Light is incident at an angle `alpha` on one planar end of a transparent cylindrical rod of refractive index `mu`. Determine the least value of `mu` so that the light entering the rod does not emerge from the curved surface of rod irrespective of the value of `alpha` `

Explanation

The ray of light is incident at A & it just gets reflected totally at B. Therefore incident angle at B is equal to critical angle given as `theta_(c)=sin^(-1) 1/n`
Sneil’s Law for refraction at A yields
`(sintheta)/(sinr) =nimpliessinr=(sintheta)/n …(1)`
Since `r + theta_(c) =90^@ implies sin(90^@ - theta_(c))=cos theta_(c)` For a ray not to come through the curved surface, `rle 90^@ - theta_(c)`
`impliessin r le sqrt(1-sin^(2)theta_(c)) le sqrt(1 - 1/n^(2)) ...(2)`
Eliminating sinr from (i) and (ii) we obtain
`(sintheta)/n le sqrt(1 - 1/n^(2)) implies sin theta le sqrt(n^(2)-1)`
`n le sqrt(1+ sin^(2)theta)`
`n_(min)= sqrt(2)`
Since `sintheta le 1`
`implies n_(min)=sqrt(2)`