Above is the video that started my problem. I wanted to show that the resistivity of aluminum decreases when you put it in liquid nitrogen. I think this video actually shows that quite well. But perhaps you like just a plain ring launcher. Here is an older style. It's bigger and a little bit more dangerous since it doesn't even have an on switch. You just plug it in and it goes (hopefully it doesn't overheat).
The problem is my over simplistic explanation of the ring launcher. I don't think my typical explanation is exactly wrong, it's just not the whole truth. Here is the way I usually explain this device.
Level 1 Ring Launcher Explanation
This launcher is basically just a coil of wire hooked up to an alternating current circuit (the iron in the middle just makes the effect greater). The first part of this demo is to show that electric currents create magnetic fields. You can show this by placing a wire directly over a compass. When the wire is connected to a battery, the compass needle moves.
Many younger kids might say "what the heck is that plastic thing?" Yes, that is a magnetic compass. It's just like the one on your phone, but this one is real. Actually, I wonder if this experiment would work with the digital compass on a smart phone. I assume it would.
Ok, but what happens if you continually change this current in the wire? Well, in that case you would create a changing magnetic field. And here is the cool part: a changing magnetic field can create an electric current. Yes, it's more complicated than that but the key word here is "can". Changing magnetic fields don't always make a current, but it does in this case.
As an added demo, you can see the effects of the induced electric current without a jumping ring. Here is a short video showing a small lightbulb with another coil of wire. When it is in the region of the changing magnetic field, the bulb lights up.
So, why does the aluminum ring jump up like that? The coil makes a changing magnetic field that then induces an electric current in the ring. This electric current in the ring then interacts with the magnetic field to make it repel. Oh, I guess I left off the small demo that show that electric currents interact with magnetic fields too.
What is wrong with this explanation?
First, let's look at changing magnetic fields. They don't always create an electric current, but they do always create an electric field. You can see this in the follow equation from Maxwell.
This is Faraday's Law. It says that the path integral of the electric field around some closed path is proportional to the time rate of change of the magnetic flux. For the case of the metal ring, since there is a closed loop of conducting material, this electric field causes a current.
The next problem has to deal with the force on a loop of current in a magnetic field. For any short segment of current, the magnetic force can be calculated as:
Just to be clear, B is the vector value of the magnetic field at the location of the small piece of wire. The small section of wire has a length dl and the current (I) is in the direction of this dl vector. Remember the direction of this force is found with the right hand rule so that it is perpendicular to both the current and the magnetic field.
This means that in a constant magnetic field, I would get some sample forces on a circular loop that would look like this:
All of these magnetic forces in this case would cancel resulting in zero net force. It actually doesn't matter about the orientation of the loop. As long as the magnetic field is constant (constant in space, not time), there will be no net force on the wire with current. Now, there can be a net torque on the loop. This is the main idea in an electric motor. But to exert a force on a loop of wire, you need a diverging magnetic field. Here is a side of that same loop but with a magnetic field that is diverging.
Ok, so it has to be a diverging field instead of a constant magnetic field. Well, there is a small problem. The shape of the coiled wired is essentially a solenoid. In our introductory physics courses, we uses this shape as an example of a configuration that creates a constant magnetic field. So, clearly there is a problem.
But wait. There is an even bigger problem. Suppose I looked straight down the axis of this solenoid with the ring. Of course, you should never actually do this. You could shoot your eye out with the ring.
I am using the typical convention to represent vectors coming out of the screen as a circle with a dot (consider it to be an arrow and you are looking at the tip). But here maybe you can see the problem. For an ideal solenoid, there is a constant magnetic field. However, there is zero magnetic field outside the solenoid. At the location of the wire with induced current, there would be no magnetic field and thus no magnetic force.
Of course this isn't actually correct. There has to be some magnetic field outside the coil. So, it must be this magnetic field on the outside of the coil that is responsible for the net force on the ring. Usually, we call these external fields fringe fields (which always makes me think of the surrey with the fringe on top).
So, this ring launcher isn't quite as simple as I thought.
More Questions and Experiments
Go back to the first ring launch video at the top of this post. In that demo, I launched an aluminum ring. Next, I launched another ring that had twice the height. The second ring obviously has twice the mass of the smaller ring (they have the same width). Which one goes higher? It turns out that the thicker ring will be launched higher. Why?
If the thicker ring is more massive, it will take a larger force to speed it up. However, since the taller ring is taller, it also has a lower resistance (wider cross sectional area). This means there will be a larger current in there creating a greater magnetic force. If you just doubled the thickness, the resistance would be half as much meaning there should be twice the current and twice the force. This double force would be just what you need to get the ring up to the same height as the shorter ring.
Why aren't they equal? I only have a guess. Remember the magnetic force pushing the ring up depends on the divergence in the magnetic field and not just the magnetic field. Since this divergence probably isn't constant in space, perhaps the top of this ring experiences a greater magnetic force than the bottom of the ring. This would mean the taller ring would have an overall advantage during the launch. I'm just guessing here.
There is another interesting question. Why does the ring shoot UP instead of down? Or perhaps there should be a modified question: what if you had a plain solenoid laying horizontally with the aluminum ring right in the middle? My guess is that the ring wouldn't go anywhere. If everything was completely symmetrical, then at the location of the ring the forces should cancel. I'm just guessing here, but I suspect for both of the versions of the ring launcher I have shown, they aren't completely symmetric.
Now for some future ideas for experiments (I am writing these down so that if I forget at least someone else can carry on).
- What is the acceleration of a ring? I could either use a high speed video or maybe a motion detector to measure the acceleration of the ring as it is launched horizontally. I suspect it's not constant but this might be difficult to measure.
- Maybe I could measure the magnetic force on the ring as a function of position (this would be another way to get the acceleration). If I put some non-conducting stick on the ring and then connected this to a force probe, it seems like I could get a value for the force exerted by the launcher. If I move the ring to different locations, this would give and expression for acceleration vs. distance.
- Maybe I could just measure the divergence in the magnetic field directly. I could use one of those Hall-Effect probes and put a constant DC current through the solenoid. Then I just just position the magnetic field sensor at different locations to determine the divergence in the field.
- What if I used that lightbulb rig to measure the induced electric current? I don't know if that would work.
- It would be fun to make a numerical model of a solenoid to estimate the fringe fields. Heck, why stop there? I could just model the whole thing numerically. If it produced a ring launch similar to real life, I would have totally dominated the whole problem.
I want to post one other thing. Remember the whole point of this started with showing that the conductivity (or maybe you prefer to deal with resistivity) of aluminum as it changed temperature? I wanted to look up a nice chart showing the resistivity (in Ohm-meters) for different temperatures. I didn't find a nice graph like I was expecting. So, I decided to make my own.
Maybe I am using it wrong, but I tried to get Wolfram Alpha to just show me the resistivity of aluminum at different temperatures. That didn't work. If you give Wolfram a particular temperature, it will give you the resistivity. That just means I can manually get a few data points to make a plot.
That looks fairly linear. However, it could be useful. If I shot the aluminum ring up at different temperatures, I should see a change in height. Since the mass of the ring doesn't change, this would just give me information about the magnetic force (the current should be inversely proportional to the resistivity).
Using Wolfram Alpha was probably silly. I suspect that Wolfram doesn't have all this resistivity data and instead has a formula it uses to calculate this value. I could have just used the formula. There is also a nice journal article that looks at resistivity of aluminum.
You can read that if you get obsessed with resistivity. Maybe it will inspire you to create your own experiments.
