Coulomb's law experiment, why consider the distance between the body's centers? In $F_e=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2}$, why did Coulomb consider the distance between the center of the charges? Why not the distance between the immediate outer surface between the two bodies? Did he assume that this "electric ability" is perhaps condensed in the center?
Does this law present a deficiency?
 A: Coulomb’s Law as you wrote it applies to point charges.
When you aren’t dealing with point charges, you can treat a continuous charge distribution as a collection of infinitesimal volume elements that can be treated as point charges, where $dq=\rho dV$, with $\rho$ being the charge density. You can then integrate the infinitesimal forces between all the infinitesimal charges.
When you have spherically symmetric charge distributions, such as two spheres with charge distributed uniformly over their surfaces, this integration happens to produce a result that looks just like Coulomb’s Law for point charges! The integral gives the product of the total charge on each sphere, divided by the square of the distance between their centers, as if the charge of each sphere were concentrated at its center. This remarkable mathematical result, which applies to for any inverse square force, is called the shell theorem.
It’s the same in electroststics as in gravity, where the Earth and Sun attract each other as if all their mass were concentrated at their centers. But what is really happening is that every atom in the Sun is attracting every atom in the Earth.
If you put charge on two conducting spheres, it is free to move around, and it will not be uniformly distributed. For example, if both spheres are positively charged, the charge will tend to move toward the side of each sphere that faces away from the other sphere. The spheres won’t have a spherically symmetric charge distribution. In this case, the force will not be the product of the total charge on each sphere divided by the square of the distance between their centers, because the shell theorem will not apply.
