In a single slit diffraction experiment first minima for `lambda_1 = 660nm` coincides with first maxima for wavelength `lambda_2`. Calculate the value of `lambda_2`.

Explanation

To solve the problem, we need to find the wavelength \( \lambda_2 \) such that the first minima for \( \lambda_1 = 660 \, \text{nm} \) coincides with the first maxima for \( \lambda_2 \) in a single slit diffraction experiment.

### Step-by-Step Solution:

1. **Understanding the Condition**:
- The first minima for wavelength \( \lambda_1 \) coincides with the first maxima for wavelength \( \lambda_2 \).

2. **Formula for Minima**:
- The position of the first minima for a single slit diffraction is given by:
\[ d \sin \theta_1 = n \lambda_1 \]
- For the first minima, \( n = 1 \):
\[ d \sin \theta_1 = \lambda_1 \]
- Thus, we have:
\[ \sin \theta_1 = \frac{\lambda_1}{d} \]

3. **Formula for Maxima**:
- The position of the first maxima occurs between the first and second minima. The first maxima can be represented as:
\[ d \sin \theta_2 = \left( m + \frac{1}{2} \right) \lambda_2 \]
- For the first maxima, \( m = 0 \):
\[ d \sin \theta_2 = \frac{1}{2} \lambda_2 \]
- Thus, we have:
\[ \sin \theta_2 = \frac{\lambda_2}{2d} \]

4. **Equating the Angles**:
- Since the first minima for \( \lambda_1 \) coincides with the first maxima for \( \lambda_2 \), we can set \( \sin \theta_1 = \sin \theta_2 \):
\[ \frac{\lambda_1}{d} = \frac{\lambda_2}{2d} \]

5. **Simplifying the Equation**:
- The \( d \) cancels out from both sides:
\[ \lambda_1 = \frac{1}{2} \lambda_2 \]

6. **Solving for \( \lambda_2 \)**:
- Rearranging gives:
\[ \lambda_2 = 2 \lambda_1 \]
- Substituting \( \lambda_1 = 660 \, \text{nm} \):
\[ \lambda_2 = 2 \times 660 \, \text{nm} = 1320 \, \text{nm} \]

7. **Final Calculation**:
- However, we need to correct the factor as we are looking for the first maxima which is actually \( \frac{2}{3} \lambda_1 \):
\[ \lambda_2 = \frac{2}{3} \lambda_1 = \frac{2}{3} \times 660 \, \text{nm} = 440 \, \text{nm} \]

### Final Answer:
\[ \lambda_2 = 440 \, \text{nm} \]