Recall our old buddy the **complex Poynting vector** for harmonic fields
(J6.132):

(13.53) |

The factor of comes from time averaging the fields. This is the energy per unit area per unit time that passes a point in space. To find the time average power per solid angle, we must relate the normal area through which the energy flux passes to the solid angle:

(13.54) |

and project out the appropriate piece of

Re | (13.55) |

where we must plug in

After a bunch of algebra that I'm sure you will enjoy doing, you will obtain:

(13.56) |

The polarization of the radiation is determined by the vector inside the absolute value signs. By this one means that one can project out each component of (and hence the radiation) before evaluating the square independently, if so desired. Note that the different components of need not have the same phase (elliptical polarization, etc.).

If all the components of
(in some coordinate system) have the same
phase, then
necessarily lies along a line and the typical angular
distribution is that of (linearly polarized) **dipole radiation**:

(13.57) |

where is measured between

(13.58) |

The most important feature of this is the dependence which is, after all, why the sky is blue (as we shall see, never fear).

**Example: A centerfed, linear antenna**

In this antenna, and

(13.59) |

From the continuity equation (and a little subtle geometry),

(13.60) |

and we find that the linear charge density (participating in the oscillation, with a presumed neutral background) is independent of :

(13.61) |

where the sign indicates the upper/lower branch of the antenna and the means that we are really treating (which cancels the related terms in the volume integral below). We can then evaluate the dipole moment of the entire antenna for this frequency:

(13.62) |

The electric and magnetic fields for in the electric dipole approximation are now given by the previously derived expressions. The angular distribution of radiated power is

(13.63) |

and the total radiated power is

(13.64) |

**Remarks.** For fixed current the power radiated increases as the square
of the frequency (at least when
, i. e. - long wavelengths
relative to the size of the antenna). The total power radiated by the antenna
appears as a ``loss'' in ``Ohm's Law'' for the antenna. Factoring out
, the remainder must have the units of resistance and is called the
**radiation resistance** of the antenna:

ohms | (13.65) |

where we do the latter multiplication to convert the resulting units to ohms. Note that this resistance is there for harmonic currents even if the conductivity of the metal is perfect. Note further that by hypothesis this expression will only be valid for small values of .

Good golly, this is wonderful. We hopefully really understand electric
dipole radiation at this point. It would be truly sublime if all radiators
were dipole radiators. Physics would be so easy. But (alas) sometimes the
current distribution has **no
moment** and there is therefore
**no dipole term**! In that case we must look at the next term or so in the
multipolar expansions

Lest you think that this is a wholly unlikely occurrance, please note that a humble loop carrying a current that varies harmonically is one such system. So let us proceed to: