In a Young’s double slit experiment, the slit separation is `0.2 cm`, the distance between the screen and slit is 1 m . Wavelength of the light used is `5000 Å`. The distance between two consecutive dark fringes (in mm ) is

Options

A. `0.25`
B. `0.26`
C. `0.27`
D. `0.28`

Answer

To solve the problem of finding the distance between two consecutive dark fringes in a Young's double slit experiment, we can follow these steps:

### Step 1: Understand the formula for dark fringes
In a Young's double slit experiment, the position of the dark fringes is given by the formula:
\[ y_n = \left(n + \frac{1}{2}\right) \frac{\lambda D}{d} \]
where:
- \( y_n \) is the position of the nth dark fringe,
- \( n \) is the order of the dark fringe (n = 0, 1, 2, ...),
- \( \lambda \) is the wavelength of light,
- \( D \) is the distance from the slits to the screen,
- \( d \) is the distance between the slits.

### Step 2: Calculate the distance between two consecutive dark fringes
The distance between two consecutive dark fringes (from \( y_n \) to \( y_{n+1} \)) can be expressed as:
\[ \Delta y = y_{n+1} - y_n = \left(n + \frac{3}{2}\right) \frac{\lambda D}{d} - \left(n + \frac{1}{2}\right) \frac{\lambda D}{d} \]
This simplifies to:
\[ \Delta y = \frac{\lambda D}{d} \]

### Step 3: Substitute the values
Given:
- \( \lambda = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} = 5 \times 10^{-7} \, \text{m} \)
- \( D = 1 \, \text{m} \)
- \( d = 0.2 \, \text{cm} = 0.2 \times 10^{-2} \, \text{m} = 2 \times 10^{-3} \, \text{m} \)

Now substituting these values into the formula:
\[ \Delta y = \frac{(5 \times 10^{-7} \, \text{m})(1 \, \text{m})}{2 \times 10^{-3} \, \text{m}} \]

### Step 4: Calculate the result
Calculating \( \Delta y \):
\[ \Delta y = \frac{5 \times 10^{-7}}{2 \times 10^{-3}} = 2.5 \times 10^{-4} \, \text{m} \]

### Step 5: Convert to mm
To convert meters to millimeters:
\[ \Delta y = 2.5 \times 10^{-4} \, \text{m} = 0.25 \, \text{mm} \]

### Final Answer
The distance between two consecutive dark fringes is **0.25 mm**.

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