Refer to Example 32.3 and use the numbers provided there....

Refer to Example 32.3 and use the numbers provided there. Further, assume that your eyes are at a height of $1.70 \mathrm{~m}$ above the water. a) Calculate the time it takes for light to travel on the path from the fish to your eyes. b) Calculate the time light would take on a straight-line path from the fish to your eyes. c) Calculate the time light would take on a path from the fish vertically upward to the water surface and then straight to your eyes. d) Calculate the time light would take on the straight-line path from the apparent location of the fish to your eyes. e) What can you say about Fermat's Principle from the results of parts (a) through (d)?

Refer to Example 32.3 and use the numbers provided there. Further, assume that your eyes are at a height of $1.70 \mathrm{~m}$ above the water.
a) Calculate the time it takes for light to travel on the path from the fish to
your eyes.
b) Calculate the time light would take on a straight-line path from the fish
to your eyes.
c) Calculate the time light would take on a path from the fish vertically upward to the water surface and then straight to your eyes.
d) Calculate the time light would take on the straight-line path from the apparent location of the fish to your eyes.
e) What can you say about Fermat's Principle from the results of parts (a) through (d)?

Key Concepts

Fermat's Principle

Fermat’s Principle is the fundamental concept stating that light travels between two points along the path that requires the least time, rather than necessarily the shortest geometric distance. This principle is used to determine the actual path light will take when transitioning between different media and underpins the derivation of the laws of reflection and refraction.

Refractive Index

The refractive index is a measure of how much the speed of light is reduced in a given medium as compared to vacuum. It plays a crucial role in optical problems by determining the speed of light in different materials, thereby affecting the time it takes for light to travel along various paths.

Optical Path Length

Optical path length is defined as the product of the physical distance traveled by light and the refractive index of the medium through which it travels. This concept is essential for analyzing situations where light moves through different media, as it provides a way to compare the effective distances, taking into account the speed alterations caused by the medium.

Geometry of Light Paths

The geometry of light paths involves the analysis of the actual trajectories light follows when moving in space, including the effects of bending due to refraction at interfaces. Understanding the configuration of these paths, such as curved versus straight-line segments, is vital for calculating the travel times according to both physical distances and optical principles.

Transcript

00:01 For this problem on the topic of geometric optics, we want to refer to the given example and use the numbers provided to calculate the time it takes for light to travel on the given path from the fish to the eyes, assuming that the eyes are at a height of 1 .7 meters above the water.

00:19 Now this problem is for the time it takes light to travel between two points which are given via the various parts.

00:26 And the situation is depicted in the diagram.

00:29 Now the time it takes for a light rate of travel its part, t is d over v, and the speed of the ray in the medium v is c over n.

00:38 So for part a, the time of travel for light along part a, ta, is l1 over v1, plus l2 over v2, which is 1 over c, times the refractive index of water and w times d1 over cosine theta 1 plus the refractive index of a and a times d2 over cosine theta 2 now if we put in our values we get this to be 1 over 3 times 10 to the power 8 meters per second so this speed of light and vacuum into the refractive index of water is 1 .33 into 1 .5 meters divided by the cosine of 32 degrees plus the refractive index of air which is 1 into 1 .7 meters divided by the cosine of 45 degrees which gives us the time along path a, t, t, a, to be 1 .59 times 10 to the minus 8 seconds.

02:19 Now for part b, the time of travel for light along part b is tb, and tb is l1 over v1 plus l2 over v2, which we can write as 1 over c into nwd1 over cosine theta 3 plus n a d2 over cosine theta 3 which we can write as n w d1 plus n a d2 divided by c cosine theta 3 the angle theta 3 can be found from the 10 of theta 3 is equal to w1 plus w2 divided by d1 plus d2, which is d1 10 theta 1 plus d2, and so we get t b1 to be n w d2, 10 theta 2, divided by d1 plus d2.

03:47 And so we get tb to be nwd1 plus n .a .d2 divided by the cosine of the arc tan, d1, plus d2, all divided by d1 plus d2, this multiplied by c.

04:31 And so if we put our values in, we get tb to be 1 .33 into 1 .5 meters and we'll suppress the units plus 1 times 1 .7, all divided by the cosine, the arc tan of 1 .5 meters, 10, of 32 degrees plus 1 .7 meters times the tern of 45 degrees all over 1 .5 meters plus 1 .7 meters and all of this multiplied by c which is 3 times 10 to the power 8 meters per second and this gives a time for the ray to travel along path b t b to be 1 .60 times 10 to the minus 8 seconds.

06:00 Now for part c, be time of travel for the light along this part, tc, is equal to nw, d1 over c, plus na over c times the square root of d2 squared plus w1 plus w1 plus...

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