Percentage of Energy: Total Internal Reflection

[Cesium]

Homework Statement

A point source of light is at depth h below the surface of a large and deep lake. Show that the fraction, f, of the light energy that escapes directly form the water surface is independent of h and is given by

f=0.5(1-\sqrt{1-1/n^2})

where n is the index of refraction of the water. Absorption within the water and reflection at the surface (except where it is total) have been neglected.

Homework Equations

n₁sin(theta1) = n₂sin(theta2)
E=hv?

The Attempt at a Solution

First step would be to find the critical angle:
nsin(theta1)=sin(90) (n2=1 for air)
theta1 = arcsin(1/n)

So anything with an incident angle greater than theta1, will be totally reflected. The problem seems to assume that all of the light with angle less than theta1 will be refracted (even though in reality some will be reflected, too).

So here's where I am going wrong: I tried to assume that all of the light would be equally split up between all angles of radiation. So out of 180 degrees, arcsin(1/n) would escape.
So my fraction and answer would be arcsin(1/n)/180, but that's obviously wrong.

I am having trouble relating energy to this. Thanks in advance.

 

[kuruman]

You have to think in 3-dimensions. The light escaping is inside a cone of half angle equal to the critical angle. Therefore the fraction escaping is the solid angle subtended by the cone divided by the total solid angle which in this case is 2π.

 

[Cesium]

Wow, thank you very much. I figured out the problem with the information you gave. I had never even heard of a solid angle before so I had some learning to do.