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The Coulomb force between charged particles is inversely proportional to the square of the distance. Yet, why don't we observe the infinite force when the distance approaches zero? Say we can bring two positively charged glass rods and make them touch each other. We don't observe a very large amount of repelling force.

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Coulomb force and gravitational force have the same mathematical form. Let's consider an analogous example with gravitational forces.

Consider two spherical planets with mass $M$ and $m$, with radius $R$ and $r$, respectively. The attractive force between them is $\frac{GMm}{d^2}$, where $d$ is the distance between their centers.

Now suppose the planets touch. What is the force? $\frac{GMm}{(r + R)^2}$, because $d = r+R$ when they touch. Is it infinite? No.

Now suppose you shrink the planets to the size of tennis balls. Will the force between them be infinite? No.

Now you can squeeze and stretch the planets, and give them the shape of rods, and the charge is still spread over a finite volume. The force between them will still be finite.

The force becomes infinite only when the two planets (or charges) are reduced to points which have zero radius. When the charges are spread over a non-zero volume (rod, sphere and so on), the force will remain finite.

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  • $\begingroup$ If you shrink Jupiter to the size of a tennis ball it becomes a black hole. $\endgroup$
    – my2cts
    Mar 5 at 9:24
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A similar statement can be made about the gravitational force. Thus when we take two point particles of mass $m$ and $m'$ arbitrarily close to each other the gravitational force between goes to infinity.

Thus either one of three things can happen:

  1. Either the gravitational inverse square law must be modified on very small scales

  2. Or a new force kicks in to stop particles of matter approaching arbitrarily closely.

  3. There is some new kind of physics

Roger Boscovitch, a Croatian physicist of the 18th C, took the second option deduced the existence of interatomic forces. This new force, although he didn't know it at the time, was an aspect of the electromagnetic field.

Similarly here, when we bring an electron arbitrarily close to a proton, quantum mechanics kicks in and the electron and proton form a bound system - a hydrogen atom. This is option 3, the new physics of quantum physics steps in.

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