Unpolarised light of intensity $32 \mathrm{Wm}^{-2}$ passes through the combination of three polaroi

Question

Unpolarised light of intensity $32 \mathrm{Wm}^{-2}$ passes through the combination of three polaroids such that the pass axis of the last polaroid is perpendicular to that of the pass axis of first polaroid. If intensity of emerging light is $3 \mathrm{Wm}^{-2}$, then the possible values of angle between pass axis of first two polaroids are _______ ${ }^{\circ}$.\

Professor PC Thomas Classes · Answer

$(30,60): I_{0}=32 \mathrm{~W} / \mathrm{m}^{2}$\ Let the angle between $P_{1}$ and $P_{2}$ is $	heta$\ $I_{1}=\frac{I_{0}}{2}=\frac{32}{2}=16 \mathrm{~W} / \mathrm{m}^{2}$\ Now, $I_{2}=I_{1} \cos ^{2} 	heta=16 \cos ^{2} 	heta$\ $\Rightarrow \quad I_{3}=I_{2} \cos ^{2}(90-	heta)=I_{2} \sin ^{2} 	heta$\ $\Rightarrow \quad 3=16 \cos ^{2} 	heta \sin ^{2} 	heta \Rightarrow 3=4(\sin 2 	heta)^{2}$\ $\Rightarrow 2 	heta=60^{\circ}$ or $	heta=30^{\circ}$\ $\Rightarrow \sin 2 	heta=\frac{\sqrt{3}}{2} \Rightarrow \sin 60^{\circ}, \sin 120^{\circ}$\ $\Rightarrow 	heta=30^{\circ}, 60^{\circ}$\