``Proof'' of the identity (Eq. (1.31))

Noting that

we see that we must show that

We introduce a small radius a such that:

For a fixed value of a,

If the function f(r) is continuous, we can make a Tayor expansion about the point r = r^¢. Jackson's text shows that it is necessary to keep only the leading term. The integral over the small sphere about r^¢ can be carried out analytically, by changing to a coordinate system centered at r^¢;

so that

We note that

If the infinitesimal value a is a << R, then (R² +a²)^3/2 » R³ and the right hand side of Eq. 8 is - 4 p. Therefore, Eq. 7 becomes,

which is consistent with Eq. 3.

Examples of solutions of the one-dimensional Poisson equation

Consider the following one dimensional charge distribution:

We want to find the electrostatic potential such that

In class, we showed that the solution is given by:

The electrostatic field is given by:

The electrostatic potential can be determined by piecewise solution within each of the four regions or by use of the Green's function G(x,x^¢) = x _< , where,

In the expression for G(x,x^¢) , x _< should be taken as the smaller of x and x^¢. It can be shown that Eq. 14 gives the identical result for F(x) as given in Eq. 12.