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The Optics Book
The law of refraction is n1sinq1= n2sinq2
To prove it, consider Fig 17 which shows two fixed points A and B in two different media and a refracting ray APB connecting them. The time t for the ray to travel from A to B is given by
t=(L1/ V1)+(L2/ V2)
Using the relation n=c/v we can write this as
where L is the optical path length defined as
For any light ray travelling through successive media, the optical path length is the sum of the products of the geometrical path length and the index of refraction of that medium. The equation ln=l/n shows that the optical wave length is equal to the length that this same number of waves would have if the medium were a vacuum. Do not confuse the optical path length with the geometrical path length which is L1+ L2 for the ray of Fig 17.
Fermat's principle requires that the time t for the light to travel the path APB must be a minimum( or a maximum or must remain unchanged) which in turn requires that x be chosen so that dt/dx=0. The optical path length in Fig 17 is
L=n1L1+ n2L2=n1(square root of a2+ x2)+n2(square root of b2+(d-x)2)
Substituting this result into t=[(n1L1+ n2L2]/c=L/c and differentiating, we obtain
=n1/2c(a2+ x2)-1/2(2x)+ n2/2c[b2+ (d-x)2]-1/2 (2)(d-x)(x-1)=0
Which we can write as
n1 [x/( square root of a2+ x2)]=n2 [(d-x)/ (square root of b2+(d-x)2)]
Comparision with Fig 17 shows that we can write as n1sinq1= n2sinq2 which is the law of refraction.
|The Optics. Made by Karen, Timothy and, César for ThinkQuest . 1999 - 2000 All rights reserved|