The two slits are 1 mm apart from each other and illuminated with a light of wavelength `5xx10^(-7)` m. If the distance of the screen is 1 m from the slits, then the distance between third dark fringe and fifth bright fringe is

Options

A. 1.2 mm
B. 0.75 mm
C. 1.25 mm
D. 0.625 mm

Explanation

To solve the problem step by step, we will calculate the distance between the third dark fringe and the fifth bright fringe in a double-slit interference pattern.

### Given Data:
- Distance between the slits, \( d = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \)
- Wavelength of light, \( \lambda = 5 \times 10^{-7} \, \text{m} \)
- Distance from the slits to the screen, \( D = 1 \, \text{m} \)

### Step 1: Calculate the position of the fifth bright fringe
The formula for the position of the \( n \)-th bright fringe is given by:
\[ y_n = \frac{n \cdot \lambda \cdot D}{d} \]
For the fifth bright fringe (\( n = 5 \)):
\[ y_5 = \frac{5 \cdot (5 \times 10^{-7}) \cdot 1}{1 \times 10^{-3}} \]
Calculating this:
\[ y_5 = \frac{25 \times 10^{-7}}{10^{-3}} = 25 \times 10^{-4} \, \text{m} = 2.5 \times 10^{-3} \, \text{m} = 2.5 \, \text{mm} \]

### Step 2: Calculate the position of the third dark fringe
The formula for the position of the \( n \)-th dark fringe is given by:
\[ y'_n = \frac{(n - 0.5) \cdot \lambda \cdot D}{d} \]
For the third dark fringe (\( n = 3 \)):
\[ y'_3 = \frac{(3 - 0.5) \cdot (5 \times 10^{-7}) \cdot 1}{1 \times 10^{-3}} \]
Calculating this:
\[ y'_3 = \frac{2.5 \cdot (5 \times 10^{-7})}{1 \times 10^{-3}} = \frac{12.5 \times 10^{-7}}{10^{-3}} = 12.5 \times 10^{-4} \, \text{m} = 1.25 \times 10^{-3} \, \text{m} = 1.25 \, \text{mm} \]

### Step 3: Calculate the distance between the third dark fringe and the fifth bright fringe
Now, we find the distance between the third dark fringe and the fifth bright fringe:
\[ \text{Distance} = y_5 - y'_3 \]
Substituting the values:
\[ \text{Distance} = 2.5 \, \text{mm} - 1.25 \, \text{mm} = 1.25 \, \text{mm} \]

### Final Answer:
The distance between the third dark fringe and the fifth bright fringe is \( 1.25 \, \text{mm} \).

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