When a polaroid sheet is rotated between two crossed polaroids then the transmitted light intensity will be maximum for a rotation of :

Solution

Physics sir · 3 steps

Split the 90° gap equally between the two pairs of sheets, and the most light gets through.

1. Set up angles. Let the middle polaroid make angle θ with the first. The last polaroid is crossed with the first, so it makes 90° - θ with the middle one. If the incident light is unpolarised with intensity I₀, the first polaroid passes I₀/2.
2. Apply Malus law twice. The middle sheet passes I₀/2 cos² θ. The last sheet passes a fraction cos²(90° - θ) = sin² θ of that. Use 2 sin θ cos θ = sin 2θ to tidy the product.

   I = I₀/2 cos² θ sin² θ = I₀/8 sin² 2θ

3. So. sin² 2θ is largest, equal to 1, when 2θ = 90°, that is θ = 45°. The maximum transmitted intensity is I₀/8, so the answer is (c).

The idea to keep. With a sheet between two crossed polaroids, the output goes as sin² 2θ. It is zero at 0° and 90° and peaks at 45°, where it is one-eighth of the unpolarised input.

Answer from the NTA final answer key. Solution by PrepWiser.