In an isosceles prism of prism angle `45^(@)`, it is found that when the angle of incidence is same as the prism angle, the emergent ray grazes the emergent surface. Find the refractive index of the material of the prism.For what angle of incidence the angle of deviation will be minimum ?

Options

A. `(2)^(1//2)`
B. `(3)^(1//2)`
C. `2`
D. `(5)^(1//2)`

Answer

D

Explanation

`i=(pi)/(2),e=(pi)/(4),A=(pi)/(4)`
`(sini)/(sin r_(1))=(sine)/(sinr_(2))=mu`
`rArr sinr_(1)=(1)/(mu) and sinr_(2)=(1)/(sqrt2mu)` Since `r_(1)+r_(2)=A=(pi)/(4) rArr r_(1)=(pi)/(4)-r_(2)`
`rArr sinr_(1) =(1)/(sqrt2)cosr_(2)- (1)/(sqrt2)sinr_(2)`
`=sqrt2 sinr_(1)+sinr_(2)=cosr_(2)`
`=(sqrt2)/(mu)+(1)/(sqrt2mu)=sqrt(1-(1)/(2mu^(2)))= (1)/(mu^(2))(2+(1)/(2)+2)=1-(1)/(2mu^(2))`
`= (1)/(mu^(2))((9)/(2)+(1)/(2))=1`
`mu^(2)=5rArr mu=sqrt5 `.