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## 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[itex]\pi[/itex] r[itex]^{2}[/itex] dr) ρ and let dU = Vdq. (Use any variable or symbol stated above along with the following as necessary: ke.)

## Homework Equations

U = [itex]\int[/itex]4[itex]\pi[/itex]r[itex]^{2}[/itex]k[itex]_{e}[/itex][itex]\frac{q}{r}[/itex]dr

[itex]\rho[/itex]=[itex]\frac{Q}{\frac{4}{3}\pi r^{3}}[/itex]

## The Attempt at a Solution

The sum of all dq is Q.

U = qV - q is test charge

U = q k[itex]_{e}[/itex][itex]\frac{Q}{r}[/itex] - equation of voltage substituted

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

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

Then I integrated both sides.