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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 expand 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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