As I was driving the other day, I noticed the little dashboard compass needle spinning whenever the car turned. Perhaps there’s a more energy efficient way to travel with a compass. How fast could you go from Brooklyn to Manhattan by sitting on the point of a giant compass needle?
***********WARNING: Math Ahead***********
Imagine a giant compass needle whose pivot is centered halfway between Manhattan and Brooklyn. Initially the needle points south so that its tip lies in Brooklyn. The needle would have to be about 8.0 km (~5.0 mi.) long. It would need to carry people, so it might look somewhat like a subway car, making it at least 3 m (~9.8 ft) in diameter.1
The torque, or rotational force, on a magnet is given by the equation,
torque = (magnetic moment) · (magnetic field) · sin(angle).
The angle in question is the angle between the magnetic moment and the magnetic field. This angle would change as the needle turns. On average, the sine of the angle would be about 0.63 as the needle passes from Brooklyn to Manhattan. The magnetic field would be the Earth’s magnetic field, which has a magnitude of roughly 5.0×10-5 T.2 The needle could be a large permanent magnet (like the ones on refrigerators), but these are not as strong as electromagnets and aren’t able to flip the north and south poles like an electromagnet can3. An electromagnet can also be turned on and off, so you could control when our magnetic transporter was about to leave the station. We can make an electromagnet by wrapping tons of wire around the needle in a helical fashion and then applying a current through it. The magnetic moment for a cylindrical coil is given by,
magnetic moment = (number of coil turns) · (current) · (area across the cylinder).
From the diameter above, the area would be about 10 m2. If the wire is 1.0 mm thick and there are 10 rows wrapped around each other, then the 8.0 km needle would have about 8 million turns in it. I’ll assume the current in the coils is 1.0 A.4
The time it takes the needle to rotate to Manhattan is given by the equation,
time = [2 π · (moment of inertia) / torque]1/2.
The moment of inertia of the needle can be computed using a formula found here. Plugging in the values for our needle, we can calculate the moment of inertia to be about 3.4×1012 kg·m2.5
By combining equations and plugging all the numbers in from about, we find that it would take about 1.1 days to get from Brooklyn to Manhattan traveling by giant compass needle.
**************************************************
There’s another interesting dilemma in this problem. Nothing in life is free, so you’d suspect that even our poorly functioning magnetic transporter must also sap energy from somewhere. Much like oil, it’s possible the source of Earth’s magnetic field would also eventually run out if we made lots of giant magnetic transporters. How long do you think it would take to sap away the Earth’s magnetic field?
[1] It would also be highly magnetic, so you couldn’t bring your credit cards.
[2] The “T” stands for “Tesla,” a unit of magnetic field.
[3] If you couldn’t flip the poles you’d never be able to get back to Brooklyn in the same way you left.
[4] I don’t have a good reason for assuming this value of current other than the fact that one of my intro physics books uses this value in a problem involving a copper wire. It would be interesting to see how this problem would work with superconducting wires, but you’d have to cool them down to very low temperatures, so the passengers might become a wee bit cranky and hypothermic.
[5] You can estimate this by assuming the needle is a 1.0 ft thick cylindrical shell made of iron.
I'm not sure I understand the question. Are you asking about the maximum torque that a one Gauss field can apply to a magnetic dipole several miles long using materials available on Earth?
ReplyDeleteFinding the maximum torque is part of the question. To calculate the torque, you'll need to estimate how large of a dipole moment a giant compass needle would have. Once you have that, it's pretty simple to figure out how long it'll take to get from point A to point B.
ReplyDeleteAlright, so supposing that you constructed the compass as a giant cable-stay bridge using carbon fibers to build the stays, holding up a much lighter compass needle, put the pivot point the West end of the Brooklyn Bridge, and carry people South to North from Hudson Ave to Corlears Park (needle is less than 2 miles long, and moves about 20 degrees). Use superconducting Niobium wires with hundreds of thousand of turns, running millions of Amps, and only intended to move the needle a few degrees so as to move the half-mile span of the Brooklyn Bridge?
ReplyDeleteIt's a good question. I'm not as familiar with superconducting wires as I should be. I'm still trying to figure out what the current in superconducting wires is. In principle, a superconductor has negligible resistance, but that would mean infinite current for any finite voltage. This, of course, is not possible. I'll let you know what happens if I find a good reference for current in a superconductor.
ReplyDelete