The distance between two coherent sources is 1 mm. The screen is placed at a distance of 1 m from the sources. If the distance of the third bright fringe is `1.2` mm from the central fringe, the wavelength of light used is

Options

A. `4000 Å`
B. `5000 Å`
C. `6000 Å`
D. `7200 Å`

Answer

To find the wavelength of light used in the given problem, we can follow these steps:

### Step 1: Understand the given data
- Distance between the two coherent sources (slits), \( d = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \)
- Distance from the slits to the screen, \( D = 1 \, \text{m} \)
- Distance of the third bright fringe from the central fringe, \( y_3 = 1.2 \, \text{mm} = 1.2 \times 10^{-3} \, \text{m} \)
- Order of the fringe, \( n = 3 \)

### Step 2: Use the formula for fringe position
The position of the \( n \)-th bright fringe in a double-slit experiment is given by the formula:
\[ y_n = \frac{n \lambda D}{d} \]
Where:
- \( y_n \) = position of the \( n \)-th fringe
- \( \lambda \) = wavelength of light
- \( D \) = distance from the slits to the screen
- \( d \) = distance between the slits

### Step 3: Rearrange the formula to find the wavelength
We can rearrange the formula to solve for the wavelength \( \lambda \):
\[ \lambda = \frac{y_n d}{n D} \]

### Step 4: Substitute the known values
Now, substituting the known values into the equation:
\[ \lambda = \frac{(1.2 \times 10^{-3} \, \text{m}) \cdot (1 \times 10^{-3} \, \text{m})}{3 \cdot (1 \, \text{m})} \]

### Step 5: Calculate the wavelength
Calculating the above expression:
\[ \lambda = \frac{1.2 \times 10^{-6}}{3} = 0.4 \times 10^{-6} \, \text{m} \]
This can also be expressed as:
\[ \lambda = 4.0 \times 10^{-7} \, \text{m} \]

### Step 6: Convert to angstroms
Since \( 1 \, \text{angstrom} = 10^{-10} \, \text{m} \):
\[ \lambda = 4.0 \times 10^{-7} \, \text{m} = 4000 \, \text{angstroms} \]

### Final Answer
The wavelength of light used is \( 4000 \, \text{angstroms} \).

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