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The robot in question is a large, worm-like mining robot built for digging tunnels. It has electromagnets positioned throughout the segments of its body, for the purpose of pulling metals out of the surrounding earth as it digs. My question is, could this robot use its electromagnets to "push off" of the earth's natural magnetic field (or even just metals in the earth's crust), and fly through the air?

It doesn't have to be prolonged flight, only for a few seconds at a time, since doing this would obviously consume a lot of power.

I would just like to know if the feat is at all possible in the first place.

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    $\begingroup$ No, not in the real world; Earth's magnetic field is waaaay too weak for this. (And only a very few metals respond to magnets with enough force, so the applications of the magnetic mining robot would be quite limited.) But in comics book physics, why not? $\endgroup$ – AlexP Dec 11 '19 at 18:00
  • $\begingroup$ No you can't on earth, but it could work on a magnetar :) $\endgroup$ – Stefano Balzarotti Dec 11 '19 at 19:01
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    $\begingroup$ @StefanoBalzarotti - though I imagine a robot in close proximity to a magnetar would have new and excitingly different problems. $\endgroup$ – jdunlop Dec 11 '19 at 19:13
  • $\begingroup$ Might I suggest an alternative where is just stores waste mining dust and uses that as propellant? $\endgroup$ – DKNguyen Dec 11 '19 at 20:15
  • $\begingroup$ Native metals are pretty rare. You'll find a lot of them in iron rich asteroids, and their handy microgravity environment would make flight much easier... $\endgroup$ – Starfish Prime Dec 11 '19 at 20:56
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Suppose the electromagnet is strong enough to interact with earth's magnetic field: it will align itself like a compass needle. It will not hover or get attracted all the way to the north or south pole. Earth's magnetic field is uniform around the magnet. Magnet's N will attract Earth's S, while magnet's S will attract Earth's N. The two forces are opposite and will not pull the magnet one way or another, just make it align like a compass.

The electromagnet can be used to repel metals only if a high-frequency alternating current is applied. Here is a demonstration.

In this example, the eddy current generated in the plate is supposed to attract the metal, but the electromagnet flips polarity too fast. The repulsive force between the coil and the plate causes the coil to hover above the plate. That would the closest thing to "fly", in my opinion.

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This was in a comment, but making it an answer:

No

An electromagnet cannot "push off" ferromagnetic metals - if the metals have a current flowing through them, it's possible (due to induced electromagnetism), but not sitting idly in the earth, and earth's magnetic field is far too weak for an electromagnet to levitate itself against. Diamagnetic materials repel magnets, but extraordinarily weakly even in refined forms. (Per the wikipedia article, a superconductor like those used in maglev trains has a magnetic susceptibility of ~-1, while the most effective non-superconducting material, pyrolitic carbon, has a susceptibility of ~-10^-4.)

(That earth's magnetic field is too weak to push against is probably for the best, as it would cause some interesting interactions between the extraordinarily powerful electromagnets in an MRI machine and the earth's field if this weren't the case.)

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Magnetic Pressure providing the lift or attraction between two magnets is defined as $$P_b = {{B^2}\over{2\mu_o}}$$ $B$ is the vector sum of the two magnetic fields expressed in Teslas.

The Earth's magnetic field is on the order of $100 nT$, roughly a million times weaker than the refrigerator magnet. Plus, as has been ably observed in other answers, it is aligned along the earth's surface, except at the very poles of our planet where it has a radial component.

So your robot couldn't use the Earth's magnetic field to levitate except may be possible at the South Pole -- since the North Pole isn't really a place you find mines. Then, if you robot could generate a field powerful enough to generate lift it would also be attracted to ferromagnetic compounds in the Earth -- which would generate a much stronger $P_b$ pulling the robot back towards Earth.

If your robot was mining a huge quartz deposit and there weren't any ferromagnetic materials for many many miles, and lightning struck the ground, then you could plausibly have enough induced magnetic and e-fields from the current in the lightning and the piezoelectric effects of the quartz to generate a burst of magnetic field that would be radial from the point the lightning hit.

Quartz is found in conjunction with gold deposits. So if your story was set in a desert with the notable absence of iron, nickel, aluminum, and other ferromagnetic materials, then with a lot of hand waving, your robot could use both the magnetic pressure and the resultant Lorenz Force from the charge and B field interacts to fly, briefly.

It is, in truth, preposterous but for your story, meh, it could work.

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You could fly with electroaerodynamic propulsion and flames.

https://en.wikipedia.org/wiki/Ion-propelled_aircraft

Electroaerodynamics is a more speculative and currently less useful branch of electrohydrodynamics, which is super cool. [https://en.wikipedia.org/wiki/Electrohydrodynamics] Basically if you have an ionized fluid or fluid containing ions, you can move it with a magnetic field. If you move the ions in a fluid you move the fluid with it. Electrohydrodynamic motors are real and move the ions in water to generate propulsion.

This has been used to move aircraft as well but you need to add ions to the air. For glider type planes you can do that with a charged wire. I propose for your robot, rather than some puny wire it generate flames.

Flames contain loads of ions. https://physics.stackexchange.com/questions/45105/does-fire-conduct-electricity-why If flames can conduct electricity then you should be able to move them with your electromagnet. You should be able to take flames you generate and blast them out behind your robot. It will look like a jet engine but you will be using your electromagnet to move.

What a cool science project that would be. I wonder...

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