Why can spheres and cylinders be treated as point charges outside them?

[jlee167]

I recently noticed that I have blindly used optimization in some problems that involve symmetrical insulating/conducting spheres and cylindrical shells.
For example, when calculating outer electric field caused by a spherical insulator/conductor, I just treated these as a simple point charge located at their center, and those ways rendered correct answers. Also, in a question involving an infinite cylindrical shell, (given charge density), I treated it as a simple line charge located at its center, and it also gave me a right answer. However, I am still not convinced how this works mathematically. Is it just a way of simplifying problem for faster calculation, or is there any theorem / definition that fully explain the validity of this simplification?
I would appreciate some help

 

[Simon Bridge]

Look up Gausses Law.

You've noticed that the "optimization" only works for simple geometries, and only outside the objects in question.