Total Charge from Charge Density?

[Slusho]

Homework Statement

Find the total charge Q given the charge density ρ(r)=ε₀A(4πδ³(r)-π²e^-λr/r

The Attempt at a Solution

I know the solution's steps start with: Q=∫ρdr=ε₀A(4π∫δ³(r)dr-λ²∫e^-λr(4πr²)/rdr)

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.

 

[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.

 

[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.

 

[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?