Almost everyone has noticed that things in glasses of water look different from how they look in air. And you've seen things in bathtubs.
But few of us have taken a close look at things that stick out of a fishtank, especially from multiple perspectives.
Have a look at the video below. An action figure is placed in a plastic tank. The tank is turned, so you can see it from multiple perspectives. Then, it's filled with water, and the tank is turned again. What do you think will happen?
What's Going On Here?
Sure, it's cool to see new things; but it's even better to use those things as an opportunity to learn how the world works.
After all...
How can your head be on a stick?
How can you have a stick for a head?
It all has to do with refraction[1]From "re-" ("back") and "frangere" ("to break"). A beam of light seems "broken" at the boundary between materials. Try remembering "fracture" if you have "refraction" as a vocabulary word on a test. - which is a technical way of saying that light changes direction when it passes from one material to another material. It's also a clear example of the fact that trigonometry is useful for describing nature, even in situations that have nothing to do with circles or triangles.
Just the concepts
If you want a conceptual understanding of how this works, you need to make some observations about light bending when it goes from water to air.[2]The camera was able to "see" Han because light from the room bounced off of him, traveled through the water and/or air, and got to the camera. The camera does not shoot out rays of light (and neither do your eyes). (See Can We Believe Our Eyes?, starting around 11:15.) (If you want to measure exactly how much, you need Snell's Law.)
Any time light passes from one material to another material, it changes direction.
Why does light change direction when it passes from one material to another? Many would say that it's because light changes speed when it passes from one material to another. But... why does light change speed when it goes to another material? And why would that mean the light bends at all? And why, if it bends, does it bend the direction it does, and by the amount it does?
These are all good questions. And you can get deep into the nature of light very quickly by following up on these questions. We won't do that here (though we may make another page, if we get enough requests). For now, let's just observe the fact that it happens, and describe what we see happening.
Have a look at the video below. When light comes "face-on" to a surface, it keeps going straight.[3]Actually, some light also gets reflected. How much gets reflected depends, in part, on the material the light starts in, the material the light goes into, and how far from "face-on" the light is at the boundary. There are equations for that too - which good ray-tracing software uses to create realistic scenes. See the discussion at The Physics Classroom. But when it arrives at an angle, it takes a sudden turn at the boundary between the two materials. And the path of the light is always closer to "face-on" to the boundary when it's in water than when it's in air.
This allows us to understand overhead views of the same situation. Light from Han's head travels straight to the camera, since it's going through air the whole way. But light from Han's neck bends when it goes from water to air - so his head and neck can't line up.
OK, light bends when it goes from one material to another. So why did you ignore the light going through the plastic?
If you look at Snell's Law, you can see that the light will indeed bend when going from air to plastic or from water to plastic. But that's only part of the story: the light also needs to refract from the plastic to the air.
It's easy to use Snell's Law and a little geometry to show that, as long as the faces of the plastic are parallel to each other, the angle between the light ray and the surface of the plastic is the same as it would have been if the light had gone straight through the air without any plastic in the way, or straight from water to air.
[insert picture]
But there is still one effect: this leads to a sideways offset between where the ray is with the plastic vs. where the ray would have been without it. The thinner the plastic, the harder it is to notice this effect. [4]There's also another effect: the colors get split a little, thanks to "dispersion."
Snell's Law
Snel's Law? Snell's Law? You may find different sources insisting on one spelling or the other. It's actually named after Willebrord Snellius, who derived it in 1621. It's been around a long time.[5]A lot longer than that. It was also described by Ibn Sahl in 984, Thomas Harriot in 1602, René Descartes in 1637, and likely others we don't know about - each probably independent of the others.
If you observe the bending for a lot of different transparent materials (air, water, glass, plastic, etc.), you find that the amount of bending is different for different materials - it's a characteristic of each material. Scientists call this characteristic the "index of refraction" and use the letter n when using it in equations. And, since it's useful to have a reference, the index of refraction of air is defined to be 1[6]The index of refraction of air is really 1.0003 while the index of refraction of vacuum (absolute nothingness) is exactly 1 (that is, 1.000000000000...). The difference is hard to notice for the types of demonstrations shown here, but it can become important: smaller differences lead to dramatic mirages, for example. (since multiplying or dividing by 1 doesn't change any number in any way). Finally, the index of refraction of every other material is greater than 1.
We don't need to know what it is in order to use it effectively. For centuries, people used tables of values for different materials, based only on measurements. That's all you need if you want to make useful eyeglasses, telescopes, binoculars, etc.
But now we know that the index of refraction is directly related to the speed of light in a material. I prefer to use the equation vmaterial = c/nmaterial. Here, vmaterial is the speed of light in a certain material, nmaterial is the index of refraction of that material, and c is the speed of light in vacuum. Since n>1 for every material (other than vacuum), light travels fastest through vacuum, and slower through everything else.[a]See [a] below.
As with many errors, it's not exactly wrong, it's incomplete.
The speed of light in vacuum is one of the most fundamental constants we know about.[b]See [b] below. Many scientists know the context so well that they don't bother mentioning "in vacuum" - but it's a really important distinction. Light travels around almost 2½ times slower in diamond than it does in air. This leads to Cherenkov radiation, among many other interesting things.
