# How much magnetic field will be required to levitate a human? [duplicate]

According to this video:

it should be possible to levitate humans because the water in our bodies is diamagnetic - but it is stated (at 1:14) "although the magnetic fields required would be enormous".

So I wonder: how strong of a magnetic field would you need to levitate a human (using only the diamagnetism of the water in our bodies)? And (maybe easier to do) - how strong was the field used to levitate the frog in the video?

There must be a way to compute the amount of magnetization that occurs, and how that results in a repulsive force; but I don't know where to begin. Any suggestions?

• Henry is right: if you can levitate a train full of humans how could you not levitate a single one?
– user66432
Mar 28 '15 at 18:26
• This is closely related to an earlier question physics.stackexchange.com/q/15747/26969 and should probably be marked as a duplicate Mar 28 '15 at 19:51
• It is hard to find similar questions, and the train thing never clicked my mind, I did have some misconceptions about magnetism, all cleared from your answer. Mar 29 '15 at 6:14
• This shouldn't have been marked as duplicate IMO. "noticeably attract" and "levitate" are quite different. Apr 1 '15 at 11:08

A bit of clicking gets you to http://www.ru.nl/hfml/research/levitation/diamagnetic/ which tells us that the frog was levitating in a field of 16 Tesla. They give the math as well:

Therefore, the vertical field gradient ∇B2 required for levitation has to be larger than $2µ_0ρg/χ$. Molecular susceptibilities χ are typically 10$^{-5}$ for diamagnetics and 10$^{-3}$ for paramagnetic materials and, since ρ is most often a few g/cm3, their magnetic levitation requires field gradients ~1000 and 10 T$^2$/m, respectively. Taking $\ell$ = 10cm as a typical size of high-field magnets and ∇B$^2$ ~ B$^2/\ell$ as an estimate, we find that fields of the order of 1 and 10T are sufficient to cause levitation of para- and diamagnetics. This result should not come as a surprise because, as we know, magnetic fields of less than 0.1T can levitate a superconductor (χ= -1) and, from the formulas above, the magnetic force increases as B2.

The key to note is that to calculate the force it's not the magnetic field itself, but the square of the gradient of the magnetic field, that matters. As an object becomes larger (like a human) your factor $\ell$ in the above becomes larger, and this means that $B$ needs to be larger too. So you would need an enormously large magnet, with fields on the order of several 10's of Tesla, to levitate a human. Even a small one.

• There's something very amusing about "even a small one," like maybe we try it first on a baby or an undergrad. Mar 28 '15 at 19:17
• @zeldredge - that scrawny kid that everyone picks on? Mar 28 '15 at 19:18
• @Floris hi your answer is THE answer,no disputing that, but just purely from curiosity and my memory of physics classes years (and years) ago, is this chain of reasoning, schematically correct? Get the potential energy of frog if field was suddenly turned off, = mgh. That's energy needed to get up to height h. Then find out how strong a magnetic field would need to be to produce that same amount of energy. Just in general terms, yes/no is fine. Bit out of practice on these problems Thanks
– user74893
Mar 28 '15 at 19:31
• "several 10's of Tesla" seems kinda vague.. is it like 20-90? The video in question makes it seem like it would be 1000s times greater than that required to levitate the frog. Even 90T doesn't seem much greater than 16T.. Apr 1 '15 at 11:06
• @laggingreflex - recognize that the force scales with the square of the gradient of the field. So you need a magnet that is capable of the gradient of this little magnet but over a much larger area. Since every part of the body will experience the gradient and thus the force, it is just a matter of maintaining the divergent field over the larger area. To maintain the divergence over something the size of the body presumably requires a stronger magnet to begin with - hard to estimate how much stronger (ratio of diameters?). This is why I believe the size of the magnet would be enormous. Apr 2 '15 at 1:01