Natural monochromatic light falls on a system of two crossed Nicol prisms between to its optical axis is insered. Find the minimum thickness of the plate at which this system will transmit `eta = 0.30` of luuminous flux if the specific rotation constant of quartz is equal to `alpha = 17 ang deg// mm`.

Explanation

If `I_(0) =` intensity of natural light then
`(1)/(2)I_(0) =` intensity of light emerging from the polarizer nicol.
Suppose the quartz plate rotates this light by `varphi`, then the analyser will transmit
`(1)/(2)I_(0)cos^(2) (90 - varphi)`
`= (1)/(2)I_(0)sin^(2) varphi`
of this intensity. Hence `eta I_(0) = (1)/(2)I_(0)sin^(2) varphi`
or `varphi = sin^(-1) sqrt (2eta)`
But `varphi = alpha d` so
`d_(min) = (1)/(alpha) sin^(-1) sqrt(2 eta)`
for minimum `d` we must take the principle value of inverse sine. Thus using `alpha = 17 ang deg//nm`.
`d_(min) = 2.99 mm`