In this problem, we are going to use Fermat\'s principle to derive Snell\'s law
ID: 2077718 • Letter: I
Question
In this problem, we are going to use Fermat's principle to derive Snell's law by minimizing the amount of time it takes for a ray of light to travel from point A to point B in the picture below. Notice that we have let x be the horizontal distance between point A and the point where the ray intersects the boundary between the two substances. a). What is the total distance traveled by the ray in the material with index of refraction n_a? What about the total distance traveled by the ray in the material with index of refraction n_b? Your answers should be in terms of D, d_A, d_B, and x. b). Use your answer to a) and the definition of the Index of refraction to find the time it takes the ray to travel from point A to point B. c). Now use calculus to find an equation that gives the minimum of the expression you found in part b) with respect to the variable x. (You do not need to solve this expression for x). d). Using trigonometry and recalling that all our angles are measured relative to the surface normal, find expressions for sin theta_A and sin theta_B in terms of the x, D, d_A, and d_B. e). Now combine your results from c) and d) to write the condition on the angles theta_A and theta_B that minimizes the time needed to travel between points A and B. This result should look familiar!Explanation / Answer
a) in medium a
d1 = sqrt(da^2 + x^2)
in medium b
d2 = sqrt(db^2 + (D-x)^2)
b) Total time taken = ta + tb
= d1/va + d2/vb where va = c/na and vb = c/nb
= sqrt(da^2 + x^2)*na/c + sqrt(db^2 + (D-x)^2)*nb/c
c) To find minimum time
dt/dx = 0
=> d/dx(sqrt(da^2 + x^2)*na/c + sqrt(db^2 + (D-x)^2)*nb/c) = 0
=> (na/c) 2 * x/sqrt(da^2 + x^2) - (nb/c) 2 * (D-x)/sqrt(db^2 + (D-x)^2) = 0
d) sin (theta A) = x/sqrt(x^2 + da^2)
sin (theta B) = (D-x)/sqrt(x^2 + da^2)
e) From c) and d)
na * sin (theta A) - nb * sin (theta B) = 0
=> na * sin (theta A) = nb * sin (theta B)
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