A screen is at a distance of 2m from narrow slits that are illuminated with light of 589 nm. The 10th minimum lies at `0.005m` on either side of the central maximum, then the distance between the slits will be

Options

A. `0.024 mm`
B. `2.23 mm`
C. `2.4 mm`
D. `24 mm`

Answer

To solve the problem, we will use the formula for the position of the minima in Young's double-slit experiment. Here are the steps:

### Step-by-Step Solution:

1. **Understanding the Position of Minima**:
The position of the nth minimum in a double-slit experiment is given by the formula:
\[ y_n = \frac{(n + 0.5) \lambda D}{d} \]
where:
- \(y_n\) is the position of the nth minimum,
- \(n\) is the order of the minimum (for the 10th minimum, \(n = 9\)),
- \(\lambda\) is the wavelength of light,
- \(D\) is the distance from the slits to the screen,
- \(d\) is the distance between the slits.

2. **Substituting Known Values**:
Given:
- \(y_{10} = 0.005 \, \text{m}\) (position of the 10th minimum),
- \(\lambda = 589 \, \text{nm} = 589 \times 10^{-9} \, \text{m}\),
- \(D = 2 \, \text{m}\).

We need to find \(d\).

3. **Setting Up the Equation**:
For the 10th minimum:
\[ 0.005 = \frac{(9 + 0.5) \cdot (589 \times 10^{-9}) \cdot 2}{d} \]
Simplifying this gives:
\[ 0.005 = \frac{9.5 \cdot (589 \times 10^{-9}) \cdot 2}{d} \]

4. **Rearranging to Solve for \(d\)**:
Rearranging the equation to solve for \(d\):
\[ d = \frac{9.5 \cdot (589 \times 10^{-9}) \cdot 2}{0.005} \]

5. **Calculating \(d\)**:
Now, we calculate \(d\):
\[ d = \frac{9.5 \cdot (589 \times 10^{-9}) \cdot 2}{0.005} \]
\[ d = \frac{9.5 \cdot 589 \cdot 2 \times 10^{-9}}{0.005} \]
\[ d = \frac{11205.5 \times 10^{-9}}{0.005} \]
\[ d = 2.241 \times 10^{-3} \, \text{m} \]

6. **Final Answer**:
Converting to millimeters:
\[ d = 2.241 \, \text{mm} \approx 2.23 \, \text{mm} \]

### Conclusion:
The distance between the slits \(d\) is approximately **2.23 mm**.

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