Try it on paper first. A hint, the key idea and the worked solution are here when you need them.
A small lamp hangs below the surface of still water in a pool (), as in the picture. Its light shines in all directions. Seen from above, light comes out of the water only inside a bright circle on the surface. Find the critical angle, the radius of the circle and its area.
Snell's law is , with the angles measured from the normal. The critical angle for light that goes from to a smaller is . Light is totally reflected when the angle of incidence is larger than . In a material, .
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Given. for the water and for the air. The depth of the lamp is .
Plan. The edge of the lit circle is where a ray meets the surface at exactly the critical angle.
Step 1. The critical angle. , so .
Step 2. The radius. The ray to the edge goes from the lamp up to the surface at the angle to the vertical. So , and we want , so .
Step 3. The area. .
Answer: , and .
Check. The circle is wider than the depth of the lamp, which is right because is larger than .
What if? Rays that meet the surface outside the circle are totally reflected. They stay in the water and light up the walls and the bottom of the pool.
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