Light passes through a system of two crossed Nicol prisms between which a quartz plate cut at right angles to its optical axis is placed. Determine the minimum thickness of the plate which allows light of wavelength `436 nm` to be complately cut off by the system and transmits half the light of wavelength `497 nm`. The specific rotation constant of quartz for these wavelength is equal to `41.5` and `31.1` angualr degress per mm respectively.

Explanation

For light of wavelength `436 nm`
`41.5^(@) xx d = k xx 180^(@) = 2k xx 90^(@)`
(Light will be completely cut off when the quartz plate rotates the plane of polarization by a multiple of `180^(@)`.) Here `d =` thickness of quartz plate is mm.
For natural incident light, half the light will be transmitted when the quartz rotates light by an odd multiple of `90^(@)`. Thus
`31.1^(@) xx d = (2k + 1) xx 90^(@)`
Now `(41.5)/(31.1) = 1.3344 ~~(4)/(3)`
Thus `k = 2` and `k' = 1` and
`d = (4 xx 90)/(41.5) = 8.67 mm`.