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 step by step, we will follow the given information and apply the necessary mathematical concepts.

### Step 1: Understand the given equation
The intensity observed in an interference pattern is given by:
\[ I = I_0 \sin^2 \theta \]
where \( I_0 \) is a constant and \( \theta \) is the angle.

### Step 2: Differentiate the intensity equation
To find the relationship between the change in intensity (\( dI \)) and the change in angle (\( d\theta \)), we differentiate the intensity equation:
\[ dI = I_0 \cdot 2 \sin \theta \cos \theta \, d\theta \]
This can be rewritten using the identity \( \sin(2\theta) = 2 \sin \theta \cos \theta \):
\[ dI = I_0 \sin(2\theta) \, d\theta \]

### Step 3: Express \( dI/I \)
Dividing both sides by \( I \):
\[ \frac{dI}{I} = \frac{I_0 \sin(2\theta) \, d\theta}{I} \]

### Step 4: Substitute \( I \)
Since \( I = I_0 \sin^2 \theta \), we can substitute this into the equation:
\[ \frac{dI}{I} = \frac{I_0 \sin(2\theta) \, d\theta}{I_0 \sin^2 \theta} \]
This simplifies to:
\[ \frac{dI}{I} = \frac{\sin(2\theta)}{\sin^2 \theta} \, d\theta \]

### Step 5: Simplify using cotangent
Using the identity \( \sin(2\theta) = 2 \sin \theta \cos \theta \):
\[ \frac{dI}{I} = 2 \cot \theta \, d\theta \]

### Step 6: Rearranging for \( d\theta \)
Rearranging gives:
\[ d\theta = \frac{dI}{2 \cot \theta \, I} \]

### Step 7: Calculate the percentage error
To find the percentage error in angle:
\[ \frac{d\theta}{\theta} \times 100 = \frac{dI}{2 \cot \theta \, I} \cdot \frac{100}{\theta} \]

### Step 8: Substitute known values
Given:
- \( dI = 0.002 \)
- \( I = 5 \)
- \( \theta = 30^\circ \) (which is \( \frac{\pi}{6} \) radians)

First, calculate \( \cot(30^\circ) \):
\[ \cot(30^\circ) = \frac{1}{\tan(30^\circ)} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \]

Now substituting:
\[ \frac{d\theta}{\theta} \times 100 = \frac{0.002}{2 \cdot \sqrt{3} \cdot 5} \cdot \frac{100}{\frac{\pi}{6}} \]

### Step 9: Simplify the expression
Calculating the right side:
\[ = \frac{0.002 \cdot 6 \cdot 100}{2 \cdot \sqrt{3} \cdot 5 \cdot \pi} \]
\[ = \frac{1.2}{10 \sqrt{3} \pi} \]

### Step 10: Final calculation
Now, we can compute the numerical value:
\[ = \frac{1.2}{10 \sqrt{3} \pi} \approx \frac{1.2}{17.32} \approx 0.0692 \]

Converting to percentage:
\[ \approx 0.0692 \times 100 \approx 6.92\% \]

### Final Answer
The percentage error in the angle \( \theta \) is approximately \( 6.92\% \).

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