I like to analyze the physics of science fiction, and so I'm going to argue that the Merrie Melodies cartoon “Compressed Hare” takes place in the far future when animals rule the world. I mean, Bugs Bunny and Wile E. Coyote walk on two legs, talk, and build stuff. How would that not be science fiction?

Let me set the scene—and I don't think we have to worry about spoiler alerts since this episode is 60 years old. The basic idea is, of course, that Wile E. Coyote has decided he should eat the rabbit. After a couple of failed attempts to capture Bugs, he comes up with a new plan. First, he's going to drop a carrot-shaped piece of iron into Bugs' rabbit hole. After the carrot is consumed (and I have no idea how that would happen), Wile E. Coyote will turn on a *giant* electromagnet and pull the rabbit right to him. It's such a simple and awesome plan, it just has to work, right?

But wait! Here's the part that I really like: While Wile E. Coyote is assembling his contraption, we see that it comes in a huge crate labeled "One 10,000,000,000 Volt Electric Magnet Do It Yourself Kit.”

In the end, you can probably guess what happens: Bugs doesn’t actually eat the iron carrot, so once the coyote turns on the magnet, it just goes zooming toward him and into his cave. And of course a bunch of other stuff gets attracted to it, too—including a lamppost, a bulldozer, a giant cruise ship, and a rocket.

OK, let's break down the physics of this massive electromagnet and see if this would have worked if Bugs had fallen for it.

What Is an Electromagnet?

There are essentially two ways to make a constant magnetic field. The first is with a permanent magnet, like those things that stick to your refrigerator door. These are made of some type of ferromagnetic material like iron, nickel, alnico, or neodymium. A ferromagnetic material basically contains regions that act like individual magnets, each with a north and south pole. If all these magnetic domains are aligned, the material will act like a magnet. (There's some very complicated stuff going on at the atomic level, but let's not worry about that right now.)

However, in this case Wile E. Coyote has an electromagnet, which creates a magnetic field with an electric current. (Note: We measure electric current in amps, which is not to be confused with voltage, which is measured in volts.) All electric currents produce magnetic fields. Normally, to make an electromagnet you would take some wire and wrap it around a ferromagnetic material, like iron, and turn the current on. The strength of its magnetic field depends on the electric current and the number of loops the wire makes around the core. It's possible to make an electromagnet without the iron core, but it won't be as strong.

When the electric current makes a magnetic field, this field then interacts with the magnetic domains in the piece of iron. Now that iron *also* acts like a magnet—the result is the electromagnet and the induced magnet attract each other.

What About 10 Billion Volts?

I don't know how the script for this episode came about, but in my mind they had a group of writers working together. Perhaps someone came up with the idea of an electromagnet and an iron carrot and everyone agreed to put that in there. Surely someone raised their hand and said, "You know, we can't just do an electromagnet. It has to be over-the-top big." Another writer must have replied, "Let's put a number there. What about 1 million volts?" Someone else interjected: "Sure, 1 million volts is cool—*but what about 10 billion volts*?"

What does 10 billion volts even mean for an electromagnet? Remember, the most important thing about an electromagnet is the electric current (in amps), not the voltage (in volts). To make a connection between voltage and current, we need to know the resistance. Resistance is a property that tells you how difficult it is to move electric charges through a wire, and it’s measured in ohms. If we know the resistance of the electromagnet wire, then we can use Ohm's law to find the current. As an equation, it looks like this:

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*R* is the resistance of the wire, and *I* is the current in the wire. I just need to estimate the resistance.

Looking at the video of the cartoon, I'm going to guess that the electromagnet wire has a diameter of 1 centimeter and is wrapped into a solenoid with a diameter of 1 meter. (A solenoid is the name for a coil of wire wrapped around a cylinder.) Let's say that the solenoid has a total of 500 loops to make the magnet. Using the circumference of a circle multiplied by the number of loops, that means that the total length of wire would be 393 meters. I can find the total resistance of the wire with the following equation:

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In this equation, ρ is the resistivity of the metal (for copper this would be 1.68 x 10-8 Ω meters), and A is the cross-sectional area of the wire, using the diameter. Using these values, the total resistance of the wire would be 0.08 ohms. That gives an electric current of 1.2 x 1011 amps.

OK, let's be realistic: A current that high would melt the wire, or at the very least make it super hot. Just to give you a comparison, when you run your vacuum cleaner, it can draw 5 to 10 amps. If you feel the power cord after you’ve vacuumed for some time, you can tell that it’s getting warm. When copper gets hot, it has an increase in the resistivity which would reduce the current. So in the cartoon, the wire in Wile E. Coyote’s electromagnet has 10 billion times the current that runs your vacuum cleaner.

Let's just modify this value and say that the electric current is 1 billion amps, which is still stupid large. That means that the electromagnet would require a 10 billion-watt power source (power = I*V). For comparison, the largest power plant on Earth is the Three Gorges Dam in China—it produces 22 billion watts. If Wile E. Coyote has a power supply that large, I don't think he needs to worry about one silly rabbit.

Could This Electromagnet Really Grab an Iron Carrot?

I'll be honest, calculating how much a magnet can pick up is never very simple. But if you have ever played with two magnets, then you should know that the attractive force is very weak when you are holding them far apart. However, when the magnets get close, the force increases quite a bit. To make this cartoon situation even more complicated, we don't have two magnets. Instead we have an electromagnet and an iron carrot.

The best way to describe both an electromagnet and a piece of iron is with a magnetic dipole moment (we use the symbol μ for this). The dipole moment basically is a way to describe the strength of a magnet, just like electric charge describes the strength of an electric interaction. For the electromagnet, the dipole moment depends on the number of loops of wire around the core, the circular cross-sectional area of the coil, and the electric current (in amps) running through the wires. Fortunately, I already have values for all of those quantities.

The magnetic moment for the carrot is a bit more difficult. In normal situations, it could have a zero magnetic moment if its magnetic domains aren't lined up. But let's just assume that under the presence of the magnetic field from the electromagnet all of its domains are aligned. In that case, I can use the magnetic dipole moment for a single iron atom and multiply it by the number of atoms in that carrot based on the molar mass of iron and Avagadro’s number. I'll skip the details, but the calculations are all in this Python code.

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Now I can use the following equation to calculate the approximate force between two magnetic dipoles:

Here the μ0/4π is just the magnetic constant, while μE is the moment for the electromagnet, and μc is the moment for the iron carrot. I still need the distance between the electromagnet and the carrot. (This is *r* in the equation above.) They don't show the exact distance between Wile E.'s cave and Bugs Bunny's hole, so I'm just going to approximate this as 500 meters.

With that, I get an attractive force of 4.05 x 10-4 newtons. That’s like the gravitational weight of something with a mass of 0.004 grams, like a single human hair. That is quite a tiny force to move a heavy iron carrot. I don't think this method would actually capture Bugs Bunny.

The main problem is the 1/r4 term in the force calculation. This means that if you double the distance between the two objects, the force will decrease by a factor of 16, which is 2 to the fourth power. Distance makes a huge difference.

Actually, it’s even worse. I assumed the carrot was a magnet. However, the magnetic moment of an actual piece of iron would depend on the strength of the magnetic field that induces it. This would make the force between the two objects even smaller as distance increases. And that makes it even less likely that this trick is going to work to get Bugs out of his hole.

As you can see, the magnetic force between two objects can be quite complicated to calculate. I guess that's why it takes a genius like Wile E. Coyote to even attempt to pull it off.

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