Two parallel slits 1 mm apart are illuminated with light of wavelength 5000 Å, from a single slit. Interference pattern is obtained on a screen placed at 1 m from the slits, what is the distance between the first bright bannd and the seventh dark band?

Options

A. 1.5 m
B. 2 mm
C. 2.25 mm
D. 2.75 mm

Explanation

To solve the problem of finding the distance between the first bright band and the seventh dark band in an interference pattern created by two parallel slits, we will follow these steps:

### Step 1: Understand the given data
- Distance between the slits (d) = 1 mm = \(1 \times 10^{-3}\) m
- Wavelength of light (λ) = 5000 Å = \(5000 \times 10^{-10}\) m = \(5 \times 10^{-7}\) m
- Distance to the screen (D) = 1 m

### Step 2: Calculate the position of the first bright band (y₁b)
The formula for the position of the m-th bright fringe in a double-slit interference pattern is given by:
\[ y_{m} = \frac{m \cdot \lambda \cdot D}{d} \]
For the first bright band (m = 1):
\[ y_{1b} = \frac{1 \cdot (5 \times 10^{-7}) \cdot 1}{1 \times 10^{-3}} = \frac{5 \times 10^{-7}}{1 \times 10^{-3}} = 5 \times 10^{-4} \text{ m} = 0.5 \text{ mm} \]

### Step 3: Calculate the position of the seventh dark band (y₇d)
The formula for the position of the n-th dark fringe is given by:
\[ y_{n} = \frac{(2n - 1) \cdot \lambda \cdot D}{2d} \]
For the seventh dark band (n = 7):
\[ y_{7d} = \frac{(2 \cdot 7 - 1) \cdot (5 \times 10^{-7}) \cdot 1}{2 \cdot (1 \times 10^{-3})} = \frac{13 \cdot (5 \times 10^{-7})}{2 \cdot (1 \times 10^{-3})} \]
Calculating this gives:
\[ y_{7d} = \frac{65 \times 10^{-7}}{2 \times 10^{-3}} = \frac{65 \times 10^{-7}}{2 \times 10^{-3}} = 3.25 \text{ mm} \]

### Step 4: Calculate the distance between the first bright band and the seventh dark band
Now, we find the distance \(y\) between the seventh dark band and the first bright band:
\[ y = y_{7d} - y_{1b} = 3.25 \text{ mm} - 0.5 \text{ mm} = 2.75 \text{ mm} \]

### Final Answer
The distance between the first bright band and the seventh dark band is **2.75 mm**.

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