The red light of wavelength 5400 Å from a distant source falls on a slit 0.80 mm wide. Calculate the distance between the first two dark bands on each side of the central bright band in the diffraction pattern observed on a screen place 1.4m from the slit.

Options

A. 1.89 mm
B. 4 mm
C. 3 mm
D. 5 mm

Answer

To solve the problem of finding the distance between the first two dark bands on each side of the central bright band in the diffraction pattern, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Wavelength of light, \( \lambda = 5400 \, \text{Å} = 5400 \times 10^{-10} \, \text{m} \)
- Width of the slit, \( d = 0.80 \, \text{mm} = 0.80 \times 10^{-3} \, \text{m} \)
- Distance from the slit to the screen, \( D = 1.4 \, \text{m} \)

2. **Understanding the Diffraction Pattern**:
- In a single-slit diffraction pattern, the positions of the dark fringes can be calculated using the formula:
\[ y_n = \frac{(n + 0.5) \lambda D}{d} \]
- Here, \( y_n \) is the distance from the central maximum to the \( n \)-th dark fringe, and \( n \) is the order of the dark fringe (for the first dark fringe, \( n = 0 \)).

3. **Calculate the Position of the First Dark Fringe**:
- For the first dark fringe (\( n = 0 \)):
\[ y_1 = \frac{(0 + 0.5) \lambda D}{d} = \frac{0.5 \times (5400 \times 10^{-10}) \times 1.4}{0.80 \times 10^{-3}} \]

4. **Perform the Calculation**:
- Substitute the values into the equation:
\[ y_1 = \frac{0.5 \times 5400 \times 10^{-10} \times 1.4}{0.80 \times 10^{-3}} \]
\[ = \frac{0.5 \times 5400 \times 1.4}{0.80} \times 10^{-7} \]
\[ = \frac{3780}{0.80} \times 10^{-7} \]
\[ = 4725 \times 10^{-7} \, \text{m} = 0.0004725 \, \text{m} = 4.725 \, \text{mm} \]

5. **Calculate the Distance Between the First Two Dark Bands**:
- The distance between the first two dark bands on each side of the central bright band is \( 2y_1 \):
\[ \text{Distance} = 2y_1 = 2 \times 4.725 \, \text{mm} = 9.45 \, \text{mm} \]

### Final Result:
The distance between the first two dark bands on each side of the central bright band is **9.45 mm**.