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Assume we have surface $S$ enclosing a volume $V$ and assume the only charge present in this setting is located on the surface, given by a surface charge density $\sigma$. My question is about the correct way to model such a situation in electrostatics and more precisely, what volume charge density $\rho$ we should take in such a case.

There is a related post (here: Relationship between surface density and volume density) where the problem is discussed in specific examples and it is explained that $\rho$ can in those examples be defined via the Dirac-Delta.

In this question however, I want to know the general "definition" for $\rho$ in such a case, without reference to specific and simple examples like spheres, planes and the like.

My idea is that in the general case of a surface charge, the volume density $\rho$ should be defined as follows: $\rho$ is a distribution with $$\int_V \varphi\rho \ \text{d}^3x = \int_{S} \varphi\sigma \ \text{da}$$ for all functions $\varphi$ of a suitable function space. Here, "distribution" is understood in the sense of mathematical distribution theory.

My question is now simple: Is this the right way to do this?

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Sure, though if you want to be super strict, then $\rho$ shouldn't be written inside an integral. You should simply say $\langle\rho,\varphi\rangle:=\int_S\sigma\varphi\,da$, i.e the value of the functional $\rho$ on the test function $\varphi$ is defined to be the integral on the right. This formulation works in any number of dimensions, not just the case of a 2D surface $S$ embedded in $\Bbb{R}^3$.

For instance, let $M$ be a $k$-dimensional embedded submanifold of $\Bbb{R}^n$ ($0\leq k\leq n$), $d\mu_k$ be induced $k$-dimensional volume measure on $M$, and let $\sigma\in L^1(M;d\mu_k)$. We can define a distribution $\delta_{M,\sigma}$ on $\Bbb{R}^n$ by setting for each test function $\varphi$ on $\Bbb{R}^n$ (i.e $\varphi\in C^{\infty}_c(\Bbb{R}^n)$ is smooth compactly supported), \begin{align} \langle\delta_{M,\sigma}, \varphi\rangle&:=\int_M\sigma\varphi\,d\mu_k. \end{align} We refer to $\delta_{M,\sigma}$ as the single-layer distribution on the submanifold $M$ with density $\sigma$. The higher derivatives of this distribution are known as multiplet-layers (the "layer" terminology comes from classical PDE solving methods of single and double layer potentials for Laplace's/Poisson's equation).

See analysis/PDE texts for more information about this sort of stuff, for example, Dieudonne's Treatise on Analysis Vol III (Chapter 17.10), or Hormander's The Analyis of Linear Partial Differential Operators I (Chapter 6.1), or the classic text by Gelfand and Shilov Generalized Functions Volume I (Chapter 3.1 is all about distributions on $\Bbb{R}^n$ concentrated lower dimensional manifolds).

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