A parallel beam of fast moving electrons is incident normally on a narrow slit. A fluorescent screen is placed at a large distance from the slit. If the speed of the electrons is increased, which of the following statements is correct?

Options

A. 1. Diffraction pattern is not observed on the screen in the case of electrons
B. 2. The angular width of the central maximum of the diffraction pattern will increase
C. 3. The angular width of the central maximum will decrease
D. 4. The angular width of the central maximum will be unaffected

Explanation

To solve the problem, we need to analyze the effect of increasing the speed of electrons on the diffraction pattern observed on a screen after passing through a narrow slit.

### Step-by-Step Solution:

1. **Understanding the Concept of Diffraction**:
- When a beam of particles (like electrons) passes through a narrow slit, it can create a diffraction pattern on a screen placed at a distance. This is due to the wave nature of particles, described by de Broglie's hypothesis.

2. **De Broglie Wavelength**:
- The wavelength (λ) of the electrons can be calculated using the de Broglie relation:
\[ \lambda = \frac{h}{p} = \frac{h}{mv} \]
where \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( v \) is its velocity.

3. **Angular Width of the Central Maximum**:
- The angular width (ω) of the central maximum in the diffraction pattern is given by:
\[ \omega = \frac{2\lambda}{d} \]
where \( d \) is the width of the slit.

4. **Substituting the Wavelength**:
- By substituting the expression for λ into the equation for ω, we get:
\[ \omega = \frac{2 \cdot \frac{h}{mv}}{d} = \frac{2h}{mdv} \]
- This shows that the angular width is inversely proportional to the velocity \( v \).

5. **Effect of Increasing Velocity**:
- If the speed of the electrons is increased (i.e., \( v \) increases), the angular width \( \omega \) decreases. This means that the central maximum will become narrower.

6. **Conclusion**:
- Based on the analysis, we conclude that the correct statement is: "The angular width of the center maximum of the diffraction pattern will decrease."

### Final Answer:
The angular width of the center maximum of the diffraction pattern will decrease.