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I'm writing a story with some sci-fi elements and want to give actual science a try before resorting to technobabble.

What properties would a magnet and a planet need to have to levitate a whole city and its inhabitants 2km above ground without moving?

Would the same properties allow for another magnet to levitate higher or would that magnet need to have different properties?

If air composition is a factor bear in mind people don't have to be human. Gravity is Earth-like.

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    $\begingroup$ Does it need to be a magnet? Flux pinning generally locks a superconductor above a magnet, not the other way around, so you might have more success with a city-sized superconductor. The consequences would probably be extraordinary, mind you, and it'd be deeply unlikely to be a stable system. $\endgroup$ – jdunlop Jan 2 at 20:33
  • $\begingroup$ It can be anything that allows a city and its people to levitate, I just assumed magnetism would be the most likely force at play. I doubt this can happen on Earth or, like you say, consequences and instability. But I was wondering if a planet with a different magnetic field, a nucleus of a different material or a (likely made-up) metal with particular magnetic properties might make this possible. $\endgroup$ – silvia bartolini Jan 2 at 21:04
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    $\begingroup$ The nature of magnetism is such that unless you change physical laws, static magnetic levitation will be unstable. And if you're changing physical laws, there's no much point in avoiding technobabble. $\endgroup$ – jdunlop Jan 2 at 21:12
  • $\begingroup$ Then if I understand correctly, it would be possible to have a ground made of such a material that would allow a massive superconductor to levitate at 2km. If by unstable you mean it will wobble, is there a way to compensate for that, like in skyscrapers for example? $\endgroup$ – silvia bartolini Jan 2 at 21:50
  • $\begingroup$ Anyone trying to answer this can find a mass and diameter of NYC in my answer, which puts some parameters for the lift needed. $\endgroup$ – SRM Jan 2 at 22:11
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This is my stab at answering your question using static magnets. There are other answers that may involve a moving planetary core or some sort of electromagnet.

I found an estimated mass of New York City at $1.136 × 10^{11}$ kilograms.

Force = mass x acceleration

Earth gravity is $9.8 m/s^s$. To lift NYC into sky, we need a magnetic field of opposite strength. Roughly, that comes out to 10*12 Newtons.

I found this calculator of neodymium magnet strength.

Diameter of Manhattan is 2.3 miles. Two neodymium magnets (one on ground, one on city base) of that diameter and 10” thick gives a repel strength of 403017.329 newtons (I used the highest quality setting possible on that site). That’s roughly 106... insufficient by six orders of magnitude.

So I went back and tried magnets that are 1km thick. Result? Ah ha! $1.756 × 10^{12}$ Newtons! Now we are in the ballpark.

Now, that gives it zero buoyancy. To lift it 2km into the sky will require some more force, but if I’m reading the graphs correctly, the added thickness isn’t that much. If anyone reading this has more info on that aspect, please add comments.

To lift NYC by static magnetic force is going to require magnets that are about 1km thick attached to the bottom of the city and on the ground. To be stable, the ones on the ground will need to be much larger than the city, so the city can drift a bit (presumably held in place by large fans or similar engines). So coat the shell of your planet in metal, then mine chunks of it to go under your cities? The engineering is left as an exercise for the reader. :-)

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    $\begingroup$ So you want a chunk of metal the size of Manhattan and one kilometer thick. That is a lot of neodymium. You may want to compute its mass... and compare it with the puny 1E11 kg mass of man-made stuff in Manhattan. $\endgroup$ – AlexP Jan 2 at 23:05
  • $\begingroup$ @AlexP my understanding from the calculator is that the force computed is lift available after accounting for lifting the magnet itself. I could be wrong about that, but the site was specifically to help people compute how much of a magnet needed to supply a given lift force. $\endgroup$ – SRM Jan 3 at 0:50
  • $\begingroup$ Presumably the force for the 1km thick magnet is "1.756 x 10^12" not "1.756 x 1012", right? $\endgroup$ – Joel Harmon Jan 3 at 20:45
  • $\begingroup$ @JoelHarmon YES! I will fix. $\endgroup$ – SRM Jan 4 at 0:27
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    $\begingroup$ @SRM That calculator isn't a "lifting force" calculator, just a repulsion force calculator, meaning it probably doesn't take mass into account. (I say "probably" because it doesn't say one way or the other, so it might.) But one thing it definitely doesn't take into account is distance. That calculator only calculates repulsion force for magnets that are touching. Magnetic field strength falls off by the square of the distance, so at 2 km, the repulsion force would be a tiny, tiny fraction of what it is when they touch. $\endgroup$ – HiddenWindshield Jan 5 at 18:19
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Magnetic fields decrease in strength in an inverse-square relationship with distance. This means to levitate a city two whole kilometers, the magnetic field would need to be astoundingly powerful and it would need to be accompanied by frankly ludicrous magnets.

