The Near Zone

Suppose that we are in the near zone. Then by definition

$$k \vert \vx - \vx' \vert << 1$$

and

$$e^{i k \vert \vx - \vx' \vert} \approx 1$$

This makes the integral equation into the ``static'' form already considered in chapter 5 (cf. equation (5.32)). We see that $-1/ 4\pi\vert \vx - \vx' \vert$ is just the Green's function for the good old Poisson equation in this approximation and can be expanded in harmonic functions just like in the good old days:

$$G_0(\vx,\vx') = \sum_L \frac{-1}{2 \ell + 1} \frac{r'^\ell}{r^{\ell+1}} Y_L(\hat{r}) Y_L(\hat{r'})^\ast.$$ (13.7)

Note Well: I will use $L \equiv (\ell,m)$ freely and without warning in this course. The sum is over all $\ell,m$ . Hopefully, by now you know what they run over. If not, read the chapter in Wyld on spherical harmonics and review Jackson as well. This is important!

This means that (if you like)

$$\lim_{ kr \rightarrow 0} \vA (\vx) = \sum_L \frac{1}{(2 \ell + 1)r^{\ell+1}} Y_L(\hat{r}) \int \vJ(\vx') r'^\ell Y_L(\hat{r'})^\ast d^3r'.$$ (13.8)

We will use expressions like this (derived from the multipolar expansion of the Green's function) frequently in what follows. For that reason I suggest that you study it carefully and be sure you understand it.

Since (for fixed r outside the source)

$$\lim_{ k \rightarrow 0} \rightarrow \lim_{ kr \rightarrow 0}$$

we see that this limit is reached (among other times) when

$$k \rightarrow 0$$

(relative to the size of the source and point of measurement)! But then the IHE turns back into the Poisson equation (or inhomogeneous Laplace equation, ILE) as it should, come to think about it. The near fields oscillate harmonically in time, but are spatially identical to the fields produced by a ``static'' current with the given spatial distribution. That's why we also call the near zone the ``static zone''.