Two slits separated by a distance of 1 mm are illuminated with red light of wavelength `6.5xx10^(-7)`m. The interference firnges are observed on a screen placed 1 m form the slits. The distance between third bright firnge and the fifth dark fringe on the same side is equal to

Options

A. 0.65 nm
B. 0.975 mm
C. 3.25 mm
D. 4.88 mm

Answer

To solve the problem, we need to find the distance between the third bright fringe and the fifth dark fringe in a double-slit interference pattern.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Distance between the slits (d) = 1 mm = \(1 \times 10^{-3}\) m
- Wavelength of light (\(\lambda\)) = \(6.5 \times 10^{-7}\) m
- Distance from the slits to the screen (D) = 1 m

2. **Formulas for Bright and Dark Fringes:**
- The position of the nth bright fringe (maxima) is given by:
\[ y_n = \frac{n \lambda D}{d} \]
- The position of the nth dark fringe (minima) is given by:
\[ y_m = \frac{(m - 0.5) \lambda D}{d} \]
where \(n\) is the order of the bright fringe and \(m\) is the order of the dark fringe.

3. **Calculate the Position of the Third Bright Fringe:**
- For \(n = 3\):
\[ y_3 = \frac{3 \lambda D}{d} = \frac{3 \times (6.5 \times 10^{-7}) \times 1}{1 \times 10^{-3}} \]
\[ y_3 = \frac{19.5 \times 10^{-7}}{10^{-3}} = 19.5 \times 10^{-4} \text{ m} = 1.95 \times 10^{-3} \text{ m} \]

4. **Calculate the Position of the Fifth Dark Fringe:**
- For \(m = 5\):
\[ y_5 = \frac{(5 - 0.5) \lambda D}{d} = \frac{4.5 \lambda D}{d} = \frac{4.5 \times (6.5 \times 10^{-7}) \times 1}{1 \times 10^{-3}} \]
\[ y_5 = \frac{29.25 \times 10^{-7}}{10^{-3}} = 29.25 \times 10^{-4} \text{ m} = 2.925 \times 10^{-3} \text{ m} \]

5. **Calculate the Distance Between the Third Bright Fringe and the Fifth Dark Fringe:**
- The distance between the two fringes is:
\[ \Delta y = y_5 - y_3 = (2.925 \times 10^{-3}) - (1.95 \times 10^{-3}) \]
\[ \Delta y = 0.975 \times 10^{-3} \text{ m} = 0.975 \text{ mm} \]

### Final Answer:
The distance between the third bright fringe and the fifth dark fringe is **0.975 mm**.