Charge distributed uniformly on sphere surface

[johann1301]

From my textbook:
"An electric charge that is uniformly distributed on the surface of a sphere, affects a different charge outside the sphere as though the whole charge was collected in the center of the sphere. This we exploit when we use Coulumbs law."

Ive tried to prove this mathematically, but can't seem to do it...
If we imagine two situations:

#1:
We imagine four protons in two pairs - each pair free to move with 2e of charge - at a distance x from each other:

Fig1:(**)---------------x---------------(**)

The the force between them would be k4e²/x²

#2:
Now we imagine that we "split" the pairs in such a way that the protons in each pair are still "attached", but the charges are separated at distance of 2P. Each proton is moved a distance P from the original point in situation #1. There is still only to pairs/particles that can move:Fig2:(*-----2p-----*)----------x---------(*-----2p-----*)

Shouldn't it be possible to prove that the force between the two pairs in the last situation also should be k4e²/x² if the original statement is correct?

(the reason for the parentheses () is to illuminate that the protons act as though the were attached to each other)

 

[jtbell]

From my textbook:
"An electric charge that is uniformly distributed on the surface of a sphere, affects a different charge outside the sphere as though the whole charge was collected in the center of the sphere. This we exploit when we use Coulumbs law."

Ive tried to prove this mathematically, but can't seem to do it...

This is part of Newton's shell theorem, applied to electrostatic forces, rather than gravitational forces which is the context in which Newton proved it.

Google for "Newton shell theorem" and you'll find many proofs of it.

The theorem applies to three-dimensional spherically symmetric objects, so I'm skeptical that your one-dimensional example is relevant.

 

[dauto]

You can also approach the problem using Gauss law.