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

Explanation

Intensity of the light transmitted through the fist polarizer `I_(1) = I_(0) //2` , Where `I_(0)` is the intensity of the incident unpolarized light . Intensity of the light transmitted through the second polarizer is `I_(2) = I_(1) cos^(2) theta "where" theta` is the angle between the characteristic direction of the polarizer sheets .
But `I_(2) = I_(0) // 3 ` (given)
`:. I_(2) = I_(1) cos ^(2) theta = (I_(0))/( 2) cos^(2) theta = (I_(0))/( 3)`
`:. cos^(2) theta = 2 // 3 rArr theta = cos ^(-1) sqrt((2)/(3))`