Coulomb's law tells us about the electric field of a point charge. This suggests the question: what is the charge density of a point charge located at point P?
This question can be answered in terms of integrals over volumes: if you integrate this density over a volume that does not contain P you get 0. If the volume contains P you get the amount of charge located there.
This means essentially that the density is 0 except at the point P. But the contribution at that point must be large enough to make a significant contribution to the integral.
This is not possible for a bounded function or for any function with our definition of the integral.
However, we do not really know that point particles are such, and have no way to distinguish experimentally between a point particle and one that has extended shape with radius on the order of 10-100 centimeters.
The integration we perform over density to get the total mass or charge in a volume V is a volume integral, which, when expressed in terms of ordinary one dimensional integrals requires three one dimensional integrals.
A phenomenon similar to the density of a point particle occurs in one dimension, where it is called a "delta function". The density of a point particle can actually be described as the product of delta functions in variables x, y and z. We therefore turn the discussion to the one dimensional situation.
Before discussing it further, we address the questions: why do we care about such matters? And why now?
And here is the answer: Coulomb's law, describes the electric field that accompanies
a point particle. We can to use this fact to determine the electric field produced
by any charge distribution characterized by charge density .
All we have to do is add up the contributions to the electric field from each
of the charged particles that give rise to that charge density.
We will soon see that in doing this we are actually solving a linear differential equation with an inhomogeneous term .
The method that we blunder into here for solving this equation can be characterized as follows: we first find the solution for a delta function inhomogeneous term for an arbitrary point P (here this is Coulomb's law independent of P). Then we exploit this solution (by integrating it), to find the solution for a general inhomogeneous term.
The solution we find is one which obeys the differential equation with appropriate zero boundary conditions. It is in general called a Green's Function for the given differential equation and those boundary conditions, since Green invented this approach. (Green was, by the way, a baker, who had a keen interest in mathematics. He was self taught in science and mathematics.) Finding it allows us to solve the same equation with an arbitrary inhomogeneous term by integration.
Here is another way to look at the same idea: you want to find the response of a given physical system to an arbitrary external stimulus whose response to a sum of stimulae is the sum of its response to each. To do this you find the response to single point stimuli at each possible point. You can then find the response to the arbitrary stimulus by (summing) integrating the product of that stimulus with the response function.
This very powerful method for solving general inhomogeneous linear differential equations, by solving such equations with delta function inhomogeneity first, means that we want to use delta functions.
And we want to use them here to generalize Coulomb's law to determine the electric field from an arbitrary charge distribution.
The one dimensional delta function can be described as the derivative of a step function, so we begin by defining step functions.
There are two standard step functions in common use.
The first, denoted by , has value 0 for negative x, and is 1 for x positive. We don't much care what its value at x = 0 is, but it should probably be either 0, or 1.
The second, often written as is 1 for x positive, -1 for x negative and 0 at 0.
These are related by , except perhaps at x = 0.
Obviously, as defined, neither of these functions has a derivative at x = 0.
Yet either one differs only slightly from simple functions that have derivatives everywhere. In fact there is no real way to tell the difference between the two.
In consequence, we may use delta functions while pretending to be using the derivative of one of these other functions.
Mathematicians were at first highly suspicious of the delta function. However they now accept it as what is called a "distribution" though not as a function.
A function that for all practical purposes is indistinguishable from is . The arctangent approaches for large negative values and for large positive ones. The factor in its argument accelerates its behavior so that it is essentially a step function. Its derivative from -1 to 1 which is perfectly well defined, is essentially .
The second function is, apart from a constant, what is called the error function. Its derivative is for k very large which has approximate properties of . This function, and the derivative of the arctangent are both huge for extremely small arguments and almost zero otherwise.
The one defined in terms of exponents and error functions gets much smaller away from zero than it gets big near zero which is a bit nice than the arctangent.
The nice properties of the delta function are that it is zero for argument other than 0, and its integral over an interval containing 0 is 1 and these properties are more or less shared by these functions except at unobservable arguments.
As a function, the delta function does not make much sense at argument 0, unless it is integrated over; its integral is a step function, and that is perfectly well defined.
When you see a delta function outside an integral, which you mostly will not, you can think of it as one of the two functions mentioned above, and not lose sleep over it.
The density of a point particle can then be described as the product of delta functions in each of the three variables x, y and z. It can be written as