Q: Unpolarized light falls on two sheets placed one on top of the other. What must be the angle between the characteristic direction of the sheets if the intensity of the transmitted light is one third of intensity of the incident beam ?

Sol: Intensity of the light transmitted through the first polarized I_{1} = I_{0}/2 , where I_{0} is the intensity of the incident unpolarized light. Intensity of the light transmitted through the second polarize is I_{2} = I_{1} cos^{2}θ

θ is the angle between the characteristic directions of the polarizer sheets.

But I_{2} = I_{0}/ 3 (given)

∴ I_{2} = I_{1} cos^{2}θ = (I_{0}/2) cos^{2} θ

I_{0}/ 3 = (I_{0}/2) cos^{2} θ

∴ cos^{2} θ = 2/3

⇒ $\large \theta = cos^{-1}\sqrt{\frac{2}{3}} $