Two polaroid `A and B` are kept in crossed position. How should a third polaroid `C` be placed between them so that the intensity of polarized light transmitted by polaroid `B` reduces to `(1)/(8)th` of the intensity of unplarised light incident on `A` ?

Answer

`45^(@)`

Explanation

Let the angle between the pass axis of the polaroids `A` and `C` be `theta`. As polaroids `A` and `B` are crossed, the angle between the pass axes of polaroids `C` and `B` is `(90 - theta)`. When unpolaroids light of intensity `I_(0)` falls on first polaroid `A`, intensity of polarised light through `A` is `I_(1) = I_(0)//2`.
This light falls on polaroid `C` at `/_ theta`.
`:.` Intensity of light transmitted through `C` is
`I_(2) = I_(1) cos^(2) theta = (I_(0))/(2)cos^(2) theta`
Light transmitted from `C` falls on polaroid `B` at an angle `(90 - theta)`. Therefore, intensity of light transmitted from `B` is
`I_(3) = I_(2)cos^(2)(90 - theta)`
`= [(I_(0))/(2)cos^(2) theta]sin^(2)theta = (I_(0))/(2)(sin theta cos theta)^(2)`
`I_(3) = (I_(0))/(8)(sin 2 theta)^(2)`
As `I_(3) = I_(0)//8`, therefore, therefore `(sin 2 theta)^(2) = 1`
`sin 2 theta = 1, or 2 theta = 90^(@) or theta = 45^(@)`