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## Homework Statement

I'm trying to find the energy stored in the field of a circular charge distribution of constant charge density.

## Homework Equations

I know that the energy stored in a field is the same as the potential energy of the system.

dU= [itex]\varphi[/itex] dq = [itex]\varphi[/itex](σ dV)

Though my book also says that dU= (1/2) ρ [itex]\varphi[/itex] dV

So I'm not sure which equation to use. (I'm using Purcell.)

## The Attempt at a Solution

If I integrate without the 1/2, I get the correct answer (8Q^2)/(3πa).

But why don't I use the 1/2?

Also, I understand that dq=σ dA, and dA=2πr^2 dr, but why do I not use the expression with the 1/2 to get the correct answer when the book says that this formula gives the energy of a system? Where does the 1/2 even come from in doing from dq to dV (or dA) as the quantity being integrated?