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From the book Thinking Physics:

Suppose the moon had a negative charge. Then it would exert a repelling force on electrons near it. But the gravitational force of the moon exerts an attracting force on the electron. Suppose the electron is one mile above the lunar surface and the attraction exactly balances the repulsion, so the electron floats.

Next, suppose the same electron was two miles above the moon. At the greater distance

[...]

c) the gravity would still balance the electrostatic force, so the electron would float

[...]

If dust could float, due to an electrostatic charge, one inch above the lunar surface, it could float at any height and so would float right off the moon! In fact, it is impossible to suspend or levitate an object by any combination of STATIC electric, gravitational or magnetic force fields because each obeys the inverse square law.

I don't understand, how does the inverse square law justifies this impossibility?

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  • $\begingroup$ It is nothing to do with the inverse square law. The basic fact is that the electrostatic force between two electrons is about $10^{40}$ times bigger than the gravitational force. Gravity is an incredibly weak force, except when very large masses are involved. $\endgroup$
    – alephzero
    Commented Aug 11, 2021 at 12:49
  • $\begingroup$ I can't see how this is relevant. That gravity is weaker than the electrostatic force doesn't make the levitation impossible. For example, a spherical body with mass m, and charge q, and with ratio q/m=G/k will generate an electric field that nullifies the gravitational field at any radius so I could put a positively charged body there and it would "levitate". $\endgroup$
    – Jon
    Commented Aug 11, 2021 at 18:31

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Suppose you have two forces, $\frac{A}{r^{2}}$ and $\frac{B}{r^{2}}$.

At distance $r_{0}$ let's suppose the "A" force is larger than the "B" force.

Then we are supposing:

$\frac{A}{r_{0}^{2}} > \frac{B}{r_{0}^{2}}$

So

$A > B$

But this means that at any $r$,

$\frac{A}{r^{2}} > \frac{B}{r^{2}}$

If they are ever equal, they must always be equal, so the object would feel zero net force from these forces, but could be agitated by something else, so it would not be held in place. If one is larger, it must always be larger, so the object would either fall or fly to infinity.

To stably suspend an object, what you'd like is a local minimum of the potential, so that any perturbation away from the minimum results in a force pushing you back. In 1-D, $F = -\nabla\phi$ and $\frac{dF}{dx} < 0$ so that moving in direction $x$ creates a force in the $-x$ direction. For gravitational and electric fields, this is only true at locations where there is material, $\nabla^{2}\phi \propto \rho$ which is why you end up having to touch material, which is no longer levitation.

Magnetic force fields don't exactly obey the inverse square law, but the main reason to discount them is that velocity is required to use them. If you allow particles to move, then you can indeed "suspend" or "levitate" them in a more loose sense of those words, like the Earth around the Sun, or charged particles trapped in magnetic fields at CERN. See Earnshaw's theorem.

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    $\begingroup$ "If they are ever equal, they must always be equal, so the object would feel zero net force from these forces, but could be agitated by something else, so it would not be held in place" So the impossibility, in this case, doesn't come from the inverse square law per se, but from the big amount of forces around everything making very unlikely a balancing of forces? $\endgroup$
    – Jon
    Commented Aug 11, 2021 at 18:24
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    $\begingroup$ To put it intuitively, if I apply 0 net force to you no matter where you are, you wouldn't say I am suspending or levitating you! But that's for the situation where the electric and gravitational force come from the same source (the moon, in your example). I shared more about stability because I thought you might be wondering about situations where there are multiple sources, like a charge "balanced" between 3 charges. The force might be zero at that single point, but it would not be stable. $\endgroup$
    – Alwin
    Commented Aug 11, 2021 at 18:47
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    $\begingroup$ Indeed. I guess my point is that this supposed impossibility that the author of the book talks about doesn't really seem impossible from a physical point of view. Just extremely unlikely (just occurred to me that in my example one tiny push would also cause the electron to move with constant velocity indefinitely as well as the net force is 0 everywhere. So more points for the unlikely aspect of something quietly levitating) $\endgroup$
    – Jon
    Commented Aug 11, 2021 at 19:03
  • $\begingroup$ Yes, his claims are true given very limiting assumptions (electric, gravitational, no time dependence, no velocity). Your intuition tells you it's possible because if you add other things (magnetic fields, quantum effects, time dependence, velocity) levitation is possible in reality, and you've likely seen many examples! $\endgroup$
    – Alwin
    Commented Aug 11, 2021 at 19:40

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