 # 2. Fermat Revisited (80 points total) An solid in the shape of an ...

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2. Fermat Revisited (80 points total) An solid in the shape of an equilateral triangle of side S increases in optical density as y increases, according to: n(y)=3.30y+1.20 where y is the vertical distance, in meters. A beam of light traveling in air (n = 1.00) in the x-y plane strikes the solid at the origin, located at the center of the base, making an x angle of incidence with the normal of O. Assume: e, n varies only with y, not with X. The light travels exclusively in the x-y plane. The block is wide enough so the light leaves through the top and not through a side. a) Use Fermat's principle and Euler's equation to find the equation for the path of the light y(x) in the tank. There will be two "undetermined" constants. (25 points) HINT: If the equations become complicated (and they will), you have two choices: - you can solve them as they are (and they are solvable!), or - you can remember the alternative ways to solve calculus of variations problems. HINT: The answer should come out as a hyperbolic function. b) Use 01 = 55.2°, and find the values for the "undetermined" constants in y(x). Write the equation for y(x) and y(x). (20 points) (c) (25 points) i. Assume S = 50.0 cm (0.500 m). Use Excel to draw a picture of the path of a light ray from y = 0 to where it hits the right side of the triangle. On the same graph, draw the right boundary of the triangle. The light ray should not go over the line; the equations change once the light enters the air. (10 points) ii. On the same graph, draw a line showing the path a light ray would follow if n = 1.20 through the entire block. Again, this ray should not pass over the boundary line. (5 points) iii. Now: expand the axes lo locate the point at which the light ray in the solid hits the right boundary. Print the graph, and give both the x-coordinate and y-coordinate to five (5) significant digits. (10 points) c) Analytically (not graphically), find the angle of incidence at the top of the block; that is, the angle between the normal and the refracted ray when the light re-enters the air. (10 points)

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