A micture of light, consisting of wavelength 590nm and an unknown wavelength, illuminates Young's double slit and gives rise to two overlapping interference patterns on the scree. The central maximum of both lights coincide. Further, it is obseved that the third bright fringe of known light coincides with the 4th bright fringe of the unknown light. From this data, the wavelength of the unknown light is:

Options

A. 393.4 nm
B. 885.0 nm
C. 442.5 nm
D. 776.8 nm

Answer

To solve the problem of finding the wavelength of the unknown light in a Young's double slit experiment, we can follow these steps:

### Step 1: Understand the given information
We have two wavelengths:
- Known wavelength \( \lambda_1 = 590 \, \text{nm} \)
- Unknown wavelength \( \lambda_2 \) (to be determined)

The central maximum of both lights coincides, and the third bright fringe of the known light coincides with the fourth bright fringe of the unknown light.

### Step 2: Write the formula for the position of bright fringes
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:
- \( y_n \) is the position of the \( n \)-th bright fringe,
- \( n \) is the order of the fringe,
- \( \lambda \) is the wavelength of the light,
- \( D \) is the distance from the slits to the screen,
- \( d \) is the distance between the slits.

### Step 3: Set up the equations for the coinciding fringes
For the known light (wavelength \( \lambda_1 \)):
- The position of the third bright fringe is:
\[ y_3 = \frac{3 \lambda_1 D}{d} \]

For the unknown light (wavelength \( \lambda_2 \)):
- The position of the fourth bright fringe is:
\[ y_4 = \frac{4 \lambda_2 D}{d} \]

Since these two positions coincide:
\[ \frac{3 \lambda_1 D}{d} = \frac{4 \lambda_2 D}{d} \]

### Step 4: Simplify the equation
We can cancel \( D \) and \( d \) from both sides of the equation:
\[ 3 \lambda_1 = 4 \lambda_2 \]

### Step 5: Solve for the unknown wavelength \( \lambda_2 \)
Rearranging the equation gives:
\[ \lambda_2 = \frac{3}{4} \lambda_1 \]

Substituting the known wavelength \( \lambda_1 = 590 \, \text{nm} \):
\[ \lambda_2 = \frac{3}{4} \times 590 \, \text{nm} \]

### Step 6: Calculate \( \lambda_2 \)
Calculating this gives:
\[ \lambda_2 = \frac{1770}{4} \, \text{nm} = 442.5 \, \text{nm} \]

### Final Answer
Thus, the wavelength of the unknown light is:
\[ \lambda_2 = 442.5 \, \text{nm} \]

---