A transparent thin plate of a polaroid is placed on another similar plate such that the angle between their axes is `30^(@)`. The intensities of the emergent and the unpolarized incident light will be in the ratio of

Options

A. `1:4`
B. `1:3`
C. `3:4`
D. `3:8`

Explanation

To solve the problem of finding the ratio of the intensity of the emergent light to the intensity of the unpolarized incident light when a transparent thin plate of a polaroid is placed on another similar plate at an angle of 30 degrees, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Incident Light**:
- Let the intensity of the unpolarized incident light be \( I_0 \).

2. **First Polaroid**:
- When unpolarized light passes through the first polaroid, the intensity of the transmitted light is reduced to half.
- Therefore, the intensity after the first polaroid, \( I_1 \), is given by:
\[ I_1 = \frac{I_0}{2} \]

3. **Second Polaroid**:
- The second polaroid is oriented at an angle of \( 30^\circ \) with respect to the first polaroid.
- According to Malus's Law, the intensity of light passing through a polaroid is given by:
\[ I_2 = I_1 \cos^2(\theta) \]
- Here, \( \theta \) is the angle between the light's polarization direction (from the first polaroid) and the axis of the second polaroid. Since the angle between the axes of the two polaroids is \( 30^\circ \), the angle to use in Malus's Law is \( 30^\circ \).

4. **Calculate the Intensity After the Second Polaroid**:
- Substitute \( I_1 \) into Malus's Law:
\[ I_2 = \left(\frac{I_0}{2}\right) \cos^2(30^\circ) \]
- The value of \( \cos(30^\circ) \) is \( \frac{\sqrt{3}}{2} \), so:
\[ I_2 = \left(\frac{I_0}{2}\right) \left(\frac{\sqrt{3}}{2}\right)^2 \]
- Simplifying this:
\[ I_2 = \left(\frac{I_0}{2}\right) \left(\frac{3}{4}\right) = \frac{3I_0}{8} \]

5. **Find the Ratio of Emergent Intensity to Incident Intensity**:
- The ratio of the emergent intensity \( I_2 \) to the incident intensity \( I_0 \) is:
\[ \text{Ratio} = \frac{I_2}{I_0} = \frac{\frac{3I_0}{8}}{I_0} = \frac{3}{8} \]

### Final Answer:
The ratio of the intensity of the emergent light to the intensity of the unpolarized incident light is \( \frac{3}{8} \).