In Young's double-slit experiment, interference pattern is observed at a distance of 1m. If light used has a frequency of `6 xx 10^(14)` Hz and slit separation distance is equal to 0.05 mm, calculate the distance between fourth bright fringe and centre.

Answer

To solve the problem of finding the distance between the fourth bright fringe and the center in Young's double-slit experiment, we can follow these steps:

### Step 1: Find the Wavelength of the Light
We know the relationship between the speed of light (c), frequency (f), and wavelength (λ) is given by the equation:
\[ c = f \lambda \]
Where:
- \( c = 3 \times 10^8 \, \text{m/s} \) (speed of light)
- \( f = 6 \times 10^{14} \, \text{Hz} \)

Rearranging the equation to find the wavelength:
\[ \lambda = \frac{c}{f} \]

Substituting the values:
\[ \lambda = \frac{3 \times 10^8}{6 \times 10^{14}} \]

Calculating this gives:
\[ \lambda = 0.5 \times 10^{-6} \, \text{m} \]
\[ \lambda = 5 \times 10^{-7} \, \text{m} \]

### Step 2: Use the Formula for the Distance of the nth Bright Fringe
The distance of the nth bright fringe from the center (central maximum) is given by the formula:
\[ y_n = \frac{n \lambda D}{d} \]
Where:
- \( y_n \) = distance of the nth bright fringe from the center
- \( n \) = order of the fringe (for the fourth bright fringe, \( n = 4 \))
- \( D \) = distance from the slits to the screen (given as 1 m)
- \( d \) = slit separation (given as 0.05 mm = \( 0.05 \times 10^{-3} \, \text{m} \))

### Step 3: Substitute the Values into the Formula
Now, substituting the values into the formula for the fourth bright fringe:
\[ y_4 = \frac{4 \times (5 \times 10^{-7}) \times 1}{0.05 \times 10^{-3}} \]

Calculating this step-by-step:
1. Calculate the numerator:
\[ 4 \times (5 \times 10^{-7}) = 20 \times 10^{-7} = 2 \times 10^{-6} \, \text{m} \]

2. Calculate the denominator:
\[ 0.05 \times 10^{-3} = 5 \times 10^{-5} \, \text{m} \]

3. Now divide the numerator by the denominator:
\[ y_4 = \frac{2 \times 10^{-6}}{5 \times 10^{-5}} = \frac{2}{5} \times 10^{-1} = 0.4 \, \text{m} \]

### Step 4: Convert to Millimeters
Since the question may require the answer in millimeters:
\[ y_4 = 0.4 \, \text{m} = 40 \, \text{mm} \]

### Final Answer
The distance between the fourth bright fringe and the center is:
\[ \text{Distance} = 40 \, \text{mm} \]

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