Two wavelength of light `lambda_(1)` and `lambda_(2)` are sent through Young's double-slit apparatus simultaneously. What must be true about `lambda_(1)` and `lambda_(2)` if the third-order bright fringe of `lambda_(1)` coincides with fifth-order dark fringe of `lambda_(2)`

Options

A. `3lambda_(1)=2lambda_(2)`
B. `2lambda_(1)=3lambda_(2)`
C. `3lambda_(1)=5lambda_(2)`
D. `5lambda_(1)=3lambda_(2)`

Answer

To solve the problem, we need to find the relationship between the two wavelengths, \( \lambda_1 \) and \( \lambda_2 \), given that the third-order bright fringe of \( \lambda_1 \) coincides with the fifth-order dark fringe of \( \lambda_2 \).

### Step-by-Step Solution:

1. **Understanding the Fringe Positions**:
- The position of the \( n \)-th bright fringe for a wavelength \( \lambda \) in Young's double-slit experiment is given by:
\[ y_n = \frac{n \lambda D}{d} \]
where \( D \) is the distance from the slits to the screen and \( d \) is the distance between the slits.

2. **Position of the Third Bright Fringe for \( \lambda_1 \)**:
- For the third-order bright fringe of \( \lambda_1 \):
\[ y_3 = \frac{3 \lambda_1 D}{d} \]

3. **Position of the Fifth Dark Fringe for \( \lambda_2 \)**:
- The position of the \( n \)-th dark fringe for a wavelength \( \lambda \) is given by:
\[ y_n = \frac{(2n-1) \lambda D}{2d} \]
- For the fifth-order dark fringe of \( \lambda_2 \):
\[ y_5 = \frac{(2 \cdot 5 - 1) \lambda_2 D}{2d} = \frac{9 \lambda_2 D}{2d} \]

4. **Setting the Two Positions Equal**:
- Since the third-order bright fringe of \( \lambda_1 \) coincides with the fifth-order dark fringe of \( \lambda_2 \), we can set the two equations equal:
\[ \frac{3 \lambda_1 D}{d} = \frac{9 \lambda_2 D}{2d} \]

5. **Simplifying the Equation**:
- Cancel \( D \) and \( d \) from both sides:
\[ 3 \lambda_1 = \frac{9 \lambda_2}{2} \]
- Multiply both sides by 2 to eliminate the fraction:
\[ 6 \lambda_1 = 9 \lambda_2 \]

6. **Finding the Relationship**:
- Rearranging gives:
\[ 2 \lambda_1 = 3 \lambda_2 \]
- This can be expressed as:
\[ \frac{\lambda_1}{\lambda_2} = \frac{3}{2} \]

### Final Answer:
The relationship between the two wavelengths is:
\[ \lambda_1 = \frac{3}{2} \lambda_2 \]