What is the homogeneous charge density (vs inhomogeneous)?

Question

Books/exams often write something like "a homogeneously charged sphere"/"a homogeneously charged rod" - I am unsure what exactly is meant.

Take the example of a sphere with radius $R$. I could interpret the statement as $dq(\vec{x})$ should be independent of the position $\vec{x}$ inside the volume. Suggesting that

$$ \rho = \frac{Q}{2\pi^2 r^2 R sin\theta} $$ is the charge density because then $$ dq=\rho dV = \frac{Q}{2\pi^2 r^2 R sin\theta} r^2 sin\theta d\theta d\phi dr = \frac{Q}{2\pi^2R} d\theta d\phi dr $$ which is independent of $\vec{x}$ and also still gives the right overall charge when integrated over the sphere. I believe this is what Jackson does when talking about equation (3.132) in his book.

On the other hand you could also say that $\rho$ itself should not depend on $\vec{x}$ and therefore it is just $\rho = \frac{Q}{\frac{4}{3} \pi R^3}$. But then I don't understand why Jackson has a factor of $r^2$ in the denominator for a uniform charge density (see the link).

As far as I can tell the different charge distributions lead to different results for electric potential etc. so it matters greatly which one is actually the uniform/equally distributed one. Can someone help me out - which one is the correct expression? Thanks!

EDIT: I finally found the right answer: The factor multiplying the area (volume) element has to be constant if the surface (volume) charge density is constant. Therefore we need no variable factors in the case of a charged sphere

Answer 1

A charge density is something that you integrate over space to get a total charge. This means that you have to be careful what you mean when you write down its functional form—the total integrated charge depends on both the charge density $\rho$ and the integration measure (that is, the Jacobian for your particular coordinate system). The uniform charge density $\rho_U$ is the one for which $Q = \int dV \rho_U \propto V$, meaning that the total charge in any region is proportional to the volume of that region. In Cartesian coordinates, that means that $\rho_U(x,y,z)$ is constant. In spherical coordinates, you need $\rho_U(r,\theta,\phi) \propto \frac{1}{r^2 \sin \theta}$ to cancel out the $r^2 \sin \theta$ in the Jacobian.

This is a subtle issue with lots of trickiness when you try to generalize, but the abstract idea of "something you can integrate" as opposed to "something that's a function" is really important in math and physics. In more abstract language they're called "differential forms", and the operation of going from a function $\rho$ to the integrate-able thing $\rho dV$ is called the "Hodge star." You should learn this stuff the regular multivariable-calculus way first, but I say this just because the whole theory is very satisfying and fills in these tricky notions super well when you get there :)