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Section 11.8 The Dirac Delta Function and Densities

The total charge/mass in space should be the same whether we consider it to be distributed as a volume density or idealize it as a surface or line density. See <<(reference to GVC:Densities)>>.

The Dirac delta function relates line and surface charge densities (which are really idealizations) to volume densities. For example, if the surface charge density on a rectangular surface is \(\sigma(x,y)\text{,}\) with dimensions \(C/L^2\text{,}\) then the total charge on the slab is obtained by chopping up the surface into infinitesimal areas \(dA = dx\, dy\) and summing up (integrating) the charge \(\sigma(x,y) dA\) on each piece, \(\int\int \sigma(x,y) \, dx \, dy\text{.}\) Equivalently, one can recognize that this surface charge density is actually a volume charge density, idealized to be concentrated at, say, \(z=0\text{.}\) Thus,

\begin{equation} \rho(x,y,z) = \sigma(x,y) \,\delta(z)\tag{11.8.1} \end{equation}

and integrating this over a solid region yields

\begin{equation} \int\!\int\!\int \rho(x,y,z) \,dz\,dx\,dy = \int\!\int\!\int \sigma(x,y) \,\delta(z) \,dz\,dx\,dy = \int\!\int \sigma(x,y) \,dx\,dy\tag{11.8.2} \end{equation}

which yields the same answer as before. Recall that \(\delta(z)\) has dimensions of inverse length, so that \(\rho\) has the correct dimensions, namely \(CL^{-3}\text{.}\)