Unpolarized light of intensity 32 `Wm^(-3)` 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 axces of the first two polarizers ? At what angle will the transmitted intensity be maximum ?

Explanation

If `theta` is the angle between the transmission axes of first polaroid `P_(1) ` and second `P_(2)` while `phi` between the transmission axes of second polaroid `P_(2)` and third `P_(3)` then according to given problem .
`theta + phi = 90^(0) or phi = (90^(0) - theta) ` . . . (1)
Now if `I_(0)` is the intensity of unpolarized light incident on polaroid `P_(1)` , the intensity of light transmitted through it ,
`I_(1) = (1)/(2) I_(0) = (1)/(2) (32) = 16 (W)/(m^(2)) . . . (2)`
Now as angle between transmission axes of polaroids `P_(1) and P_(2) is theta` , in a accordance with Malus law, intensity of light transmitted through `P_(2)` will be
`I_(2) = I_(1) cos ^(2) theta = 16 cos^(2) theta ` . . . (3)
And as angle between transmission axes of `P_(2) and P_(3) is phi ` , light transmitted through `P_(3)` be
`I_(3) = I_(2) cos^(2) phi = 16 cos^(2) theta cos^(2) phi` . . . (4)
According to given problem , `I_(3) = 3 W //m^(2)`
So `4 (sin 2 theta) = 3 i.e., sin 2 theta = (sqrt(3) //2) or 2 theta = 60^(0) , i . e ., theta = 30^(0)` .