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Lets say you are flying in a plane headed due west at 1000 km/h (278 m/s) with an altitude of 10km. According to http://www.ngdc.noaa.gov/geomag-web/#igrfwmm, at 10km altitude, Earth's magnetic field is about $2*10^{-5}$ Tesla.

Let's say an object has a mass $m$ in kilograms and has a negative charge of magnitude $q$ in Coulombs. In the airplane it would experience a Lorentz force determined by $F=qv \times B$ of about $0.00556*q$ Newtons. This would result in an acceleration (given by $F = m*a$) of $0.00556*\frac{q}{m}$ meters per second squared in an upward direction (negative charges going west in a magnetic field pointing north experience an upward force). The object would also experience a downward acceleration of about $9.8$ meters per second squared.

Therefore if the object had a charge to mass ratio of roughly $$\frac{q}{m}= 1763 \text{ Coulombs per kilogram}$$ then it would float in mid air. So my first observation is that free moving negatively charged ions in the air which have a charge to mass ratio orders of magnitude above this would experience a strong upward force and positively charged ions would experience a strong downward force. This would quickly create an electric field from a positive charge buildup on the bottom of the plane and a negative charge buildup on the top. The electric field would grow until the electric field force (pushing in the opposite direction of the Lorentz force) balanced the Lorentz force.

Therefore, lets say you have the following available to you:

1) An airplane that does not generate an electric field while flying by somehow being built completely from a non conductive material and being filled with a vacuum and somehow generating no static build up from friction with the air outside. This possibly magical airplane is not of central importance to the question so don't focus on it. I just want to provide a vacuum environment with no electric field.

2) A Van de Graaff generator capable of generating very large static charges on objects.

3) Any materials you wish that you think can be charged with a large charge to mass ratio. For example: Aerogel, copper, aluminum, ceramics, graphene sheets, carbon nanotubes or something else which may be able to hold a lot of charge and be very light. I am guessing that a high surface area to weight ratio is probably the best bet. NOTE: Capacitors will not work. Capacitors store a large amount of charge, but it is positive and negative, so the net Lorentz force on a capacitor would be $0$.

If for some strange reason it is easier to place a large positive charge on an object by electron depletion, you are allowed to make the plane fly east and use a Van de Graaff generator which generated positive static charge.

So my question is; With modern materials, is it possible to construct and charge an object like this so that it will float in the airplane?

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  • $\begingroup$ You've put a lot of thought into this. I don't think it's possible to put that much charge on an object. A back-of-the envelope calculation suggest that if the material were carbon you would have to add or remove an electron from every fifteenth atom. Check my work on that. I can't imagine how it would be possible to do such a thing. $\endgroup$ – garyp Aug 21 '16 at 10:48
  • $\begingroup$ From my rough estimation. A carbon atom weighs 2E-26 kg and the charge of one electron is 1.6E-19 C so with only one extra electron on 4000 carbon atoms it would achieve a charge to mass ratio of 2000 C/kg $\endgroup$ – Andrew Aug 21 '16 at 11:00
  • $\begingroup$ That's right. Now take a cube of carbon having an excess charge density of one atom per 4000. How many neutral atoms are there between one charged atom and the next? $\endgroup$ – garyp Aug 21 '16 at 11:25
  • $\begingroup$ Looking at en.wikipedia.org/wiki/Carbon_nanofoam it would seem each electron would be 6 nm from the closest other extra electron. It may not be possible, but I would like a starting point to figure out the limit of static charge an object can hold. I could check experimentally, but this would be difficult for me to get all the supplies. Is there a non experimental way to determine the maximum amount of static charge an object can hold? $\endgroup$ – Andrew Aug 21 '16 at 13:25

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