Total charge from charge density

1. Aug 6, 2015

Slusho

1. The problem statement, all variables and given/known data
Find the total charge Q given the charge density ρ(r)=ε0A(4πδ3(r)-π2e-λr/r

3. The attempt at a solution

What I don't understand is where that 4πr at the end comes from. That last step is only distributing the integrals except for putting that 4πr in there, so it seems to come from nowhere.

2. Aug 6, 2015

TSny

Could be a misprint.

ρ is a volume charge density. So, Q =∫ρdV = ∫ρd3r where dV and d3r are different ways to express a volume element.

Q ≠ ∫ρdr.

3. Aug 6, 2015

Slusho

Hmm. Maybe it's a way of simplifying the problem to be one-dimensional from three-dimensional due to inherent symmetry? I doubt it's a typo as this comes from a solutions manual with plenty of errata documents available.

4. Aug 6, 2015

TSny

The integrand must have the dimensions of charge. But ρdr does not have the dimensions of charge. The quantity dr has the dimensions of length. You need to multiply ρ by volume in order to get charge.

For problems of spherical symmetry, you can take the volume element d3r to be a spherical shell of radius r and thickness dr. This would be appropriate for the second term in your expression for ρ. Can you express the volume of the shell in terms of r and dr?