# superconductor levitating in earth's magnetic field? [duplicate]

Possible Duplicate:
Can superconducting magnets fly (or repel the earth's core)?

I've seen superconductors levitating on magnets. But is it possible for superconductors to levitate on Earth from Earth's magnetic field?

• Another duplicate that is arguably better written than the original. Certainly the answer here is superior. Any thoughts from on merging them or leaving them separate? Commented Nov 18, 2011 at 15:07

The lift generated by magnetic field B on a superconductor of area S is:

$$F = \frac{B^2S}{2\mu_0}$$

disregarding lateral forces and assuming superconducting cylinder (or similar shape) with area S at the top and bottom and height h, we need three forces to remain in the equilibrium: magnetic pressure on top, bottom and gravity force:

$$F_{b} - F_{t} = F_{g}$$

denoting density of the superconductor as ρ, Earth' gravity as g and magnetic field at the top and bottom of the object as Bt and Bb, we have

$$\frac{1}{2\mu_0}(B_{b}^2-B_{t}^2)=\rho gh$$

assuming the vertical rate of change of magnetic field is nearly constant and denoting the average magnetic field as B, we have

$$-B\frac{dB}{dz}=\mu_{0}\rho g$$

Compare with diamagnetic levitation (superconductor's magnetic susceptibility is -1).

Now, Earth magnetic field is between 25 to 65 μT. For the derivative I have found this survey from British Columbia with upper point on the scale being 2.161 nT/m. Assuming this to be the maximum for vertical derivative we get the required density of 1.1394e-08 kg/m3. For comparison air density at the sea level at 15C is around 1.275 kg/m3, so required density is 8 orders of magnitude smaller.

Even assuming a very high vertical derivative where B goes from its maximum 65 μT to 0 on 1 m of height results in density required of 0.00034272 kg/m3.

• so it's possible for levitation if the superconductor has maximal area and minimal density? What about metallic hydrogen? I hear that it is the lightest metal in the universe. Though I can't find any numbers.... Commented Nov 17, 2011 at 21:33
• Well, you certainly couldn't keep on decreasing the thickness beyond London penetration depth. Commented Nov 17, 2011 at 21:51
• thats like in the nm right? Which would be enough in terms of thickness for levitation. Commented Nov 17, 2011 at 22:12
• since a sheet so thin would break easily, it should theoretically work in the same way if instead I had superconductor powder. I'm assuming that each particle is like a very small thin square sheet, so given the right density, these particles can levitate under gravity, even when in close proximity to each other. Is this assumption correct? Commented Nov 17, 2011 at 22:22
• just saw your update about the magnetic field obstacle. so in effect, given the perfect material, levitation cannot be achieved practically? quite a shame.... Commented Nov 17, 2011 at 22:32