Distance of `5^("th")` dark fringe from centre is 4 mm. If D = 2 m, `lambda=600` nm, then distance between slits is :

Options

A. 1.35 mm
B. 2.00 mm
C. 3.25 mm
D. 10.35 mm

Explanation

To solve the problem, we need to find the distance between the slits (d) given the distance of the 5th dark fringe from the center (y), the distance from the slits to the screen (D), and the wavelength of light (λ).

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Distance of the 5th dark fringe from the center (y) = 4 mm = 4 × 10^(-3) m
- Distance from the slits to the screen (D) = 2 m
- Wavelength (λ) = 600 nm = 600 × 10^(-9) m

2. **Use the Formula for the Position of Dark Fringes:**
The distance of the nth dark fringe from the center in a double-slit experiment is given by:
\[ y_n = \frac{(2n - 1) \lambda D}{2d} \]
For the 5th dark fringe (n = 5):
\[ y_5 = \frac{(2 \times 5 - 1) \lambda D}{2d} = \frac{9 \lambda D}{2d} \]

3. **Set Up the Equation:**
We know that \(y_5 = 4 \times 10^{-3}\) m. Therefore:
\[ 4 \times 10^{-3} = \frac{9 \lambda D}{2d} \]

4. **Substitute the Known Values:**
Substitute λ and D into the equation:
\[ 4 \times 10^{-3} = \frac{9 \times (600 \times 10^{-9}) \times 2}{2d} \]

5. **Simplify the Equation:**
The equation simplifies to:
\[ 4 \times 10^{-3} = \frac{9 \times 600 \times 10^{-9} \times 2}{2d} \]
Cancel out the 2 in the numerator and denominator:
\[ 4 \times 10^{-3} = \frac{9 \times 600 \times 10^{-9}}{d} \]

6. **Rearranging to Solve for d:**
Rearranging gives:
\[ d = \frac{9 \times 600 \times 10^{-9}}{4 \times 10^{-3}} \]

7. **Calculate d:**
\[ d = \frac{5400 \times 10^{-9}}{4 \times 10^{-3}} = \frac{5400}{4} \times 10^{-6} = 1350 \times 10^{-6} = 1.35 \times 10^{-3} \text{ m} \]
Converting to mm:
\[ d = 1.35 \text{ mm} \]

### Final Answer:
The distance between the slits (d) is **1.35 mm**.