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Section3Refraction of Particles

We will now set up an experiment using a mechanical analogue to the refraction of light and examine its consequences. To begin, consider the natural philosophers of the seventeenth and eighteenth centuries, many of whom believed in the particular, and especially corpuscular, nature of light. For them, light could be accelerated or decelerated in a direction perpendicular to the interface between two mediums. In our mechanical analogue, we will represent the two media by two horizontal, planar surfaces at different heights, and our light particle will be represented by a rolling steel ball. On a given medium, the light ball will travel at an (essentially) constant velocity, while our interface will be represented by a short, sloping plane that will cause the light ball to accelerate as it passes from the higher to lower medium.

Our experiment will proceed as follows:

Arrange two horizontal surfaces so that they are level and connect them by a short but not very steep plane. Place one sheet of white paper on each of the surfaces. Over each sheet of white paper, place one sheet of carbon paper face down. Use a short launching ramp to start a steel ball rolling along the upper surface. Repeat this procedure to obtain tracks of different angles of incidence at the interface. It is necessary to start the ball from the same height on the ramp each time to ensure a consistent speeds, especially as the speed of light in a given medium is relatively constant and so we must try to emulate that if our data is to be at all helpful. For each track, measure the angles of incidence and refraction from the normals to the horizontal edges of the sloping plane.

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Figure3.1Experimental \(\theta_i\) vs \(\theta\) for a mechanical analogue to light travel.

We can see that this data follows a similar pattern as that of refraction, and as such our “index of refraction”, which is a ratio in the speeds of the two mediums is 1.6. As such, we can say that our upper plane is analogous to water while the lower is analogous to air in our refractive experiments. As such, we would expect the speed of light to be faster in air than in water.

We have shown that light will travel paths of extreme time where the time is dependent upon both the media being traveled within, as well as the path taken. As such, we have the general form for the Principle of Extremes, whereby all true paths will be such that \(n_i \cdot \theta_i = n_r \cdot \theta_r\text{.}\) This unifies our rules for planar, spherical, and elliptical mirrors as well as refraction of light.