In YDSE, the slits are separated by `0.5 mm` and screen is placed `1.0 m` away. It is found that the ninth bright fringe is at a distance of `8.835 mm` from the second dark fringe. Find the wavelength of light used.

Explanation

To solve the problem step-by-step, we will use the principles of Young's Double Slit Experiment (YDSE) to find the wavelength of light used.

### Step 1: Understand the given data
- Distance between the slits, \( d = 0.5 \, \text{mm} = 0.5 \times 10^{-3} \, \text{m} \)
- Distance from the slits to the screen, \( D = 1.0 \, \text{m} \)
- Distance between the 9th bright fringe and the 2nd dark fringe, \( y = 8.835 \, \text{mm} = 8.835 \times 10^{-3} \, \text{m} \)

### Step 2: Formulate the position of the bright and dark fringes
The position of the \( n \)-th bright fringe is given by:
\[ y_n = \frac{n \lambda D}{d} \]
For the 9th bright fringe (\( n = 9 \)):
\[ y_9 = \frac{9 \lambda D}{d} \]

The position of the \( m \)-th dark fringe is given by:
\[ y_m = \frac{(m - 0.5) \lambda D}{d} \]
For the 2nd dark fringe (\( m = 2 \)):
\[ y_2 = \frac{(2 - 0.5) \lambda D}{d} = \frac{1.5 \lambda D}{d} \]

### Step 3: Set up the equation
We know that the distance between the 9th bright fringe and the 2nd dark fringe is given by:
\[ y_9 - y_2 = y \]
Substituting the expressions for \( y_9 \) and \( y_2 \):
\[ \frac{9 \lambda D}{d} - \frac{1.5 \lambda D}{d} = 8.835 \times 10^{-3} \]

### Step 4: Simplify the equation
Combine the terms:
\[ \frac{(9 - 1.5) \lambda D}{d} = 8.835 \times 10^{-3} \]
\[ \frac{7.5 \lambda D}{d} = 8.835 \times 10^{-3} \]

### Step 5: Solve for the wavelength \( \lambda \)
Rearranging the equation gives:
\[ \lambda = \frac{8.835 \times 10^{-3} \cdot d}{7.5 \cdot D} \]
Substituting the values of \( d \) and \( D \):
\[ \lambda = \frac{8.835 \times 10^{-3} \cdot (0.5 \times 10^{-3})}{7.5 \cdot 1} \]
\[ \lambda = \frac{8.835 \times 0.5 \times 10^{-6}}{7.5} \]
\[ \lambda = \frac{4.4175 \times 10^{-6}}{7.5} \]
\[ \lambda = 0.589 \times 10^{-6} \, \text{m} \]

### Step 6: Convert to Angstroms
To convert meters to Angstroms:
\[ \lambda = 0.589 \times 10^{-6} \, \text{m} = 5890 \, \text{Å} \]

### Final Answer
The wavelength of light used is \( 5890 \, \text{Å} \).

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