Furthermore, magnetic fields aren't one directional (there are no magnetic monopoles). This means that inside your city the magnetic field is just as strong as below it, but in the opposite direction (simplified). This has several consequences that make it completely infeasible for building a city on top of:

  • No ferromagnetic material would be usable anywhere near the city. With the magnetic field being powerful enough to levitate a city at 2km, bringing anything with iron in it within sight of the city would be a no-go. This includes all electric motors (washing machines, elevators, electric cars) among other things.

  • Even non-ferromagnetic materials like aluminum (which is paramagnetic) would need to be banned from the city because at high enough field strength, induced magnetism becomes an issue too. It's also possible that moving metal through the field would induce eddy currents powerful enough to melt the material

  • Water is diamagnetic which means that in a high enough magnetic field, it is affected too along with other biological processes in the human being. There are experiments where frogs are levitated in magnetic fields. A field powerful enough to lift a city would slowly kill any people who approach the city, creating and "aura of death" or a "keep out zone" if you want to put it nicer. There hasn't been a lot of research into the effects of extremely powerful magnetic fields on humans, but there are several interesting possibilities:

    • Humans contain a couple grams of iron. In a powerful magnetic field, the iron is simply pulled out of the human through the skin, cells, neuron-connections, and whatever else might be in the way

    • Electrical impulses are important for thinking in humans and using powerful magnets close to people's skulls is even an emerging type of therapy (simplified). It's possible that when approaching death-city, people would start suffering neurological problems, probably starting out with light hallucinations or fine motor control issues and eventually escalating to seizures and death

    • Maybe, the magnetic field simply crushes the human or throws them so far in the air that they die when they fall back down.

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  • $\begingroup$ And the force between two magnets is tiny in the far field (where the inverse square approximation holds), very much smaller than would be naively thought given the large force in the near field and the inverse square law. $\endgroup$ – AlexP Jan 2 at 23:02
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Buckminstr Fuller suggested floating cities lifted by hot air.

Cloud Nine is the name Buckminster Fuller gave to his proposed airborne habitats created from giant geodesic spheres, which might be made to levitate by slightly heating the air inside above the ambient temperature.1

Geodesic spheres (structures of triangular components arranged to make a sphere) become stronger as they become bigger, due to how they distribute stress over their surfaces. As a sphere gets bigger, the volume it encloses grows much faster than the mass of the enclosing structure itself. Fuller suggested that the mass of a mile-wide geodesic sphere would be negligible compared to the mass of the air trapped within it. He suggested that if the air inside such a sphere were heated even by one degree higher than the ambient temperature of its surroundings, the sphere could become airborne. He calculated that such a balloon could lift a considerable mass, and hence that 'mini-cities' or airborne towns of thousands of people could be built in this way.

A Cloud Nine could be tethered, or free-floating, or maneuverable so that it could migrate in response to climatic and environmental conditions, such as providing emergency shelters.2

https://en.wikipedia.org/wiki/Cloud_Nine_%28tensegrity_sphere%29[1]

Also see articles about ficitonal floating cities and islands for ideas.

https://en.wikipedia.org/wiki/Floating_cities_and_islands_in_fiction[2]

https://enacademic.com/dic.nsf/enwiki/1738945[3]

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