The Hertzian dipole

(1075) |

(1076) |

The magnetic vector potential generated by a current distribution
is given by
the well-known formula (see Sect. 4.12)

(1077) |

(1078) |

(1079) |

In the region ,

(1080) |

(1081) |

(1082) |

(1083) |

The scalar potential is most conveniently evaluated using the Lorentz gauge condition (see Sect. 4.12)

(1085) |

(1086) |

Given the vector and scalar potentials, Eqs. (1084) and (1087),
respectively, we can
evaluate the associated electric and magnetic fields using (see Sect. 4.12)

(1088) | |||

(1089) |

Note that we are only interested in

(1090) |

(1091) |

The average power crossing a spherical surface (whose radius is much greater than
) is

(1092) |

(1093) |

Note that the energy flux is radially outwards from the source. The total power flux across is given by

(1095) |

(1096) |

Recall that for a resistor of resistance the average ohmic heating power is

(1097) |

(1098) |

(1099) |

In the theory of electrical circuits, an antenna is conventionally represented as a resistor whose resistance is equal to the characteristic radiation resistance of the antenna plus its real resistance. The power loss associated with the radiation resistance is due to the emission of electromagnetic radiation. The power loss associated with the real resistance is due to ohmic heating of the antenna.

Note that the formula (1100) is only valid for . This suggests
that
for most Hertzian
dipole antennas: *i.e.*, the radiated power is
swamped by the ohmic losses. Thus, antennas whose lengths are much less than
that of the emitted radiation tend to be extremely inefficient.
In fact, it is necessary
to have in order to obtain an efficient antenna. The simplest
practical antenna is the *half-wave antenna*, for which . This
can be analyzed as a series of Hertzian dipole antennas
stacked on top of one another, each
slightly out of phase with its neighbours. The characteristic radiation resistance
of a half-wave antenna is

(1101) |

Antennas can also be used to receive electromagnetic radiation. The incoming wave
induces a voltage in the antenna, which can be detected in an electrical
circuit
connected to the antenna. In fact, this process is equivalent to the emission
of electromagnetic waves by the antenna viewed in reverse. It
is easily demonstrated that antennas most readily detect electromagnetic radiation
incident from those directions in which they preferentially emit radiation.
Thus, a Hertzian dipole antenna is unable to detect radiation incident along
its axis, and most efficiently detects radiation incident in the plane perpendicular
to this axis. In the theory of electrical circuits, a receiving antenna is represented
as a voltage source in series with a resistor. The voltage source,
, represents
the voltage induced in the antenna by the incoming wave. The resistor,
, represents the power re-radiated by the antenna (here,
the real resistance
of the antenna is neglected). Let us represent the detector circuit as a single
load resistor , connected in series with the antenna. The question is:
how can we choose so that the maximum power is extracted from the
wave and transmitted to the load resistor? According to Ohm's law:

(1102) |

The power input to the circuit is

(1103) |

(1104) |

(1105) |

(1106) |

(1107) |

(1108) |

For a Hertzian dipole antenna interacting with an incoming wave whose electric
field has an amplitude , we expect

(1109) |

(1110) |

(1111) |

(1112) |

(1113) |

(1114) |

For a properly aligned half-wave antenna,

(1115) |