Singularity of Coulomb s law Coulomb's law says electrostatic field intensity varies as $1/r²$ i.e ha singularity at location of charge.
now if a charge distribution is accumulation of many point charges. then for a charged metallic sphere why do we get finite potential at surface? it should hav been infinity. since it is whwre charges are located.
please answer as simply possible.
 A: At a given point on the sphere, the rest of the surface is a positive distance away so contributes a finite potential. In theory you can compute the potential at a point by integrating over all these contributions, although that's not the simplest method. It shouldn't surprise you too much to learn the answer is finite, even though one point's contribution is infinite. (As a simple analogy, $x^{-1/2}$ diverges at $0$ but has finite $\int_0^1 dx$ integral.)
Suppose the sphere has radius $a$ and surface charge density $\sigma$. A concentric sphere of radius $r$ encloses charge $0$ if $r<a$ and $4\pi\sigma a^2$ if $r\ge a$. We need to find the electric field $E$, then use $E=-\dfrac{dV}{dr}$. By Gauss's law, the electric field strength at $r\ge a$ satisfies $4\pi r^2E=\dfrac{4\pi}{\epsilon_0}\sigma a^2$ so $E=\dfrac{\sigma}{\epsilon_0}\left(\dfrac{a}{r}\right)^2$, whereas for $r<a$ there is no electric field. We thus have a constant potential inside the sphere; this constant can be chosen arbitrarily. The most popular convention sets the potential to $0$ at $r=\infty$, so outside the sphere $V=\dfrac{\sigma a^2}{\epsilon_0r}$. Note this implies a surface potential of $\dfrac{\sigma a}{\epsilon_0}$.
A: In short, the coulomb law which you correctly cite, applies only to point charges. If you go close to the electrons, this approximation is no longer valid and you have to integrate the charge distribution in space using Gauss' law. 
The charge of the metal sphere can be imagined as being made up of individual electrons. These electrons are a finite distance apart and they are not point charges but 'smeared' over a very small but measurable volume. If the surface is defined as going through the center of all the outermost electrons, the electric field will still never be infinite, because the electrons are not point charges but best imagined as charge distributions, little clouds. 
