The Crystal Ball Mystery · AI Generator #GEN-71A7EFCE
Physics · optics
Problem
A magician places a small LED light source at the bottom of a clear crystal sphere of radius and refractive index . The sphere sits on a black table in a dark room. When the LED is turned on, observers notice that only a circular region on the bottom of the sphere appears to glow, while the curved sides of the sphere remain completely dark. The LED emits light uniformly in all directions. Calculate the radius of the glowing circular region on the bottom surface of the sphere.Intuitive explanation
Total internal reflection is the key here—light rays from the LED can only escape the sphere if they hit the bottom surface at an angle shallow enough to refract out rather than bounce back inside. Beyond a critical angle, all light is trapped by the sphere's high refractive index. The glowing circle's radius is determined by finding which rays from the LED can actually exit through the bottom surface; rays hitting at steeper angles undergo total internal reflection and never reach the table, leaving the sides dark.
Intuitive explanation
Total internal reflection is the key here—light rays from the LED can only escape the sphere if they hit the bottom surface at an angle shallow enough to refract out rather than bounce back inside. Beyond a critical angle, all light is trapped by the sphere's high refractive index. The glowing circle's radius is determined by finding which rays from the LED can actually exit through the bottom surface; rays hitting at steeper angles undergo total internal reflection and never reach the table, leaving the sides dark.