In a double-slit experiment, two parallel slits are illuminated first bylight of wavelength 400 nm and then by light of unknown wavelength. The fourth-order dark fringe resulting from the known wavelength of light falls in the same place on the screen as the second-order bright fringe from the unknown wavelength. The value of unknown wavelength of light is

Options

A. 900 nm
B. 700 nm
C. 300 nm
D. none of these

Explanation

To solve the problem, we need to find the unknown wavelength of light in a double-slit experiment where the fourth-order dark fringe from a known wavelength (400 nm) coincides with the second-order bright fringe from the unknown wavelength.

### Step-by-Step Solution:

1. **Understand the Conditions for Dark and Bright Fringes**:
- The condition for dark fringes in a double-slit experiment is given by:
\[ y_d = \frac{(m + \frac{1}{2}) \lambda D}{d} \]
where \(y_d\) is the position of the dark fringe, \(m\) is the order of the dark fringe, \(\lambda\) is the wavelength of light, \(D\) is the distance from the slits to the screen, and \(d\) is the distance between the slits.
- The condition for bright fringes is given by:
\[ y_b = \frac{m' \lambda D}{d} \]
where \(m'\) is the order of the bright fringe.

2. **Identify the Known Values**:
- Given that the known wavelength \(\lambda = 400 \, \text{nm}\) and the fourth-order dark fringe (\(m = 4\)) coincides with the second-order bright fringe (\(m' = 2\)) of the unknown wavelength.

3. **Set Up the Equation**:
- For the fourth-order dark fringe:
\[ y_d = \frac{(4 + \frac{1}{2}) \cdot 400 \, \text{nm} \cdot D}{d} = \frac{(4.5) \cdot 400 \, \text{nm} \cdot D}{d} \]
- For the second-order bright fringe:
\[ y_b = \frac{2 \cdot \lambda' \cdot D}{d} \]
- Since these two positions are the same, we can set them equal:
\[ \frac{(4.5) \cdot 400 \, \text{nm} \cdot D}{d} = \frac{2 \cdot \lambda' \cdot D}{d} \]

4. **Simplify the Equation**:
- Cancel \(D/d\) from both sides:
\[ 4.5 \cdot 400 \, \text{nm} = 2 \cdot \lambda' \]

5. **Solve for the Unknown Wavelength**:
- Rearranging gives:
\[ \lambda' = \frac{4.5 \cdot 400 \, \text{nm}}{2} \]
- Calculate \(\lambda'\):
\[ \lambda' = \frac{1800 \, \text{nm}}{2} = 900 \, \text{nm} \]

6. **Final Result**:
- The value of the unknown wavelength \(\lambda' = 900 \, \text{nm}\).

### Conclusion:
The unknown wavelength of light is **900 nm**.