When describing a physical system, one would often like to describe some of its components as pointlike particles, i.e. with no spatial extent. Obviously, all objects have a finite size, but this size can be much smaller than the length scales relevant to the problem at hand. All the attributes of the particle (mass, charge, etc.) can then be assigned to a space point.
However convenient that description might be conceptually, casting it in mathematical language is problematic, the usual notion of a function being of little use: the charge density (for instance) corresponding to a point-like particle would be zero everywhere except at the one point where the particle resides. At this point, it should presumably be infinite —in some sense— so that the integral of the density in the vicinity of the particle should yield its total charge. Furthermore, this "function" would vary from zero to infinity infinitely fast; clearly, there is no sense in talking about its derivative. Of course, one could avoid these infinities by writing the density as a localized function (e.g. a Gaussian), that becomes more and more peaked while maintaining a constant integral. This limiting process might be useful in some circumstances, but it is in general more cumbersome than simply fixing a small and finite size for the particle.
In order to retain the advantage of a point-like model, one needs to define a distribution, or generalized function, which extends the concept of a "well-behaved" function to cover such extreme situations. Instead of going through a limiting process, we will simply denote the charge density of a point-like particle with total charge \(q\) placed at point \(\mathbf{r}_0\) by \(q\delta(\mathbf{r} - \mathbf{r}_0)\), where the distribution \(\delta(\mathbf{r})\) is known as the Dirac delta function. Of course, this mere designation is not sufficient unless complemented by a procedure for using the distributions in the equations of physics. Fortunately, the formalism is quite simple. As an example, let us find the force applied to the point-like charge described above by a space-varying electric field \(\mathbf{E}(\mathbf{r})\). Since the particle is located in \(\mathbf{r}_0\), it will experience the field \(\mathbf{E}(\mathbf{r}_0)\), so that the force is simply \(\mathbf{F} = q \mathbf{E}(\mathbf{r}_0)\). For a continuous charge distribution, \(\rho (\mathbf{r})\), the force must be written as an integral:
\begin{equation}
\mathbf{F} = \int \mathbf{E}(\mathbf{r}) \rho (\mathbf{r}) \mathrm{d}^3 \mathbf{r}
\label{eq:Felec}
\end{equation}
Let us replace formally in equation \eqref{eq:Felec} the function \(\rho (\mathbf{r})\) by the distribution:
\begin{equation}
q\delta(\mathbf{r} - \mathbf{r}_0)
\label{eq:diracdistr}
\end{equation}
We have seen above what the force is in this case:
\begin{equation}
\mathbf{F} = \int \mathbf{E}(\mathbf{r}) q\delta(\mathbf{r} - \mathbf{r}_0) \mathrm{d}^3 \mathbf{r} = q \mathbf{E}(\mathbf{r}_0)
\label{eq:Felec1}
\end{equation}
and, after dividing both sides by the constant \(q\):
\begin{equation}
\int \mathbf{E}(\mathbf{r}) \delta(\mathbf{r} - \mathbf{r}_0) \mathrm{d}^3 \mathbf{r} = \mathbf{E}(\mathbf{r}_0)
\label{eq:defDirac}
\end{equation}
yielding a definition of sorts for the Dirac delta function. Its significance is in line with our intuitive formulation above: from the entire field \(\mathbf{E}(\mathbf{r})\), it extracts its value at the relevant point \(\mathbf{r}_0\). It is therefore a sampling tool1 that can be used if the field is "smooth enough" in some mathematical sense that we will not discuss any further here. However, on physical grounds it should be clear that the pointlike model is no longer useful if the field varies significantly over the size of the particle, because this length scale becomes relevant for the problem and should be taken into account explicitly by writing equation \eqref{eq:Felec} with the detailed charge distribution \(\rho (\mathbf{r})\).
The distribution \eqref{eq:diracdistr} describes a charge density: it must therefore have units of \(\dfrac{\mbox{charge}}{\mbox{volume}}\), implying that the Dirac delta has units of reciprocal volume. This is also apparent from equation \eqref{eq:defDirac}.
