An Unpolarised light of Intensity `I_0` is incident on two polaroids placed co-axially such that `(I_0)/2` intensity is received on screen. By what angle the polaroid placed close to screen should be rotated so that intensity on screen becomes `3/8I_0`

Options

A. `30^@`
B. `60^@`
C. `45^@`
D. `90^@`

Answer

To solve the problem, we need to analyze the behavior of light as it passes through two polaroids.

### Step-by-step Solution:

1. **Understanding the Initial Conditions**:
- We have unpolarized light of intensity \( I_0 \) incident on the first polaroid.
- The intensity of light after passing through the first polaroid is given by the formula for unpolarized light passing through a polaroid, which is \( \frac{I_0}{2} \).

2. **Intensity After the First Polaroid**:
- The intensity after the first polaroid is:
\[ I_1 = \frac{I_0}{2} \]

3. **Setting Up the Second Polaroid**:
- The second polaroid is initially aligned with the first one (coaxial). Therefore, the intensity after the second polaroid, without any rotation, would also be \( I_1 = \frac{I_0}{2} \).

4. **Rotating the Second Polaroid**:
- We need to rotate the second polaroid by an angle \( \theta \). The intensity of light transmitted through the second polaroid is given by Malus's Law:
\[ I_2 = I_1 \cos^2(\theta) \]
- Substituting \( I_1 \):
\[ I_2 = \frac{I_0}{2} \cos^2(\theta) \]

5. **Setting the Final Intensity**:
- According to the problem, we want the intensity on the screen to be \( \frac{3I_0}{8} \):
\[ \frac{I_0}{2} \cos^2(\theta) = \frac{3I_0}{8} \]

6. **Solving for \( \cos^2(\theta) \)**:
- Dividing both sides by \( I_0 \) and simplifying:
\[ \frac{1}{2} \cos^2(\theta) = \frac{3}{8} \]
- Multiplying both sides by 2:
\[ \cos^2(\theta) = \frac{3}{4} \]

7. **Finding \( \theta \)**:
- Taking the square root of both sides:
\[ \cos(\theta) = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]
- The angle \( \theta \) that satisfies this equation is:
\[ \theta = 30^\circ \]

### Final Answer:
The angle by which the polaroid placed close to the screen should be rotated is \( 30^\circ \).

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