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A clear sheet of polaroid is placed on t...

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

A

`16 : 25`

B

`9 : 25`

C

`4 : 5`

D

`8 : 25`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to calculate the ratio of the intensity of the emergent light to that of the unpolarized incident light when two polaroids are placed with their axes at an angle of \( \sin^{-1} \left( \frac{3}{5} \right) \). ### Step-by-Step Solution: 1. **Understanding the Incident Light**: - The incident light is unpolarized. The intensity of unpolarized light is denoted as \( I \). 2. **Intensity After First Polaroid**: - When unpolarized light passes through a polaroid, the intensity of the transmitted light is reduced to half of the incident intensity. - Therefore, the intensity after the first polaroid is: \[ I_1 = \frac{1}{2} I \] 3. **Finding the Angle**: - We are given that the angle \( \theta \) between the axes of the two polaroids is \( \sin^{-1} \left( \frac{3}{5} \right) \). - To find \( \cos \theta \), we can use the Pythagorean identity: \[ \cos^2 \theta + \sin^2 \theta = 1 \] - Given \( \sin \theta = \frac{3}{5} \), we find \( \cos \theta \): \[ \cos^2 \theta = 1 - \left( \frac{3}{5} \right)^2 = 1 - \frac{9}{25} = \frac{16}{25} \] \[ \cos \theta = \frac{4}{5} \] 4. **Intensity After Second Polaroid**: - The intensity transmitted by the second polaroid is given by Malus's Law: \[ I_2 = I_1 \cos^2 \theta \] - Substituting \( I_1 \) and \( \cos^2 \theta \): \[ I_2 = \left( \frac{1}{2} I \right) \left( \frac{4}{5} \right)^2 = \frac{1}{2} I \cdot \frac{16}{25} = \frac{8}{25} I \] 5. **Calculating the Ratio**: - The ratio of the intensity of the emergent light \( I_2 \) to the intensity of the incident light \( I \) is: \[ \text{Ratio} = \frac{I_2}{I} = \frac{\frac{8}{25} I}{I} = \frac{8}{25} \] ### Final Answer: The ratio of the intensity of the emergent light to that of the unpolarized incident light is \( \frac{8}{25} \).
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