Ring Around the 'Rora

Recently I started reading a page called Michigan Aurora Chasers, which shares pictures of the aurora taken in our current home state. The pictures are incredible, but I was really interested by a post that came up discussing Newton's Rings, an effect that can sometimes appear when viewing light from a monochromatic source through a series of lenses, like a camera.

Wikipedia has an example of the effect in a microscope, viewing a sodium lamp:

Wikipedia

For aurora viewers, this happens due to using a flat filter over their curved camera lens. When the light passes through the filter, some will bounce between the lens and filter one or more times, changing the phase. This light can then interfere with the light that passed straight through, producing the dark fringes seen above. The extra distance traveled by the light changes depending on how far from the center of the lens it hits:

The wavelength of light also changes how these rings will appear, since the total phase change from bouncing once from each surface is φ = 4πd/λ, where d is the distance between the filter and lens, and λ is the wavelength. We can scan through the visible wavelengths to see how the pattern of fringes changes (thanks to John D. Cook for the wavelength/RGB conversion):

Due to the spherical shape of the lens, as we get farther from the center, the distance changes more rapidly. This means that if we add up several wavelengths (since true monochromatic light is rare in nature), we see that the rings are only visible near the center of the image, as in the aurora photos from the link at the top:

Our area of Michigan is a bit too far south to get to see the aurora in our own sky, so it's been great to get to see the amazing pictures the group members post. On top of that, they introduced me to this really neat optical effect – Thanks Michigan Aurora Chasers!

Buoys Will Be Buoys

Through my many, many physics classes, I've never done much with fluid mechanics, and I've often thought of questions about how such systems work. Recently I was thinking about Archimedes' principle, which is behind the famous exclamation of "Eureka!" and I wondered whether I could derive it using the physics that I do know.

The principle states that an object placed in water (or other fluid) will displace a volume of equal mass. The way I thought about this was that the displaced water has to go on top of the water surrounding the object. This increases the potential energy of the water, since

where m is the mass, g is the acceleration due to gravity, and h is the height. Lowering the object decreases its energy, but pushes more water up, increasing that energy. I figured I could map out the energy of the system by placing an object at a given height, filling a set volume of water around it, then measuring the energy using the equation above. To start with, I tried a simple box, with a density 90% that of water:

Plots of gravitational potential energy are really handy, since you can think of them like a hill: an object will roll down to the lowest point. The knee in this graph shows where the box floats, and the different slopes on either side show that it's easier to push the box further into the water than to lift it out.

The box is the simplest case, since it's easy to calculate the amount of water displaced, but I was curious to try the same method on more complicated shapes, specifically a ball and a boat, which I approximated as a hollow box:

These plots show the grid of water/air/object I made to calculate the energy. The labels give the fractional density of the object that I used. Since the boat is just a thin frame filled with air, the frame can be much denser than water and still float. If you look carefully at the border between the air and water, you can see a wrinkle – This is due to the resolution of the grid I used. The water is not able to spread perfectly evenly.

Now to Archimedes: For the various cases above, we can measure the displaced volume of water by counting the cells below the water level that belong to the object or air. We can compare that to the total weight of the object – Count its cells and multiply by its density. We can do that for each of the 3 shapes, and vary the size and density:

Aside from a bit of error from the wrinkles I mentioned, the points fall on the line, just as Archimedes predicted! This is one case where considering energy did make things easier.

Flattered or Flattened

Last night I got an interesting question from my nephew Phineas: Do 2D objects exist in our world? How do they work? In my exhausted state, I managed to cobble something together about electrons in a conductor, but I felt I owed him a more complete answer once I thought a bit more.

First let's discuss a bit what it means for something to be 2D (or 3D, or even 0D). The number of dimensions refers to the extent of an object, or the space it takes up. However, we can talk about embedding an object of lower dimension in a higher-dimension space. For example, we can put a plane, which is 2D, in a 3D space:

The equation for this particular plane is z = 1, with x & y between -1 and 1. That means we can pick a point on the plane, like (0.5, -0.5, 1), but if we go even a tiny bit off of z, like (0.5, -0.5, 0.99999), we'll no longer be on the plane.

Even if it's mathematically possible, it's still hard to believe such a thing could exist in the real world. That's where my example from above comes in: Imagine a metal ball that we charge up with some free electrons. Metals conduct electricity, so the electrons can move around freely, but they also are trying to stay as far from each other as possible since like charges repel. If you imagine trying to find the largest separation between a set of points in a sphere, you'll see that the solution requires all the points to be on the surface. This surface is 2D, even if it's curved and embedded in 3D, like the plane above. You can tell it's 2D because you only need 2 coordinates to position yourself on the surface of a sphere, like latitude and longitude.

Now that answer may not be satisfying, since if we imagine ourselves like the electrons, moving around the surface of the Earth, we're not 2D, just the surface we're walking on. However, based on the measurements that have been made in particle colliders, electrons act as single points, or 0D objects. In response to this, Phineas asked, If electrons are 0D how do they exist? They must exist to become 1D, or 2D, or even 3D! I still don’t understand how if something is 2 or 1D how it exists. This is fair point – If something has fewer than 3 dimensions, it will have a volume of zero, so how can it have mass? If they do have mass, then they should have infinite density, which would make them a black hole. My feeling is that the answer lies in the fact that electrons, and similar sub-atomic particles, are quantum objects, and quantum mechanics works very differently from anything we're used to our own lives.

These questions are really at the limits of what we know right now – While looking up info to write this post, I found an article from 2020 about the discovery of 2D particles. If you're curious about this stuff, Phineas, you can become a physicist and maybe explain it to your UncOrion some day!