Unpolarized light of intensity I₀ first passes through a polarizer whose transmission axis is at 30° to the horizontal and then through an analyser whose axis is at 90°. A third polarizer with axis at 60° is inserted between them. What is the ratio of the output intensity with the third polarizer to the output intensity without it?
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Question

Unpolarized light of intensity I₀ first passes through a polarizer whose transmission axis is at 30° to the horizontal and then through an analyser whose axis is at 90°. A third polarizer with axis at 60° is inserted between them. What is the ratio of the output intensity with the third polarizer to the output intensity without it?

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G1 Physics · Accepted Answer

Without the third polarizer, the first polarizer transmits I₁ = I₀ ÷ 2. The angle between the first polarizer at 30° and analyser at 90° is 60°. By Malus' law, I(without) = (I₀ ÷ 2)cos²60° = (I₀ ÷ 2)(1 ÷ 4) = I₀ ÷ 8. With the third polarizer at 60°, the angle between successive transmission axes is 30° in each step. Therefore, I(with) = (I₀ ÷ 2)cos²30° cos²30° = (I₀ ÷ 2)(3 ÷ 4)(3 ÷ 4) = 9I₀ ÷ 32. Thus, I(with) ÷ I(without) = (9 ÷ 32) ÷ (1 ÷ 8) = 9 ÷ 4. Correct option: C.