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 upolarized 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, we need to determine the ratio of the intensity of the emergent light after passing through two polaroids to the intensity of the unpolarized incident light. We will use Malus's Law to find the solution.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two polaroids placed one over the other, with their transmission axes at an angle of \(30^\circ\). The incident light is unpolarized.

2. **Intensity of Unpolarized Light**:
Let the intensity of the unpolarized light be \(I_0\). When unpolarized light passes through a polaroid, the intensity of the transmitted light is reduced to half. Therefore, after passing through the first polaroid, the intensity \(I_1\) is given by:
\[ I_1 = \frac{I_0}{2} \]

3. **Applying Malus's Law**:
When the polarized light from the first polaroid passes through the second polaroid, we apply Malus's Law. According to Malus's Law, the intensity of light transmitted through a second polarizer is given by:
\[ I_2 = I_1 \cos^2(\theta) \]
where \(\theta\) is the angle between the transmission axes of the two polaroids. Here, \(\theta = 30^\circ\).

4. **Calculating the Intensity After the Second Polaroid**:
Substitute \(I_1\) and \(\theta\) into the equation:
\[ I_2 = \left(\frac{I_0}{2}\right) \cos^2(30^\circ) \]
We know that \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), so:
\[ I_2 = \left(\frac{I_0}{2}\right) \left(\frac{\sqrt{3}}{2}\right)^2 \]
\[ I_2 = \left(\frac{I_0}{2}\right) \left(\frac{3}{4}\right) = \frac{3I_0}{8} \]

5. **Finding the Ratio**:
Now, we need to find the ratio of the emergent intensity \(I_2\) to the incident intensity \(I_0\):
\[ \text{Ratio} = \frac{I_2}{I_0} = \frac{\frac{3I_0}{8}}{I_0} = \frac{3}{8} \]

### Conclusion:
The ratio of the intensities of the emergent light to the unpolarized incident light is:
\[ \text{Ratio} = 3 : 8 \]