Fermat's Principle
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Physics 5B Winter 2009 Fermat’s Principle and the Laws of Reflection and Refraction Fermat’s principle states that “light travels between two points along the path that requires the least time, as compared to other nearby paths.” From Fermat’s principle, one can derive (a) the law of reflection [the angle of incidence is equal to the angle of reflection] and (b) the law of refraction [Snell’s law]. This is problem 32-81 on page 864 of Giancoli. The derivations are given below. (a) Consider the light ray shown in the figure. A ray of light starting at point A reflects off the surface at point P before arriving at point B, a horizontal distance l from point A. We calculate the length of each path and divide the length by the speed of light to determine the time required for the light to travel between the two points. Derivation of the laws of reflection and refraction ( ) 2 2 2 2 2 1 x h x h t c c − + + = + l To minimize the time we set the derivative of the time with respect to x equal to zero. We also use the definition of the sine as opposite side over hypotenuse to relate the lengths to the angles of incidence and reflection. ( ) ( ) ( ) ( ) 2 2 2 2 1 2 1 2 1 2 2 2 2 2 1 2 0 sin sin x dt x dx c x h c x h x x x h x h θ θ θ θ − − = = + → + − + − = → = → = + − + l l l l 1 h 2 h x x − l 1 θ 2 θ
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Fermat's Principle
Transcript of Fermat's Principle
Physics 5BWinter 2009
Fermats Principle and the Laws of Reflection and Refraction Fermats principle states that light travels between two points along the path that requires the least time, as compared to other nearby paths.From Fermats principle, one can derive (a) the law of reflection [the angle of incidence is equal to the angle of reflection] and (b) the law of refraction [Snells law]. This is problem 32-81 on page 864 of Giancoli.The derivations are given below. (a) Consider the light ray shown in the figure.A ray of light starting at point A reflects off the surface at point P before arriving at point B, a horizontal distance l from point A.We calculate the length of each path and divide the length by the speed of light to determine the time required for the light to travel between the two points. Derivation of the laws of reflection and refraction ( )222 221x hx htc c ++= +l To minimize the time we set the derivative of the time with respect to x equal to zero.We also use the definition of the sine as opposite side over hypotenuse to relate the lengths to the angles of incidence and reflection. ( )( )( )( )2 2 22121 2 1 22 2 22120 sin sinxdt xdxc x hc x hxxx hx h = = + + += = =+ +llll 1h2hx x l12(b) Now we consider a light ray traveling from point A to point B in media with different indices of refraction, as shown in the figure.The time to travel between the two points is the distance in each medium divided by the speed of light in that medium. ( )222 2211 2x hx htcn cn ++= +l To minimize the time we set the derivative of the time with respect to x equal to zero.We also use the definition of the sine as opposite side over hypotenuse to relate the lengths to the angles of incidence and reflection. ( )( )( )( )2 21 11 1 2 22 2 2 2 2 22 21 12 20 sin sinn x n xd t n x n xn ndxc x h x hc x h x h = = + = =+ + + +l ll l 2008 Pearson Education, Inc., Upper Saddle River, NJ.All rights reserved.This material is protected under all copyright laws as the currently exist.No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 1n1h12n2h2l-x l
