A point source of light is placed at a distance `h` below the surface of a large deep lake.
(a) Show that the fraction `f` of the light energy that escapes directly from the water surface is independent of `h` and is given by `f=(1)/(2)-(1)/(2n)sqrt(n^(2)-1)` where `n` is the index of refraction of water.
(Note: Absorption within the water and reflection at the surface, except where it is total, have been neglected)
(b) Evaluate this ratio for `n=4//3`.

Answer

(b) `(4-sqrt7)//8`

Explanation

Let the point source of power (`P` watts) be situated at `L`.
All the rays (from source) entering the cone of half angle `C` (critical angle) about `LM` escape out of water. The power crossing the ring of radus `x` and with `dx` on surface of water is `dp=(I cos theta)2pix dx=(P)/(4pi(h sec theta)^(2)) cos theta2pix dx` (`I=` intensity of light at point on annular ring.) `x=h tan theta dx=h sec^(2)theta dtheta`
`therefore dP=(P)/(2) sin theta dtheta` `therefore` power escaping out of water is
`P=intdP=int_(0)^(C)(P)/(2) sinthetad theta=(P)/(2)(1-cosC)`
where `C` is critical angle
`therefore` fraction of power escaped `=(1)/(2)(1-cos C)=(1)/(2)[1-(sqrt(n^(2)-1))/(n)]`
(b) for `n=4//3,f=(1)/(2)[1-(sqrt(((4)/(3))^(2)-1))/((4)/(3))]`