Some theories suggest the speed of light in vacuum changes over time, or from place to place - but no one has convincingly observed that.
If you want some interesting reading, look up "metamaterials" that have a "negative index of refraction".
When calculating the interactions of light with a surface (whether reflection or refraction), scientists measure angles from the "normal line". In the context of math, science, and engineering, the word "normal" usually means "perpendicular"[7]From the Latin norma for "carpenter's square" which carpenters use to make sure things are perpendicular. Never call it the "natural line" or "usual line" - it has a totally different meaning. I teach my students to build an autocorrect function into their brains: "normal" → "perpendicular".
If you measure carefully, you find, for light traveling from any material (typically called "material 1") to any other material (typically called "material 2")
n1 sinθ1 = n2 sinθ2
where n1 and n2 are the index of refraction of the two materials, and θ1 and θ2 are the angles the light makes from the normal line, in each of those materials.
If you'd measure from the surface, you'd just use cosine instead of sine in Snell's Law. Isn't that easier than dealing with the normal line?
In science and engineering, you need to specify exactly what you mean. If you program a drone to travel 200 meters but aren't careful about exactly what direction it needs to go, you're going to have a problem.
Look at the image below: There's a ray approaching the surface, and there is a good, unique way of saying what angle it makes with the surface. But if you only say the angle leaving is, say, 60°, then everything in the cone satisfies that answer.
[insert image]
The ray doesn't get to choose which line along that cone it gets to go - it's only going to go one way. Which way seems obvious to us - the one directly on the other side. But how do you define that mathematically?
A good way that everyone can agree on is: there's only one plane that contains the ray of light approaching the surface and the line perpendicular to the surface at the point the ray hits the surface. The refracted ray[7a]and the reflected ray must be in this plane.
[insert image]
Think about how the sine function works: when θ = 0, sinθ = 0. As θ gets bigger, sinθ gets bigger. (We only consider from zero to 90°, since a ray can't be more than 90° from the normal line - that's the definition of "normal".)
nair ≈ 1.00 and nwater ≈ 1.33. So, sinθair must be greater than sinθwater[8](except when θ = 0, when they're both equal to zero). That means that a ray in water that was headed straight toward your eye will bend away from the normal line[9]"perpendicular-to-the-surface line" (and toward the surface) when it passes into the air - so you don't actually see that ray. Instead, you actually see a ray that was originally closer to the normal line.
Using these equations, you can calculate exactly how much of an offset there should be for any real-world situation. Or, you could design a ray-tracing program to create realistic refractions - but that could make things (like the photos and videos above) that people might not believe!
Look at the video carefully. As water filled the tank, Han's head didn't appear to get smaller - instead, his body appeared to get bigger. (We'll explain this for when you're looking face-on to the tank, since that's easier; you should be able to extend this to a more general case.)
Once again, this can be explained using Snell's Law. As shown in the image, Han's body looks wider when it's under the water than it does when it's in air. (If you draw more rays, you see that it's magnified in all dimensions, with slight distortions.) The rays have to follow those paths because θair must be greater than θwater, because nair is less than nwater.
n1 sinθ1 = n2 sinθ2
How To Do It
It's tempting to think this is hard to do, since you may not have seen it before. Actually, it's really easy.
Step 1: Get a clear container able to hold water that has flat, vertical sides
I used a display case you can find most craft/hobby stores. You could use a fish tank, a food storage tub, a plastic shoebox, whatever.
Step 2 (optional): If you want to rotate it easily, put it on a turntable or lazy susan
Step 3: Put something in it that will have part of it under water, and part above the water
Step 4: Have your eyes or camera at the level of the water
Step 5: Look at the object from different perspectives
That's it! You'll probably be able to do it with what you have at home now.
How To See Rays of Light
For the second video, you could see the path of the light in the air and in the water. That's really useful if you want to study Snell's Law carefully, or just understand what's going on better.
Step 1: Set the container on something that scatters light well
A piece of paper is great - but anything flat and mostly white works well.
Step 2: Get a source of light
I used the type of laser line generators used to make sure pictures are hanging straight on a wall; you can buy them for around $10. If you don't want to get one, you could use an LED flashlight that can be narrowed to a bright square beam, or a laser pointer, or make a beam using a bright light behind a narrow vertical slit.
Step 3: Arrange things so you can you can see the light on the paper in the air
If you used a laser line or a light behind a slit, you don't need to do any work for this. If you used a flashlight, you might need to hold it at just the right height, and angle it down just a little bit. If you're using a laser pointer, you can see the path of the laser through the air using smoke or fog.
Step 4: Arrange things so you can you can see the light on the paper in/under the container.
If you arranged things well in Step 3, you don't have any more work to do. You can also see the path of the light through the water itself, fully three-dimensionally, by putting in some dilute particles that will scatter the light back toward your eyes. My favorite: a tiny bit of coffee creamer.[10]And I do mean tiny. A few specks will usually do the job, once you mix it well. It should be almost impossible to tell that there's any creamer in the water without shining a laser through it.
