Two coherent sources are placed 0.9 mm apart and the fringes are observed one metre away. The wavelength of monochromatic light used if it produces the second dark fringes at a distance of 10 mm from the central finge will be

Options

A. `6 xx 10^(-4) cm`
B. `6 xx 10^(-6) cm`
C. `6 xx 10^(-7) cm`
D. `1.2 xx 10^(-4) cm`

Answer

To find the wavelength of the monochromatic light used in the interference pattern, we can follow these steps:

### Step 1: Understand the Problem
We have two coherent sources separated by a distance \(d = 0.9 \, \text{mm} = 0.9 \times 10^{-3} \, \text{m}\). The screen is placed at a distance \(D = 1 \, \text{m}\) from the sources. The second dark fringe is observed at a distance \(y = 10 \, \text{mm} = 10 \times 10^{-3} \, \text{m}\) from the central fringe.

### Step 2: Use the Formula for Dark Fringes
The position of the dark fringes in a double-slit interference pattern is given by the formula:
\[ y = \left(n - \frac{1}{2}\right) \frac{\lambda D}{d} \]
where \(n\) is the order of the dark fringe, \(\lambda\) is the wavelength, \(D\) is the distance to the screen, and \(d\) is the distance between the slits.

### Step 3: Substitute Known Values
For the second dark fringe, \(n = 2\). We can substitute the known values into the formula:
\[ 10 \times 10^{-3} = \left(2 - \frac{1}{2}\right) \frac{\lambda \cdot 1}{0.9 \times 10^{-3}} \]
This simplifies to:
\[ 10 \times 10^{-3} = \left(\frac{3}{2}\right) \frac{\lambda}{0.9 \times 10^{-3}} \]

### Step 4: Solve for Wavelength \(\lambda\)
Rearranging the equation to solve for \(\lambda\):
\[ \lambda = \frac{10 \times 10^{-3} \cdot 0.9 \times 10^{-3}}{\frac{3}{2}} \]
Calculating the right-hand side:
\[ \lambda = \frac{10 \times 0.9}{\frac{3}{2}} \times 10^{-6} = \frac{9}{1.5} \times 10^{-6} = 6 \times 10^{-6} \, \text{m} \]

### Step 5: Convert to Centimeters
To convert meters to centimeters:
\[ \lambda = 6 \times 10^{-6} \, \text{m} = 6 \times 10^{-4} \, \text{cm} \]

### Step 6: Conclusion
Thus, the wavelength of the monochromatic light used is:
\[ \lambda = 6 \times 10^{-4} \, \text{cm} \]

### Final Answer
The correct option is \(6 \times 10^{-4} \, \text{cm}\).

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