A beam of monochromatic light of wavelength `lambda` falls normally on a diffraction grating of line spacing d. If `theta` is the angle between the second-order diffracted beam and the direction of the incident light, what is the value of `sin theta` ?

Options

A. `(lambda)/(d)`
B. `(d)/(lambda)`
C. `(2 lambda)/(d)`
D. `(2d)/(lambda)`

Explanation

To solve the problem, we will use the diffraction grating formula, which relates the angle of diffraction to the wavelength of light and the spacing between the grating lines.

### Step-by-Step Solution:

1. **Understand the Problem**: We have a monochromatic light beam of wavelength \( \lambda \) that falls normally on a diffraction grating with line spacing \( d \). We need to find \( \sin \theta \) for the second-order diffraction (where \( n = 2 \)).

2. **Use the Diffraction Grating Formula**: The formula for diffraction grating is given by:
\[ d \sin \theta = n \lambda \]
where:
- \( d \) is the spacing between the grating lines,
- \( \theta \) is the angle of the diffracted beam,
- \( n \) is the order of the diffraction (in this case, \( n = 2 \)),
- \( \lambda \) is the wavelength of the light.

3. **Substitute the Values**: For the second-order diffraction, we substitute \( n = 2 \) into the equation:
\[ d \sin \theta = 2 \lambda \]

4. **Solve for \( \sin \theta \)**: Rearranging the equation to solve for \( \sin \theta \):
\[ \sin \theta = \frac{2 \lambda}{d} \]

5. **Final Answer**: Therefore, the value of \( \sin \theta \) for the second-order diffracted beam is:
\[ \sin \theta = \frac{2 \lambda}{d} \]