Total internal reflection, underwater light

gigli

1. The problem statement, all variables and given/known data
A point source of light is at the bottom of a koi pond, at a depth of 0.525 meters. What is the radius of the circle of light formed on the water's surface? Take the index of refraction of water to be 1.33. Hint: Some of the light emitted experiences total internal reflection inside the water.

2. Relevant equations
sin(crit angle) = n2/n1
n1>n2
n for air = 1.00
n for water = 1.33

3. The attempt at a solution
So the critical angle for water to air is:
arcsin(1/1.33) = 48.7534666 degrees

This is where it all ends. There is no formula for distance or light radius. I am completely stuck at this point and cannot find anywhere in my book where I could derive the answer. I would deeply appreciate some help with this one!

Doc Al

Draw yourself a diagram of the light source with rays emanating in all directions. For various ray angles, what happens when the light reaches the water surface? At what angle with the vertical will they be totally reflected. Use a bit of trig to figure out the radius of the circle of light.

gigli

Wouldn't the radius of light only be a point at the surface if it is only a point at the bottom? I'm afraid I'm still not grasping this.

Doc Al

Did you draw a diagram of what's happening with the light rays?

Doc Al

Think of the point source of light sending out zillions of light rays in all directions. What happens to a light ray that goes straight up? One that goes 5 degrees from the vertical? 10 degrees? Etc.

gigli

Yes. When the angle is not quite to the critical angle it shines out of the water. when it reaches the critical angle it shines across the surface of the water, but none shines into the air, and when it is past the critical angle no light shines out of the water only deeper back into the water.

I think I see why it stays in the water and all that, but I fail to see how there is a radius of light at the surface, when it is only a point form the source. Sorry if I am being rather thick.

gigli

Ooooooooooooooooh. I got it now. Thank you so much.

The light was shining a cone straight up, but past 48 degrees it was reflecting back into the water. I was thinking it was a laser beam aimed at 48 degrees.

Thanks again! I got the right answer

Doc Al

Good. Now imagine a line going straight up from the light source to the water's surface. That spot on the surface will be the center of a circle. Think of all those rays of light hitting the surface. What's the farthest from the center that the light can reach when it passes the surface?
It's not a point from the source, it's a "point source" of light. That just means that the light source is small enough to treat as being a single point, instead of some extended light source. Imagine it as a small light bulb. (As compared, perhaps, to a 3 foot long flourescent light tube--which would not make a good "point source".)

Doc Al

Excellent. I knew it would click, sooner or later.

gigli

That last little bit really clarified what I was looking for. Thanks for the help Doc Al. Guess I'll try and get cracking on another stumper.