Snell's Law Suppose that a light sourc ? e at? is in a medium in which light travels at a?speed ? ? 1and the?point B ? is in a medium in which light travels at a?speed ? ? 2 (see figure). Using Fermat's Principle, which slates that light travels along the path that requires the minimum travel time (Exercise 59), show that the path taken between ? poin ? t? and B ? sat?isfie?s (sin? 1)? ?1 ?= (si? 2)?/?2.
Solution Step 1 Consider that a light source at Ais in medium, in which light travels at a speedV and1the pointBis in medium in which the light travels at a speedV . 2 Consider a light ray travelling from point Ato pointB, which has different refractive indices in different media. Now, the refractive index of any medium is defined as the speed of the light in vacuum to the speed of the light in that medium. In the above figure, Let, In right-angled triangleAMOuse the Pythagorean Theorem to obtain Similarly, in right-angled triangleBNC use the Pythagorean Theorem to obtain The time travel between the two points is the distance in each medium divided by the speed of light in that medium. That is, time travel is equal to Step 2 To obtain the minimum travel time, differentiate the expression of time with respect tox and equate to zero Equate the above expression equal to zero In right-angled triangles AMOandBNO, use the trigonometric formula of cosine function Substitute the values in the following equation to obtain the following: