If the angle between the pass axis of polariser and analyser is `45^(@)`, write the ratio of intensities of original light and the transmitted light after passing through analyser.

Options

A. `(I)/(2)`
B. `(I)/(3)`
C. `I`
D. `(I)/(4)`

Explanation

To solve the problem of finding the ratio of the intensities of original light (I₀) and the transmitted light (I) after passing through a polarizer and an analyzer at an angle of 45 degrees, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Polarization**:
- When light passes through a polarizer, it becomes polarized, meaning it vibrates in a specific direction. The intensity of the polarized light after passing through the polarizer is given by Malus's Law.

2. **Applying Malus's Law**:
- The intensity of light after passing through a polarizer is given by the formula:
\[ I = I_0 \cos^2(\phi) \]
- Here, \(I_0\) is the intensity of the original unpolarized light, and \(\phi\) is the angle between the light's initial polarization direction and the axis of the polarizer.

3. **Substituting the Angle**:
- In this case, the angle \(\phi\) is given as 45 degrees. Therefore, we substitute \(\phi = 45^\circ\):
\[ I = I_0 \cos^2(45^\circ) \]

4. **Calculating \(\cos^2(45^\circ)\)**:
- We know that \(\cos(45^\circ) = \frac{1}{\sqrt{2}}\). Thus:
\[ \cos^2(45^\circ) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

5. **Substituting Back into the Intensity Equation**:
- Now substitute \(\cos^2(45^\circ)\) back into the intensity equation:
\[ I = I_0 \cdot \frac{1}{2} \]
- This simplifies to:
\[ I = \frac{I_0}{2} \]

6. **Finding the Ratio of Intensities**:
- The ratio of the original intensity \(I_0\) to the transmitted intensity \(I\) is:
\[ \frac{I_0}{I} = \frac{I_0}{\frac{I_0}{2}} = 2 \]

### Final Result:
The ratio of the intensities of the original light to the transmitted light after passing through the analyzer is:
\[ \frac{I_0}{I} = 2 \]