Question

A light beam starting from a point 퐴 in the air takes the path of least time in order to reach a point 퐵 under water. Use this principle to prove Snell’s law. Assume the speed of light is 푐/푛 in water, where 푐 is the speed of light in air and 푛 is the refractive index

Answer 1

starting from point A and reach point B. assume speed of light is c/n in water, where c is spped of light, n is refractive index

Answer 2

its a differnetial equations problem

Answer 3

Tough one. Begin by setting up a diagram with a horizontal line representing the border between the air and the water. A beam of light will strike this line at an angle and be bent downward by the refraction of the water. Draw a line perpendicular to the border between the air and the water (in other words, the normal) at the point where the light crosses that border. Choose a point on the "air" side of the light path and another point on the "water" side. Choose variables for the vertical distances between these points and the air-water border. You also need a variable for the horizontal distance between your "air" point and your "water" point, and another for the horizontal distance between your "air" point and the point where the beam of light crosses the air-water boundary. Now you're in position to create a formula for the total distance traveled by the light, which will be the sum of two paths, one from air-point to boundary, and one from boundary to water-point. Find the derivative of that formula, and you should see that the derivative equates to zero when the sines of the angles of incidence are proportional to the speed of light in the medium.