1. In undergraduate-level expositions this is often called a test charge, without going into the detail.↩
However convenient that description might be conceptually, casting it in mathematical language is problematic, the usual notion of a function being of little use: the charge density (for instance) corresponding to a point-like particle would be zero everywhere except at the one point where the particle resides. At this point, it should presumably be infinite —in some sense— so that the integral of the density in the vicinity of the particle should yield its total charge. Furthermore, this "function" would vary from zero to infinity infinitely fast; clearly, there is no sense in talking about its derivative. Of course, one could avoid these infinities by writing the density as a localized function (e.g. a Gaussian), that becomes more and more peaked while maintaining a constant integral. This limiting process might be useful in some circumstances, but it is in general more cumbersome than simply fixing a small and finite size for the particle.
In order to retain the advantage of a point-like model, one needs to define a distribution, or generalized function, which extends the concept of a "well-behaved" function to cover such extreme situations. Instead of going through a limiting process, we will simply denote the charge density of a point-like particle with total charge \(q\) placed at point \(\mathbf{r}_0\) by \(q\delta(\mathbf{r} - \mathbf{r}_0)\), where the distribution \(\delta(\mathbf{r})\) is known as the Dirac delta function. Of course, this mere designation is not sufficient unless complemented by a procedure for using the distributions in the equations of physics. Fortunately, the formalism is quite simple. As an example, let us find the force applied to the point-like charge described above by a space-varying electric field \(\mathbf{E}(\mathbf{r})\). Since the particle is located in \(\mathbf{r}_0\), it will experience the field \(\mathbf{E}(\mathbf{r}_0)\), so that the force is simply \(\mathbf{F} = q \mathbf{E}(\mathbf{r}_0)\). For a continuous charge distribution, \(\rho (\mathbf{r})\), the force must be written as an integral:
\begin{equation}
\mathbf{F} = \int \mathbf{E}(\mathbf{r}) \rho (\mathbf{r}) \mathrm{d}^3 \mathbf{r}
\label{eq:Felec}
\end{equation}
Let us replace formally in equation \eqref{eq:Felec} the function \(\rho (\mathbf{r})\) by the distribution:
\begin{equation}
q\delta(\mathbf{r} - \mathbf{r}_0)
\label{eq:diracdistr}
\end{equation}
We have seen above what the force is in this case:
\begin{equation}
\mathbf{F} = \int \mathbf{E}(\mathbf{r}) q\delta(\mathbf{r} - \mathbf{r}_0) \mathrm{d}^3 \mathbf{r} = q \mathbf{E}(\mathbf{r}_0)
\label{eq:Felec1}
\end{equation}
and, after dividing both sides by the constant \(q\):
\begin{equation}
\int \mathbf{E}(\mathbf{r}) \delta(\mathbf{r} - \mathbf{r}_0) \mathrm{d}^3 \mathbf{r} = \mathbf{E}(\mathbf{r}_0)
\label{eq:defDirac}
\end{equation}
yielding a definition of sorts for the Dirac delta function. Its significance is in line with our intuitive formulation above: from the entire field \(\mathbf{E}(\mathbf{r})\), it extracts its value at the relevant point \(\mathbf{r}_0\). It is therefore a sampling tool1 that can be used if the field is "smooth enough" in some mathematical sense that we will not discuss any further here. However, on physical grounds it should be clear that the pointlike model is no longer useful if the field varies significantly over the size of the particle, because this length scale becomes relevant for the problem and should be taken into account explicitly by writing equation \eqref{eq:Felec} with the detailed charge distribution \(\rho (\mathbf{r})\).
The distribution \eqref{eq:diracdistr} describes a charge density: it must therefore have units of \(\dfrac{\mbox{charge}}{\mbox{volume}}\), implying that the Dirac delta has units of reciprocal volume. This is also apparent from equation \eqref{eq:defDirac}.
1. In undergraduate-level expositions this is often called a test charge, without going into the detail.↩
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