A ray of light is incident on the surface of a transparent sphere of `R.I. mu = (sqrt(7))/(2)`. After refraction is is internally reflected and refracted out of the sphere as shown in the figure. Find the angle of incidence 'I' so that the total deviation produced is minimum. 'C' is the centre of the sphere.
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Answer

`60^(@)`

Explanation

From figure total deviation
`delta=(i-r)+(180^(@)-2r)+(i-r)`
`=2(i-r)+(180^(@)-2r)`
`=2i + 180^(@) - 4r`
for minimum `delta`
`(d delta)/(di)=2+0-4 (dr)/(di)=0`
`rArr (dr)/(di)=(1)/(2)`....(1)
by snell law at refracting surface
`1 sini =mu sin r`
`rArr mu sin r =sini`
`rArr mu(cos r)(dr)/(di)=cos i`
`rArr mu(cos r)xx (1)/(2)=cos i`
`rArr cos r=(2 cos i)/(mu)`
`:. sqrt(1-sin^(2)r)=(2 cos i)/(mu)`
`rArr sqrt(1-((sin i)/(mu))^(2)) =(2 cosi)/(mu) rArr i = 60^(@)`.