To solve the problem of finding the separation between the fourth bright fringes of two wavelengths in Young's double slit experiment, we can follow these steps:
### Step 1: Understand the formula for fringe separation
In Young's double slit experiment, the position of the nth bright fringe from the central maximum is given by the formula:
\[ y_n = \frac{n \lambda D}{d} \]
where:
- \( y_n \) = position of the nth bright fringe
- \( n \) = fringe order (in this case, \( n = 4 \))
- \( \lambda \) = wavelength of light
- \( D \) = distance from the slits to the screen
- \( d \) = separation between the slits
### Step 2: Identify the given values
From the problem statement:
- Slit separation, \( d = 2 \, \text{mm} = 2 \times 10^{-3} \, \text{m} \)
- Distance to the screen, \( D = 1.2 \, \text{m} \)
- Wavelengths:
- \( \lambda_1 = 6500 \, \text{Å} = 6500 \times 10^{-10} \, \text{m} \)
- \( \lambda_2 = 5200 \, \text{Å} = 5200 \times 10^{-10} \, \text{m} \)
### Step 3: Calculate the position of the fourth bright fringe for both wavelengths
Using the formula for \( n = 4 \):
1. For \( \lambda_1 = 6500 \, \text{Å} \):
\[
y_{4,1} = \frac{4 \cdot 6500 \times 10^{-10} \cdot 1.2}{2 \times 10^{-3}}
\]
\[
y_{4,1} = \frac{4 \cdot 6500 \cdot 1.2}{2} \times 10^{-7}
\]
\[
y_{4,1} = \frac{31200}{2} \times 10^{-7} = 15600 \times 10^{-7} \, \text{m} = 1.56 \times 10^{-3} \, \text{m} = 1.56 \, \text{mm}
\]
2. For \( \lambda_2 = 5200 \, \text{Å} \):
\[
y_{4,2} = \frac{4 \cdot 5200 \times 10^{-10} \cdot 1.2}{2 \times 10^{-3}}
\]
\[
y_{4,2} = \frac{4 \cdot 5200 \cdot 1.2}{2} \times 10^{-7}
\]
\[
y_{4,2} = \frac{24960}{2} \times 10^{-7} = 12480 \times 10^{-7} \, \text{m} = 1.248 \times 10^{-3} \, \text{m} = 1.248 \, \text{mm}
\]
### Step 4: Calculate the separation between the two fringes
The separation between the fourth bright fringes for the two wavelengths is:
\[
\Delta y = y_{4,1} - y_{4,2} = 1.56 \, \text{mm} - 1.248 \, \text{mm} = 0.312 \, \text{mm}
\]
### Final Answer
The separation between the fourth bright fringes of the two wavelengths is:
\[
\Delta y = 0.312 \, \text{mm}
\]
