A ray of light is incident on the surface of a sphere of refractive index `sqrt(7)/(2)` other half of sphere is silvered. After refraction it is reflected and then refracted out of sphere again such that the total deviation is minimum then total deviation of the ray will be `(sin41_(0)-sqrt((3)/(7)))`

Options

A. `102^(0)`
B. `98^(0)`
C. `136^(0)`
D. `163^(0)`

Answer

C

Explanation

deviation at 'P' `= i-r`
deviation at 'Q' `= pi-2r`
deviation at 'R' =i-r
Total deviation
`delta =2(i-r)+pi-2r`
`=pi+2i-4r`
for minimum deviation
`(ddelta)/(di)=0 implies 2-4(dr)/(di)=0`
`implies (dr)/(di)=(1)/(2)`
`therefore musin r= sin i`
`mucos r dr = cosi di`
` (dr)/(di)=(cos i)/(mu cos r)=(1)/(2)`
` mu^(2) cos^(2)r=(cos^(2)i)4`
`mu^(2)(1-sin^(2)r)=4cos^(2)i`
`mu^(2)(1-(sin^(2)i)/(mu^(2)))=4cos^(2)i`
`implies cos^(2)i=(1)/(4)`
`cosi=(1)/(2)`
`i=60`
`thereforesinr=sqrt((3)/(7))impliesr=41^(0)`
`therefore delta =pi+2i-4r`
`=180+2(60)-4(41)`
`=136^(0)`