Optics

Step 1: Malus's Law and Polarization
When unpolarized light of intensity is incident on a polarizer, the transmitted intensity after the first polarizer is given by: Step 2: Intensity after the Analyser
According to Malus's Law, the intensity of light transmitted through an analyser is: where is the angle between the transmission axis of the analyser and the polarizer.
Step 3: Substituting Values
Given that , we substitute: Since , we get: Step 4: Conclusion
Thus, the intensity of the light emerging from the analyser is .

Step 1: Lens Maker’s Formula in Air
The focal length of a convex lens in air is given by the lens maker's formula:
where is the refractive index of the lens material.
Step 2: Lens Maker’s Formula in a Liquid Medium
When the lens is placed in a liquid with refractive index , the new focal length is: Since the focal length doubles ( ), we equate: Simplifying, Step 3: Solving for Rearranging, Since is 80% of , Substituting, Step 4: Conclusion
Thus, the refractive index of the liquid is .