Direction of the first secondary maximum in the Fraunhofer diffraction pattern at a single slit is given by (a is the width of the slit)

Options

A. (a) `asin theta=lambda/2`
B. (b) `a cos theta=(3lambda)/(2)`
C. (c) `asin theta=lambda`
D. (d) `a sin theta=(3lambda)/(2)`

Explanation

To find the direction of the first secondary maximum in the Fraunhofer diffraction pattern at a single slit, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a single slit of width \( a \).
- The light passing through the slit creates a diffraction pattern on a screen placed at a distance \( D \).

2. **Condition for Secondary Maxima**:
- The condition for the position of secondary maxima in a single slit diffraction pattern is given by:
\[ a \sin \theta = \left( n + \frac{1}{2} \right) \lambda \]
- Here, \( n \) is an integer representing the order of the maximum (0, 1, 2,...), and \( \lambda \) is the wavelength of the light.

3. **Finding the First Secondary Maximum**:
- For the first secondary maximum, we set \( n = 0 \):
\[ a \sin \theta = \left( 0 + \frac{1}{2} \right) \lambda = \frac{\lambda}{2} \]

4. **Solving for \( \sin \theta \)**:
- Rearranging the equation gives:
\[ \sin \theta = \frac{\lambda}{2a} \]

5. **Conclusion**:
- The direction of the first secondary maximum is given by the angle \( \theta \) such that:
\[ \sin \theta = \frac{\lambda}{2a} \]

### Final Answer:
The direction of the first secondary maximum in the Fraunhofer diffraction pattern at a single slit is given by:
\[ \sin \theta = \frac{\lambda}{2a} \]