Lifting Stones
12th graders are usually already aware of refractive lifting. They have stuck pencils into water or have reached for something under water that was not in the position they expected. What is subtle perhaps is that as ripples wash toward shore, the lifting effect is intensified, and the objects appear to shake or move, and are magnified for a moment as the ripples pass over. We believe that ripples are local gatherings of more water than their immediate surroundings, and so it seems that this lifting effect depends on the amount of water through which we are looking. How can we measure the visual depth to the stone to compare with the dry depth to the tangible stone? In the field this would be nearly impossible. However, in the lab we can set up a controlled pond (fish tank) and do a beautiful demonstration to solve this very problem. As the students are probably aware (or quickly determine), we cannot simply insert a ruler into the water to measure the depth to a quarter resting there. The ruler’s inches will be compressed (lifted) so that they won’t correspond to inches on dry land or out of the tank. Could we simply find a conversion factor for the underwater ruler? No, in fact, deeper inches appear more compressed than shallower ones. The whole visual scale of the submerged ruler is affected and cannot be easily related to a tangible one outside the water. This realization has a connection to Special Relativity, which we cover at the end of the block when we will be forced to accept the compression of lengths and time (!) between reference frames under special conditions. Quite a dilemma–we would like to characterize the lifting effect by comparing the visual quarter to its tangible counterpart, but how? How do we carry the tangible measurements from out of the water into the visual, refracted space in a way that is usefully comparable?

Think back to something we already have determined, a reflected image is a visual copy of its tangible counterpart with dimensional symmetry. Therefore, we have an extremely elegant way of seeing both a ruler with measurements that correspond exactly to those in dry space (air) and seeing the refracted quarter on the bottom of the fish tank–we can reflect a ruler held just at the water’s surface into the water and compare the two depths (we can measure the drained fish tank to determine its dry, visually unaltered depth, or insert a ruler and mark the water surface on it). What else will we need to know? If I move while observing the quarter, does it remain in the same place, or is it lifted different amounts? Obviously, as with the ripples, the more water through which we look, the more lifting occurs. The students will figure out that we need to know one more measurement and to determine a way to mark that (distance to the ‘piercing point’ through which we are looking). We used a framing square with a string-suspended nail set exactly to point to the water surface. The observer would direct an assistant to position the square until it pointed to the place through which he was looking. This way (as opposed to simply using the edge of the tank) we were able to look through several different angles and compare results for the refractive index.

Here we have already made practical use of the first visual lawfulness of the block–the exact correspondence of the reflected space to the tangible one. The beauty of this solution can hardly be overstated. From the two depths it is possible to derive mathematical relationships–the familiar Snell’s Law. Our measurements in the fish tank yielded results for water that are very close to those listed in textbooks. After a few examples, students will notice that the angle to the surface normal in the less-dense medium is always greater than in the denser. Looking at a sketch, we might predict something unusual–that at some point the exterior angle will be 90o while the interior one is not. “Freddy the Fish” will not be able to see out of the lake! Some students may have experienced this while diving, and we can now discuss total internal reflection. From this, a discussion of fiber optics is a nice application.

What is the index of refraction really about? An optical phenomenon has indicated a material property of the medium into which we are looking. Each transparent medium into which we might look has a certain material property that we can quantify with the index of refraction. It may have to do with density, but regardless, it is an index assigned to various materials of an optical characteristic. It is a relative scale. Remember the triboelectric series from 11th grade? The temperature scale of 9th? There is no absolute scale for indexing the refractive effect of various materials, it is simply arranged relative to air (the most common medium through which we look). Implied by our observations is the further conclusion that the amount of material through which we look is an important factor, in other words, the amount of visual lifting is both dependent on the material and on the amount of that material.

Note that both reflection and refraction indicate the presence of ripples–a disturbance in the reflection indicates surface disturbance, while an intensification of the lifting indicates changes in depth. Finally it is possible, though very difficult, to see colored fringes around the stones at the pond edge. Most students have a hard time experiencing this. Some students have commented that they can see “rainbows” at the light/dark edges of the ripples on the water under certain conditions (in certain positions relative to the sun). We will draw out the colors in a more effective way next.