# Marconi's law

Marconi's law is the relation between length of antennas and maximum signaling distance of radio transmissions. Guglielmo Marconi enunciated at one time an empirical law that, for simple vertical sending and receiving antennas of equal height, the maximum working telegraphic distance varied as the square of the height of the antenna. It has been stated that the rule was tested in experiments made on Salisbury Plain in 1897, and also by experiments made by Italian naval officers on behalf of the Royal Italian Navy in 1900 and 1901. Captain Quintino Bonomo gave a report of these experiments in an official report.[citation needed]

## Description

If H is the height (i.e. the length) of the antenna and D the maximum signalling distance, then we have, according to Marconi's law

$H=c{\sqrt {D}}$ ,

where c is some constant.

 c D Apparatus 0.17–0.19 60 kilometres (37 mi) Marconi's original apparatus 0.15–0.16 60 kilometres (37 mi) Same, with longer sending spark 0.12–0.14 136 kilometres (85 mi) Marconi's improved apparatus, with jigger in receiver 0.23–0.15 143 kilometres (89 mi) The same, but with Italian Navy telephonic receiver

Marconi's law can be deduced theoretically as follows:

Hertz has shown that distances large compared with its length the magnetic force of a linear oscillator varies inversely as the distance. The maximum value of the current set up in any given receiving antenna varies as its length, also as the magnetic force of the waves incident on it, and as the maximum value of the current in the transmitting antenna. Hence, if $M$ is the magnetic force of the waves incident on a receiving antenna of height H, and if $D$ is the distance between the sending and receiving antenna, and if $I_{1}$ and $I_{2}$ are the maximum values of the currents in the sending and receiving antennæ, we have—

$M\propto {\frac {I_{1}}{D}}$ and $I_{2}\propto MH$ Hence $I_{2}\propto {\frac {I_{1}H}{D}}$ Also, since for a given charging voltage the current $I_{1}$ in the sending antenna varies very nearly as its capacity—that is, as its height—and if the sending antenna has the same height, $H$ , as the receiving aerial, we have—

$I_{1}\propto H$ But $I_{2}\propto {\frac {I_{1}H}{D}}$ Therefore $I_{2}\propto {\frac {H^{2}}{D}}\propto$ some constant

For any given receiving apparatus a certain constant minimum value of the maximum current in the receiving antenna is necessary to cause a signal. Therefore it follows that, with given receiving and sending apparatus, we must have ${\frac {H^{2}}{D}}$ a constant, or—

$H=c{\sqrt {D}}$ That is, the maximum signalling distance with given apparatus will vary as the square of the height of the antenna.

The above law is, however, much interfered with by the nature of the surface over which the propagation takes place.