Asymptotic properties in the Zones

In the near zone we get:

$$\mbox{\boldmath$B$} = \mu_0\mbox{\boldmath$H$}= \frac{i\omega\mu_0}{4\pi}(\hat{\mbox{\boldmath$n$}}\times \mbox{\boldmath$p$}) \frac{1}{r^2}$$ (11.123)

$$\mbox{\boldmath$E$} = \frac{1}{4\pi \epsilon_0}[3\hat{\mbox{\boldmath$n$}}(\hat{\mbox{\boldmath$n$}}\cdot \mbox{\boldmath$p$}) - \mbox{\boldmath$p$}] \frac{1}{r^3}$$ (11.124)

and can usually neglect the magnetic field relative to the electric field (it is smaller by a factor of $kr « 1$). The electric field is that of a ``static'' dipole (J4.13) oscillating harmonically.

In the far zone we get:

$$\mbox{\boldmath$B$} = \mu_0 \mbox{\boldmath$H$}= \frac{ck^2\mu_0}{4\pi}\left(\hat{\mbox{\boldmath$n$}}\times \mbox{\boldmath$p$}\right) \frac{e^{ikr}}{r}$$ (11.125)

$$\mbox{\boldmath$E$} = \frac{ic}{k}\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$B$}= c \left(\mbox{\boldmath$B$}\times \hat{\mbox{\boldmath$n$}}\right).$$ (11.126)

This is transverse EM radiation. Expanded about any point, it looks just like a plane wave (which is how ``plane waves'' are born!). We are most interested, as you know, in the radiation zone and so we will focus on it for a moment.