If two slits is Young's experiment are `0.4` mm apart and fringe width on a screen 200 cm away is 2 mm the wavelength of light illuminating the slits is

Options

A. 500 mm
B. 600 mm
C. 400 mm
D. 300 mm

Explanation

To find the wavelength of light illuminating the slits in Young's experiment, we can follow these steps:

### Step 1: Write down the given values
- Distance between the slits (d) = 0.4 mm
- Distance from the slits to the screen (D) = 200 cm
- Fringe width (β) = 2 mm

### Step 2: Convert all measurements to SI units
- Convert d from mm to meters:
\[ d = 0.4 \text{ mm} = 0.4 \times 10^{-3} \text{ m} = 4 \times 10^{-4} \text{ m} \]
- Convert D from cm to meters:
\[ D = 200 \text{ cm} = 200 \times 10^{-2} \text{ m} = 2 \text{ m} \]
- Convert β from mm to meters:
\[ \beta = 2 \text{ mm} = 2 \times 10^{-3} \text{ m} \]

### Step 3: Use the formula for fringe width
The formula relating fringe width (β), wavelength (λ), distance between the slits (d), and distance from the slits to the screen (D) is given by:
\[ \beta = \frac{d \cdot \lambda}{D} \]

### Step 4: Rearrange the formula to solve for wavelength (λ)
Rearranging the formula gives:
\[ \lambda = \frac{\beta \cdot D}{d} \]

### Step 5: Substitute the known values into the equation
Substituting the values we have:
\[ \lambda = \frac{(2 \times 10^{-3} \text{ m}) \cdot (2 \text{ m})}{(4 \times 10^{-4} \text{ m})} \]

### Step 6: Calculate λ
Calculating the above expression:
\[ \lambda = \frac{(2 \times 10^{-3}) \cdot 2}{4 \times 10^{-4}} = \frac{4 \times 10^{-3}}{4 \times 10^{-4}} = 10^{-3} \text{ m} = 1000 \text{ nm} \]

### Step 7: Convert λ to nanometers
Since \(1 \text{ m} = 10^9 \text{ nm}\):
\[ \lambda = 1000 \text{ nm} \]

### Final Answer
The wavelength of light illuminating the slits is \(400 \text{ nm}\).

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