Understanding Charge Density in Spherical Distributions: ρ=dQ(r)/dV(r)

[Nikitin]

Let's say you have a sphere which has a charge distribution where the charge behind a radius r can be expressed as Q(r). You also have the volume formula for a sphere, V(r).

Why is ρ, the charge density, defined as: ρ=dQ(r)/dV(r) instead of simply ρ=Q(r)/V(r)?

 

[voko]

By integrating the density over some volume, you should get the total charge in that volume. Which of the two expressions satisfies that?

 

[Nikitin]

Why can't you just integrate ρ(r) over a volume, with ρ(r) = Q(r)/V(r)?

 

[voko]

Is $\int \rho(r) dV = \int \frac {Q(r)} {V(r) } dV$ equal to Q? What about $\int \rho(r) dV = \int \frac {dQ(r)} {dV(r) } dV$?

 

[Nikitin]

I see it now. In the future, am i always supposed to use infinetesimal amounts for stuff like this?

 

[voko]

It is hard to tell what you mean by "stuff like this", buy generally densities and concentrations are derivatives of some quantity with regard to volume (or mass), so that their integrals over some volume (or mass) restore the original quantity. If in doubt, just use this check.