If `I_0` is the intensity of the principal maximum in the single slit diffraction pattern. Then what will be its intensity when the slit width is doubled?

Options

A. `I_(0)pi//omega`
B. `I_(0)//2`
C. `2I_(0)`
D. `4I_(0)`

Explanation

To solve the problem, we need to analyze how the intensity of the principal maximum in a single slit diffraction pattern changes when the slit width is doubled.

### Step-by-Step Solution:

1. **Understanding the Initial Condition**:
- Let the initial slit width be \( a \).
- The intensity of the principal maximum is given as \( I_0 \).

2. **Central Maximum Width**:
- The width of the central maximum in a single slit diffraction pattern is inversely proportional to the slit width. The angular width of the central maximum is given by:
\[ \theta = \frac{2\lambda}{a} \]
- Here, \( \lambda \) is the wavelength of the light used.

3. **Condition When Slit Width is Doubled**:
- If the slit width is doubled, the new width becomes \( 2a \).
- The new angular width of the central maximum becomes:
\[ \theta' = \frac{2\lambda}{2a} = \frac{\lambda}{a} \]
- This indicates that the width of the central maximum is halved when the slit width is doubled.

4. **Intensity and Area Relationship**:
- The intensity of the central maximum is related to the area over which the light is distributed. When the slit width increases, the area increases, and thus the intensity decreases.
- The intensity is inversely proportional to the area of the slit:
\[ I \propto \frac{1}{A} \]
- If the width of the slit is doubled, the area \( A \) of the slit becomes \( 2a \times L \) (where \( L \) is the length of the slit), resulting in an increase in area.

5. **Calculating the New Intensity**:
- Since the area has doubled, the new intensity \( I_1 \) can be related to the original intensity \( I_0 \) as follows:
\[ I_1 = k \cdot \frac{I_0}{2} \quad \text{(where \( k \) is a proportionality constant)} \]
- However, we also know that the intensity at the principal maximum is proportional to the square of the amplitude of the wave. When the slit width is doubled, the amplitude effectively increases, leading to:
\[ I_1 = 4I_0 \]

6. **Conclusion**:
- Therefore, when the slit width is doubled, the intensity of the principal maximum becomes:
\[ I_1 = 4I_0 \]

### Final Answer:
The intensity of the principal maximum when the slit width is doubled is \( 4I_0 \).