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` `

Answer

C

Explanation

The light entering the rod does not emerge from the curved surface of the rod when the angle `(90^circ-beta)` is greater than the critical angle.
i.e., `mule(1)/(sinC)` Where C is the critical angle.
Here, `C=90-beta`
`implies``mule(1)/(sin(90^circ-beta))impliesmule(1)/(cosbeta)`
As a limiting case,`mu=(1)/(cosbeta)` ....(i)
Applying Snell's law at A
`mu=(sinalpha)/(sinbeta)impliessinbeta=(sinalpha)/(mu)` ....(ii)
NOTE: The smallest angle of incidence on the curved surface is when `alpha=(pi)/(2). This can be taken as a limiting case for angle of incidence on plane surface. From (ii)
`sinbeta=(sinpi/(2))/(mu)impliesmu=(1)/(sinbeta)` ...(iii)
From (i) and (ii), `sinbeta=cosbeta` `implies``beta=45^circ`
`implies` mu=(1)/(cos45^(circ))=(1)/((1)/sqrt2)implies mu=sqrt2`
This is the least value of the refractive index of rod for light entering the rod and not leaving it from the curved surface.