electromagnetism - Relationship between surface density and volume density

Often in an E&M problem, I'm having to "chop" an extended object into an infinite sum of smaller extended objects which I know more about to find a potential or electric field or whatever. The part I'm having trouble with is how to convert the surface density (line density) to volume density (surface density) and vice versa.

In one problem I did recently, I just thought maybe I should just collapse a dimension in volume density $\rho$ to get surface density $\sigma$ via: $\rho dr = \sigma$. It gave me the right answer, but I doubt that's the right way to think about it and that that formula will always work.

Right now I'm trying to "cut" a cylinder of uniform volume density $\rho$ into disks of uniform surface density $\sigma$. I thought maybe the right approach would be to relate the total charges. I've got $$Q_{\text{cylinder}}=\int \rho d\tau=\rho \pi r^2 h\\ \text{and}\\ Q_{\text{disk}}=\int \sigma dS=\sigma \pi r^2\; .$$ However then I'm at a loss as to how I should relate $Q_\text{cylinder}$ and $Q_\text{disk}$. What's the general process here?

Answer

I have personally also contemplated this issue, and have come up with a simple solution that is satisfactory, to me at least. I'm sure this can also be found in many textbooks. In general, we have

$$\tag{$\star$} Q=\int \rho\ d\tau$$

because we are considering a three dimensional space. Intuitively, we feel it should be possible to talk about a three dimensional charge distribution in every case. The question is how to conceptualize this when discussing surface, line or point charges. The solution comes in the form of the Dirac delta distribution (or function, depending who you ask).

Let's take a look at an example: Consider a 2-sphere of radius $R$, with some charge distribution $\sigma(\theta,\phi)$ on it. What is the three dimensional charge distribution $\rho(r,\theta,\phi)$ corresponding to this situation? Like I said, we have to use the Dirac delta:

$$\rho(r,\theta,\phi)=\delta(r-R)\sigma(\theta,\phi)$$

Now, $(\star)$ gives us: $$ Q=\int\rho\ d\tau=\int_{0}^{2\pi}\int_{0}^\pi\int_0^\infty \rho\ r^2\sin\theta\ dr\ d\theta\ d\phi=R^2\int_{0}^{2\pi}\int_{0}^\pi\sigma\ \sin\theta\ d\theta\ d\phi$$

Similarly, when considering a line or point charge, one uses two or three Dirac delta's to describe the distribution in 3-space.