A jewel thief decides to hide a stolen diamond by placing it at the bottom of a crystal-clear fountain. He places a circular piece of wood on the surface of the water and anchors it directly above the diamond at the bottom of the fountain, as shown below. If the fountain is x = 1.65 m deep, find the minimum diameter of the piece of wood that would prevent the diamond from being seen from outside the water.
The outer edge of the circular wood piece should be at a location such that rays from the diamond to the surface correspond to the critical angle for total internal reflection. Let R be the radius of the wood piece and =x the depth. The incidence angle of a ray striking the edge will be
arctan R/x. This must equal the critical angle, which is arcsin 1/N, where N is the index of refraction.
R/x = tan (critical angle)
The critical angle is arcsin 1/1.33 = 48.7 degrees. Its tangent is 1.14
Therefore R/x = 1.14
The diameter you want us 2R. You know that x = 1.65 m.
Rays that go beyond the edge of the wood disc will exceed the critical and will be totaly refkected at the water's surface. Therefore the diamond cannot be seen from above.