Show that when the unpolarized light passe through polariser then the intensity emerging light is half that of the intensity the incident light.

Explanation

If the intensity of light on the polariser is I then the intensity of emergent light (from Malu law) is `I=I_(0)cos^(2)theta` where `theta` is an angle betwee pass-axis of both polaroid.
When unpolarized light incident on a polaroid the intensity of emerging light is equal to th average intensity between `theta=0` to `2pi`.
`:.` Average intensity,
`ltIgt""=I_(0)ltcos^(2)thetagt`
`=I_(0)int_(0)^(2pi)(cos^(2)theta)/(2pi)d""theta`
`=(I_(0))/(2pi)int_(0)^(2pi)[(1+cos2theta)/(2)]d"theta`
`(I_(0))/(2pi){[theta]_(0)^(2pi)+[(sin2theta)/(2)]_(theta)^(2pi)}`
`=(I_(0))/(4pi){(2pi-0)+(sin4pi-sin0)/(2)}`
`=(I_(0))/(4pi){(2pi+0)-0}`
`=(I_(0))/(4pi)xx2pi`
`=(I_(0))/(2)`
Hence, the intensity of emerging light is half that of the intensity of the incident light that means 50%.