How Does Smearing Charge dQ Uniformly on a Sphere Maintain Constant Energy?

[hasan_researc]

A sphere of radius r contains a charge Q distributed uniformly throughout its volume. Charge dQ is brought in from infinity and deposited uniformly over the surface. The potential energy of the charge dQ is Q dQ/4\pi\epsilon₀.

This follows naturally from the result for the potential of 2 point charges after noting
(a) E due to Q is the same as a point charge and
(b) smearing dQ around the sphere (i.e., tangential to the radial direction) doesn’t change the energy.

Why/how does "smearing dQ around the sphere (i.e., tangential to the radial direction) not change the energy"?

Thanks in advance for any help!

[N B : I am a first year physics undergraduate.]

 

[hasan_researc]

I am sorry for the poor latex code. The potential energy expression should read Q dQ/4\pi\epsilon_{0}

 

[hasan_researc]

Mistaken for the second time now. The potential energy expression should read Q dQ/4\pi\epsilon_{0}r