A direct-vision prism is made out of three prisms, each with a refracting angle of `phi=60^(@)` , attached to each other as shown in Figure. Light of a certain wavelength is incident on the first prism. The angle of incidence is `30^(@)` and the ray leaves the third prism parallel to the direction of incidence. The refractive index of the glass of the first and third prisms, is 1.5. Find the refractive index of the material of the middle prism. `(sqrt(6)=2.45)`

Answer

`(2sqrt(6)-1)/(2)`

Explanation

For middle prism, `r+r=A`
`2r=60^(@)`
`rArr r=30^(@)`
Refraction from interface of `I^(st)` surface
`1 sin30^@=1.5 sintheta`
`1.5sin(60-theta)=musin30^(@)`
`rArr sintheta=1//3 and cos theta=sqrt(8)//3`
Refraction from interface of `I^(st)` & `II^(st)` prism,
`1.5sin60^(@) costheta-1.5cos60^(@)sintheta=musin30^(@)`
`(1.5)xx(sqrt(3))/(2)xx(2sqrt(2))/(3)-1.5xx(1)/(2)xx(1)/(3)=muxx(1)/(2)`
`sqrt((3)/(2))-(1)/(4)=mu(1)/(2)`
`rArr mu=sqrt(6)-(1)/(2)=(2sqrt(6)-1)/(2)=1.95` ltbr