A clear sheet of polaroid is placed on the top of similar sheet so that their axes make an angle `sin^(-1)(3/5)` with each other. The ratio of intensity of the emergent light to that of unpolarised incident light is

Options

A. `16 : 25`
B. `9 : 25`
C. `4 : 5`
D. `8 : 25`

Answer

To solve the problem, we need to find the ratio of the intensity of the emergent light to that of the unpolarized incident light when two polaroid sheets are placed at an angle of \(\sin^{-1}(3/5)\) with respect to each other.

### Step-by-Step Solution:

1. **Understanding the Incident Light**:
The incident light is unpolarized. The intensity of the unpolarized light is denoted as \(I_0\).

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

3. **Angle Between the Polarizers**:
The angle between the transmission axes of the two polarizers is given as \(\theta = \sin^{-1}(3/5)\).

4. **Calculating \(\cos \theta\)**:
To use Malus's law, we need to find \(\cos \theta\). We can calculate it using the identity:
\[ \cos^2 \theta = 1 - \sin^2 \theta \]
Given \(\sin \theta = \frac{3}{5}\):
\[ \sin^2 \theta = \left(\frac{3}{5}\right)^2 = \frac{9}{25} \]
Therefore,
\[ \cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25} \]
Thus,
\[ \cos \theta = \sqrt{\frac{16}{25}} = \frac{4}{5} \]

5. **Second Polarizer**:
The intensity of light after passing through the second polarizer can be calculated using Malus's law:
\[ I_{\text{emergent}} = I_1 \cdot \cos^2 \theta \]
Substituting the values we have:
\[ I_{\text{emergent}} = \left(\frac{I_0}{2}\right) \cdot \left(\frac{16}{25}\right) \]
Simplifying this gives:
\[ I_{\text{emergent}} = \frac{16 I_0}{50} = \frac{8 I_0}{25} \]

6. **Finding the Ratio**:
The ratio of the intensity of the emergent light to that of the unpolarized incident light is:
\[ \frac{I_{\text{emergent}}}{I_0} = \frac{\frac{8 I_0}{25}}{I_0} = \frac{8}{25} \]

### Final Answer:
The ratio of the intensity of the emergent light to that of the unpolarized incident light is:
\[ \frac{8}{25} \]