Question 182 of 191

Unpolarized light of intensity $I$ is incident on a system of two polarizers, $A$ followed by $B$. The intensity of emergent light is $\dfrac{I}{2}$ . If a third the polarizer $C$ is placed between $A$ and $B$ the intensity of emergent light is reduced to $\dfrac{I}{3}$ . The angle between the polarizers $A$ and $C$ is $\theta$ , then

A$\cos ⁡\theta ={\left(\dfrac{2}{3}\right)}^{\dfrac{1}{4}}$B$\cos ⁡\theta ={\left(\dfrac{1}{3}\right)}^{\dfrac{1}{4}}$C$\cos ⁡\theta ={\left(\dfrac{1}{3}\right)}^{\dfrac{1}{2}}$D$\cos ⁡\theta ={\left(\dfrac{2}{3}\right)}^{\dfrac{1}{2}}$

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Answer: (A) $\cos ⁡\theta ={\left(\dfrac{2}{3}\right)}^{\dfrac{1}{4}}$

A and B have the same alignment of the transmission axis. Let's assume $c$ is introduced at $\theta$ angle, $\Rightarrow \dfrac{I}{2}{\cos }^{2}\theta \times {\cos }^{2}\theta =\dfrac{I}{3}$$\Rightarrow {\cos }^{4}\theta =\dfrac{2}{3}$
$\Rightarrow \cos \theta ={\left(\dfrac{2}{3}\right)}^{\dfrac{1}{4}}$.