Pierre de Fermat (16011665) showed that whenever light

Chapter 35, Problem 84

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Pierre de Fermat (16011665) showed that whenever light travels from one point to another, its actual path is the path that requires the smallest time interval. This statement is known as Fermats principle. The simplest example is for light propagating in a homogeneous medium. It moves in a straight line because a straight line is the shortest distance between two points. Derive Snells law of refraction from Fermats principle. Proceed as follows. In Figure P35.84, a light ray travels from point P in medium 1 to point Q in medium 2. The two points are, respectively, at perpendicular distances a and b from the interface. The displacement from P to Q has the component d parallel to the interface, and we let x represent the coordinate of the point where the ray enters the second medium. Let t 5 0 be the instant the light starts from P. (a) Show that the time at which the light arrives at Q is t 5 r1 v1 1 r2 v 2 5 n1"a 2 1 x 2 c 1 n2"b 2 1 1d 2 x2 2 c (b) To obtain the value of x for which t has its minimum value, differentiate t with respect to x and set the derivative equal to zero. Show that the result implies n1x "a 2 1 x 2 5 n2 1d 2 x2 "b 2 1 1d 2 x2 2 (c) Show that this expression in turn gives Snells law, n1 sin u1 5 n2 sin u2