Two coherent sources are placed 0.9 mm apart and the fringes are observed one metre away. If it produces the second dark fringe at a distance of 10 mm from the central fringe. The wavelength of monochromatic light is `x xx 10^(-4) cm`, What is the value of x ?

Answer

To solve the problem step by step, we will use the formula for the position of dark fringes in an interference pattern created by two coherent sources.

### Step 1: Understand the given parameters
- Distance between the two coherent sources (d) = 0.9 mm = 0.9 x 10^(-3) m
- Distance from the sources to the screen (D) = 1 m
- Distance of the second dark fringe from the central maximum (L) = 10 mm = 10 x 10^(-3) m
- We need to find the wavelength of the monochromatic light in the form x x 10^(-4) cm.

### Step 2: Identify the formula for dark fringes
The position of the nth dark fringe in a double-slit interference pattern is given by the formula:
\[ L = \frac{(2n - 1) \lambda D}{2d} \]
where:
- \( L \) = distance of the dark fringe from the central maximum
- \( n \) = order of the dark fringe (for the second dark fringe, n = 2)
- \( \lambda \) = wavelength of the light
- \( D \) = distance from the sources to the screen
- \( d \) = distance between the two coherent sources

### Step 3: Substitute the known values into the formula
For the second dark fringe (n = 2):
\[ L = \frac{(2 \cdot 2 - 1) \lambda D}{2d} \]
This simplifies to:
\[ L = \frac{3 \lambda D}{2d} \]

### Step 4: Rearrange the formula to solve for the wavelength (λ)
Rearranging gives us:
\[ \lambda = \frac{2dL}{3D} \]

### Step 5: Substitute the values into the equation
Now, substituting the values we have:
- \( d = 0.9 \times 10^{-3} \) m
- \( L = 10 \times 10^{-3} \) m
- \( D = 1 \) m

So,
\[ \lambda = \frac{2 \cdot (0.9 \times 10^{-3}) \cdot (10 \times 10^{-3})}{3 \cdot 1} \]
\[ \lambda = \frac{2 \cdot 0.9 \cdot 10^{-6}}{3} \]
\[ \lambda = \frac{1.8 \times 10^{-6}}{3} \]
\[ \lambda = 0.6 \times 10^{-6} \text{ m} \]

### Step 6: Convert the wavelength to cm
To convert meters to centimeters:
\[ \lambda = 0.6 \times 10^{-6} \text{ m} = 0.6 \times 10^{-4} \text{ cm} \]

### Step 7: Identify the value of x
From the problem statement, we have:
\[ \lambda = x \times 10^{-4} \text{ cm} \]
Thus, comparing:
\[ x = 0.6 \]

### Final Answer
The value of \( x \) is \( 6 \).

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