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:
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
angle of incidence with the normal of O. Assume:
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.
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.
These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.