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 angle incidence at the curved surface of the cylindrical rod is given to be ` (90^(@) - beta)`
Light entering the rod at angle of incidence `alpha` on one planar end will not emerge from the curved surface if angle of incidence `(90^(@) - beta)` is greater than the ciritical angle. Refractive index is denoted by a `mu`.
`:. mu lt=1/(sinC)` or `mu lt= 1/(sin(90^(@)-beta))`
or `Plt= 1/cosp, `, As a limiting case,
A.=`1/(cos beta)` ...........(1)
According to Snell’s law , `mu = (sin alpha)/(sin beta)`
If `alpha = pi/2 `as a limiting case on planar end of rod, or `mu = 1/(sin beta)` .......(2)
From (1) and (2), considering the limiting case,
`cos beta = sin beta` or `beta=45^(@)` `:. mu = 1/(cos45^(@)) =sqrt2`
Hence refractive index `mu = sqrt2`
`:. ` The least value of `mu` so that the fight entering the rod does not emerge from the curved surface is `sqrt2`.