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The thin lens equation is 1/f = 1/do + 1/di, where f is focal length, do is object distance and di is image distance. A 50 mm lens with an object at 200 mm produces an image at 66.7 mm behind the lens. When the object is at exactly twice the focal length (100 mm for a 50 mm lens), the image forms at the same distance on the other side with a magnification of -1 (same size, inverted).

Snell's law relates the angle of incidence to the angle of refraction: n1 times sin(theta1) = n2 times sin(theta2). Light entering water (n = 1.33) from air (n = 1.00) at 45 degrees refracts to about 32 degrees. Total internal reflection occurs when light goes from a denser to a less dense medium at angles above the critical angle - for water to air, the critical angle is about 48.8 degrees.

Magnification equals the negative ratio of image distance to object distance (M = -di/do). A converging lens with an object at 30 cm and image at 60 cm gives M = -2, meaning the image is twice as large and inverted. For a camera with a 200 mm telephoto lens focused at 10 meters, the magnification is about -0.02 - the image on the sensor is 50 times smaller than the real object.

The mirror equation is identical to the thin lens equation: 1/f = 1/do + 1/di. For a concave mirror with a 20 cm focal length and an object at 30 cm, the image forms at 60 cm in front of the mirror (real, inverted, magnified 2 times). Convex mirrors always produce virtual, upright, reduced images - a car side mirror with f = -100 cm shows objects at roughly 1/3 their actual angular size.

Air has a refractive index of 1.0003, water is 1.33, crown glass is 1.52, flint glass is 1.62 and diamond is 2.42. Higher refractive index means light travels slower and bends more when entering the material. Diamond's high index is what creates its brilliance - light entering at most angles undergoes total internal reflection, bouncing around inside before exiting through the top facets.