Is it a misnomer when an electrostatics problem specifies a conductor to have $\rho$ distributed uniformly throughout

shouldn't it just be $\sigma$ spread across the surface area

\sigma for surface charge

oh right

and you're more-or-less right i'd say---i'd say it's just "charge" which is uniformly distributed, and for a surface that's a uniform sigma

hmm okay. So the picture should still be charge on the surface. It might just be a way for the question to not "give away" that fact.

right

or shorthand for it

sometimes we get so used to saying it the quick way that we forget that it's not obvious

Even in the classical picture, it's not implied that $\mathbf{E}=0$ in a conductor btw

well, you do have to say something about electrostatic equilibrium

it has to be assumed since it's not impossible for charge to exist along the axes/points of symmetry for the conductor

i.e., a single point charge in the center of a sphere, or a continuous line of charge along the cylindrical axis of symmetry for a cylindrical conductor

true

there's probably some semantics about what the right 'definition' is

the formal definition i'd probably fall back on is "a conductor is what you get if you start with a dielectric material and take $\epsilon\to \infty$."

@SillyGoose

oh i think the implication was that it wasn't a conductor but an insulator btw

also that is adorable @Relativisticcucumber

@RyderRude the domain of the functor is the group. It isn't the category of groups

Any group is a category with a single object

@Obliv in reality it is very logical in electrostatics. The definition of conductor is a material where charge is free to move, at equilibrium (which is were electrostatics works) no charge should be moving and for that to happen you need $\vec{E} = 0$ inside the conductor or else the charge would be free to move

For the level of the discourse it is extremely clear that it is some kind of misuderstanding. It should be an insulator. But when you learn further and need to model what the electrons inside a metal is doing, then we would treat all the freed electrons in a statistical thermodynamics model, separate from all the ions. The first approximation is then to smear the ions out into a uniform charge density background. Later, when you learn crystalline physics, then you can upgrade to crystalline

ionic background

@Relativisticcucumber amazing