Why does the inverse square law make impossible for an object to levitate throught a combination of distance action forces?

Question

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

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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?

Answer 1

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.