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 calculating 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, \( \lambda = 5400 \, \text{Å} = 5400 \times 10^{-10} \, \text{m} = 5.4 \times 10^{-7} \, \text{m} \)
- Width of the slit, \( d = 0.80 \, \text{mm} = 0.80 \times 10^{-3} \, \text{m} = 8.0 \times 10^{-4} \, \text{m} \)
- Distance from the slit to the screen, \( D = 1.4 \, \text{m} \)

2. **Calculate the Fringe Width (\( \beta \)):**
The formula for the fringe width in a single-slit diffraction pattern is given by:
\[ \beta = \frac{\lambda D}{d} \]
Substituting the values we have:
\[ \beta = \frac{(5.4 \times 10^{-7} \, \text{m})(1.4 \, \text{m})}{8.0 \times 10^{-4} \, \text{m}} \]

3. **Perform the Calculation:**
\[ \beta = \frac{(5.4 \times 1.4)}{8.0} \times 10^{-7 + 0 + 4} \, \text{m} \]
\[ = \frac{7.56}{8.0} \times 10^{-3} \, \text{m} \]
\[ = 0.945 \times 10^{-3} \, \text{m} = 0.945 \, \text{mm} \]

4. **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 twice the fringe width:
\[ \text{Distance} = 2 \beta = 2 \times 0.945 \, \text{mm} = 1.89 \, \text{mm} \]

5. **Final Answer:**
The distance between the first two dark bands on each side of the central bright band is \( 1.89 \, \text{mm} \).