If the distance between the first maxima and fifth minima of a double-slit pattern is 7 mm and the slits are separated by 0.15 mm with the screen 50 cm from the slits, then wavelength of the light used is

Options

A. 600 nm
B. 525 nm
C. 467 nm
D. 420 nm

Explanation

To solve the problem, we need to find the wavelength of light used in a double-slit interference pattern given the distance between the first maxima and fifth minima, the slit separation, and the distance from the slits to the screen.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Distance between the first maxima and fifth minima (D) = 7 mm = 0.007 m
- Slit separation (d) = 0.15 mm = 0.00015 m
- Distance from the slits to the screen (L) = 50 cm = 0.5 m

2. **Understand the Positions of Maxima and Minima:**
- The position of the m-th maxima (y_m) is given by the formula:
\[ y_m = \frac{m \lambda L}{d} \]
- The position of the n-th minima (y_n) is given by the formula:
\[ y_n = \frac{(n + \frac{1}{2}) \lambda L}{d} \]

3. **Calculate the Positions:**
- For the first maxima (m = 1):
\[ y_1 = \frac{1 \cdot \lambda \cdot 0.5}{0.00015} \]
- For the fifth minima (n = 5):
\[ y_5 = \frac{(5 + \frac{1}{2}) \lambda \cdot 0.5}{0.00015} = \frac{5.5 \lambda \cdot 0.5}{0.00015} \]

4. **Find the Distance Between the First Maxima and Fifth Minima:**
- The distance between the first maxima and fifth minima is:
\[ D = y_5 - y_1 \]
- Substituting the expressions for \(y_5\) and \(y_1\):
\[ D = \left(\frac{5.5 \lambda \cdot 0.5}{0.00015}\right) - \left(\frac{1 \cdot \lambda \cdot 0.5}{0.00015}\right) \]
- Simplifying gives:
\[ D = \frac{(5.5 - 1) \lambda \cdot 0.5}{0.00015} = \frac{4.5 \lambda \cdot 0.5}{0.00015} \]

5. **Set the Equation Equal to the Given Distance:**
- We know that \(D = 0.007\) m, so:
\[ 0.007 = \frac{4.5 \lambda \cdot 0.5}{0.00015} \]

6. **Solve for Wavelength (\(\lambda\)):**
- Rearranging the equation gives:
\[ \lambda = \frac{0.007 \cdot 0.00015}{4.5 \cdot 0.5} \]
- Calculating this:
\[ \lambda = \frac{0.00000105}{2.25} = 4.67 \times 10^{-7} \text{ m} = 467 \text{ nm} \]

### Final Answer:
The wavelength of the light used is approximately **467 nm**.