Electric Potential Energy Spherical Shells

[JosephK]

Homework Statement

A solid sphere of radius R has a uniform charge density ρ and total charge Q. Derive an expression for its total electric potential energy. Suggestion: Imagine that the sphere is constructed by adding successive layers of concentric shells of charge dq = (4$\pi$ r$^{2}$ dr) ρ and let dU = Vdq. (Use any variable or symbol stated above along with the following as necessary: ke.)

Homework Equations

U = $\int$4$\pi$r$^{2}$k$_{e}$$\frac{q}{r}$dr

$\rho$=$\frac{Q}{\frac{4}{3}\pi r^{3}}$

The Attempt at a Solution

The sum of all dq is Q.

U = qV - q is test charge
U = q k$_{e}$$\frac{Q}{r}$ - equation of voltage substituted

dQ = dq k$_{e}$$\frac{Q}{r}$ -small potential energy with respect to small charge

dQ = 4$k_{e}\pi\rho\frac{Q}{r} r^2 dr$ - dq plugged in

Then I integrated both sides.

 

[Delphi51]

I'm having a little trouble following that.
It seems to me the dQ for the spherical shell is 4πR²ρ*dR.
The work done to bring dQ in from infinity to R is dU = kQ/R*dQ.
And Q up to radius R is 4/3*πR³ρ.
Combined, dU = 16/3π²k ρ²R⁴dR
Check carefully; I make mistakes.

 

[JosephK]

Thank you