavelength into the fringe width formula, we get:
\[ \beta = \frac{d \cdot \left(\frac{h}{mv}\right)}{D} = \frac{dh}{mvD} \]

4. **Analyzing the Effect of Speed on Fringe Width**:
From the formula \(\beta = \frac{dh}{mvD}\), we can see that:
- As the speed \(v\) of the electrons increases, the term \(mv\) in the denominator increases.
- This results in a decrease in fringe width \(\beta\).

5. **Conclusion for Statement I**:
Therefore, Statement I is true: If the speed of electrons is increased, the fringe width decreases.

6. **Analyzing Statement II**:
Statement II states that the de Broglie wavelength of the electron is inversely proportional to the speed of the electrons. From the de Broglie wavelength formula \(\lambda_b = \frac{h}{mv}\), it is clear that as \(v\) increases, \(\lambda_b\) decreases.

7. **Conclusion for Statement II**:
Thus, Statement II is also true: The de Broglie wavelength is inversely proportional to the speed of the electrons.

### Final Conclusion:
Both statements are true.