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

Options

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

Answer

To solve the problem, we need to analyze the behavior of a parallel beam of fast-moving electrons as it passes through a narrow slit and how the speed of the electrons affects the diffraction pattern observed on a fluorescent screen.

### Step-by-Step Solution:

1. **Understanding Electron Behavior**:
- Electrons, although they are particles, exhibit wave-like behavior due to their wave-particle duality. This means they can create a diffraction pattern when they pass through a slit.

**Hint**: Remember that particles like electrons can behave like waves, leading to phenomena such as diffraction.

2. **Setting Up the Problem**:
- We have a narrow slit and a fluorescent screen placed at a large distance from the slit. When electrons pass through the slit, they will create a diffraction pattern on the screen.

**Hint**: Visualize the setup: a slit and a screen far away where the diffraction pattern will be observed.

3. **Analyzing the Diffraction Pattern**:
- The central maximum of the diffraction pattern is the brightest spot directly in line with the slit. The angular width of this central maximum is important for understanding how the pattern changes with the speed of the electrons.

**Hint**: Recall that the central maximum is the brightest part of the diffraction pattern.

4. **Using the Diffraction Formula**:
- The angular width (θ) of the central maximum can be derived from the formula:
\[ d \sin \theta = n \lambda \]
For the first minimum (n=1), this simplifies to:
\[ d \sin \theta = \lambda \]
For small angles, we can approximate \(\sin \theta \approx \theta\):
\[ d \theta = \lambda \quad \Rightarrow \quad \theta = \frac{\lambda}{d} \]

**Hint**: Remember that for small angles, the sine of the angle can be approximated by the angle itself in radians.

5. **Relating Wavelength to Speed**:
- According to de Broglie's hypothesis, the wavelength (λ) of an electron is given by:
\[ \lambda = \frac{h}{p} = \frac{h}{mv} \]
where \(h\) is Planck's constant, \(m\) is the mass of the electron, and \(v\) is its velocity. This shows that the wavelength is inversely proportional to the speed of the electron.

**Hint**: Remember the de Broglie wavelength formula and how it relates momentum and velocity.

6. **Impact of Increasing Speed**:
- As the speed of the electrons increases, the wavelength decreases (since \(\lambda \propto \frac{1}{v}\)). Since the angular width (θ) is directly proportional to the wavelength (θ ∝ λ), an increase in speed results in a decrease in the angular width of the central maximum.

**Hint**: Think about how increasing speed affects the wavelength and subsequently the angular width of the diffraction pattern.

7. **Conclusion**:
- Therefore, if the speed of the electrons is increased, the angular width of the central maximum of the diffraction pattern will decrease.

**Final Answer**: The correct statement is that the angular width of the central maximum of the diffraction pattern will decrease as the speed of the electrons increases.