Question

I don't understand Question 5-9

Any information on how to answer these?

The speed of light depends on the medium through which it travels, and it is generally slower in the denser media. Fermat's principle in optics asserts that light travels from one point to another along a path for which the time of travel is minimum. In a uniform medium, where the speed of light remains constant, "shortest time" means "shortest path", and the ray of light follows a straight line. In this question you are asked to derive the Snell's Law of Refraction: sin theta_1/c_1 = sin theta_2/c_2 which relates the angle of incidence theta_1 of a ray of light passing through a uniform medium 1 (e.g. air) where the speed of light is c_1 and the angle of refraction theta_2 when ray of light continues passing through a uniform medium 2 (e.g. water) where the speed of light is c_2. To guide you through your discovery, follow the following steps. Suppose the light ray starts at the point A = (0, a) in an uniform medium 1 where the speed of light is c_1 and then passes through an uniform medium 2 where the speed of light is c_2 reaching point B = (d, -b). The line separating the two media is the x-axis; see the figure below. What will be the shortest path of the light ray from A to B? (It will consist of two segments.) What is the time t_1 needed for the light ray to trawl along the first segment of the path (in the medium 1)? What is the time t_2 needed for the light ray to travel along the second segment of the path (in the medium 2)? What is the total time t that takes the light ray to trawl from A to B? How Fermat's principle applies to t? Find dt/dx. The expression in the answer to item above should have two terms. In each, express part of the term using sin theta_1 and sin theta_2, respectively. (Recall your Trigonometric Identities.) Now is the tricky part, so we use this as a fact. It can be shown (for this we would need a bit more calculus knowledge) that the function dt/dx has a zero at a unique x on (0, d). Use the fact from item above to justify your discovery, i.e., prove (1).

Answer 1

5) Fermat's principle says that the path taken by the light is the path that requires the least time to travel. Thus Fermat's principal relate the distance to the least time 6) The value of is dt d(+t,) dx ax 7) We can write this as dt dt sine sine dx c 8) From first derivative test we know that a function has its maxima or minima at the point where the derivative of the function is zero. From Fermat's we know that the path is a minima and hence we have0to be there in the range 0 to d in x axis dt ax 9) We have dt sin&, sin 6 sina sin θ. sin sin6,