Definition of radius of sight optics

[Werg22]

"A fish is 3 meters underwater. What is the fish's sight radius of the external world?"

My question is what is the definition of this radius? What does it mean?

 

[Doc Al]

If you consider the refraction of light entering the water, you'll realize that any light that makes it to the fish's eye must enter the water within a circle at the water's surface. Find the radius of that circle.

 

[andrevdh]

If you look up towards the surface from within a swimming pool you will find that a circular region appears on the surface of the pool where you can see what is outside of the pool. Outside of this circular region you cannot see what is above the surface. This circle is centered on a point right above you and its existence can be explained with total internal reflection.

 

[Werg22]

I'm not sure if I understand (a quick drawing might be helpfull) but if I do, I have to calculate the distance between the fish and a point on the surface of the water where a ray that is parrallel to a straight line between the fish and that same point is not refracted?

 

[andrevdh]

A ray traveling from the eye of the fish to point p on the surface are incident at the surface of the ocean at the critical incident angle $\alpha_c$. Beyond this point total internal reflection will take place and rays cannot enter from above the surface into the eye of the fish.

 

[Doc Al]

andrevdh has provided a picture that will surely help, but I suggest you try the following to understand what's going on. Imagine the fish's head to be replaced by a light bulb emitting light in all directions. Trace the path of several of these rays as they refract into the air (apply Snell's law of refraction) and see what you can deduce. Start with rays that are nearly vertical (small angle of incidence) and then slowly increase the angle of incidence and see what happens to the refracted ray in air. You'll find, as andrevdh states, that after some critical angle of incidence the light from the bulb (on the fish's head) won't make it out of the water.

Then realize that the light coming from outside the water to the fish's eye must follow these same paths but in reverse.