Intensity obseverd in an interferecne pattern is `I = I_(0) sin^(2) theta`. At `theta = 30^(@)`, Intensity `I = 5 +- 0.002`. The pecentage error in angle is

Options

A. `4 sqrt 3 xx 10^(-2) %`
B. `(4)/(pi) xx 10^(-2) %`
C. `(4 sqrt 3)/(pi) xx 10^(-2) %`
D. `sqrt 3 xx 10^(-2) %`

Explanation

To solve the problem, we need to find the percentage error in the angle \( \theta \) given the intensity \( I = I_0 \sin^2 \theta \) and the intensity value at \( \theta = 30^\circ \).

### Step-by-Step Solution:

1. **Understand the given formula**:
The intensity in the interference pattern is given by:
\[ I = I_0 \sin^2 \theta \]

2. **Differentiate the intensity with respect to \( \theta \)**:
To find the relationship between the change in intensity \( dI \) and the change in angle \( d\theta \), we differentiate:
\[ dI = I_0 \cdot 2 \sin \theta \cos \theta \cdot d\theta \]
This can be simplified using the double angle identity:
\[ dI = I_0 \sin(2\theta) d\theta \]

3. **Express \( dI/I \)**:
We can express the relative change in intensity:
\[ \frac{dI}{I} = \frac{I_0 \sin(2\theta) d\theta}{I_0 \sin^2 \theta} = \frac{2 \cos \theta}{\sin \theta} d\theta \]
Thus,
\[ \frac{dI}{I} = 2 \cot \theta \, d\theta \]

4. **Relate \( d\theta \) to \( dI \)**:
Rearranging gives:
\[ d\theta = \frac{dI}{2 \cot \theta \cdot I} \]

5. **Substitute known values**:
At \( \theta = 30^\circ \):
- \( I = 5 \)
- \( dI = 0.002 \)
- \( \cot(30^\circ) = \sqrt{3} \)

Plugging in these values:
\[ d\theta = \frac{0.002}{2 \cdot \sqrt{3} \cdot 5} \]

6. **Calculate \( d\theta \)**:
\[ d\theta = \frac{0.002}{10\sqrt{3}} = \frac{0.0002}{\sqrt{3}} \text{ radians} \]

7. **Convert \( \theta \) to radians**:
Convert \( 30^\circ \) to radians:
\[ \theta = \frac{\pi}{6} \text{ radians} \]

8. **Calculate the percentage error**:
The percentage error in \( \theta \) is given by:
\[ \frac{d\theta}{\theta} \times 100 = \frac{\frac{0.0002}{\sqrt{3}}}{\frac{\pi}{6}} \times 100 \]

9. **Simplify the expression**:
\[ = \frac{0.0002 \cdot 6 \cdot 100}{\pi \sqrt{3}} = \frac{1.2}{\pi \sqrt{3}} \times 100 \]

10. **Final calculation**:
\[ \text{Percentage error} = \frac{120}{\pi \sqrt{3}} \approx \frac{120 \cdot 100}{3.14 \cdot 1.732} \approx \frac{12000}{5.441} \approx 2200.5 \text{ (approximately)} \]

### Final Result:
The percentage error in angle \( \theta \) is approximately:
\[ \frac{4\sqrt{3}}{25\pi} \times 10^{-2} \text{ percent} \]