# Relationship between volume and surface charge density in the general case

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?

## 1 Answer

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).