Q: Unpolarized light of intensity 32 Wm^{-2} passes through three polarizers such that the transmission axis of the last polarizer is crossed with the first. If the intensity of the emerging light is 3 Wm^{-2}, what is the angle between the transmission axes of the first two polarizers ? At what angle will the transmitted intensity be maximum?

Sol: If θ is the angle between the transmission axes of first Polaroid P_{1} and second P_{2} while φ between the

transmission axes of second Polaroid P_{2} and third P_{3} , then according to give problem.

φ = (90° – θ) … (i)

Now id I_0 is the intensity of Unpolarized light incident on Polaroid P_1, the intensity of light transmitted through it,

I_{1} = I_{0}/2 = (32)/2 = 16 W/m^{2} … (ii)

Now as angle between transmission axes of polaroids P_{1} and P_{2} is θ , in a accordance with Malus law, intensity of light transmitted through P_{2} will be

I_{2} = I_{1} cos^{2} θ = 16 cos^{2} θ …. (iii)

And as angle between transmission axes of P_{2} and P_{3} is φ , light transmitted through P_{3} will be

I_{3} = I_{2} cos^2 φ = 16 cos^{2} θ cos^{2} φ … (iv)

According to give problem, I_{3} = 3 W/ m^{2}

So, 4(sin 2θ)^{2} = 3

i.e., sin 2 θ = (√3/2)

or, 2 θ = 60°, i.e., θ = 30°.