# Dipole field strength in free space

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Dipole field strength in free space, in telecommunications, is the electric field strength caused by a half wave dipole under ideal conditions. The actual field strength in terrestrial environments is calculated by empirical formulas based on this field strength.

## Power density

Let N be the effective power radiated from an isotropic antenna and p be the power density at a distance d from this source[1]

${\displaystyle {\mbox{p}}={\frac {N}{4\cdot \pi \cdot d^{2}}}}$

Power density is also defined in terms of electrical field strength;

Let E be the electrical field and R be the impedance of the free space

${\displaystyle {\mbox{p}}={\frac {E^{2}}{R}}}$

The following relation is obtained by equating the two,

${\displaystyle {\frac {N}{4\cdot \pi \cdot d^{2}}}={\frac {E^{2}}{R}}}$

or by rearranging the terms

${\displaystyle {\mbox{E}}={\frac {{\sqrt {N}}\cdot {\sqrt {R}}}{2\cdot {\sqrt {\pi }}\cdot d}}}$

## Numerical values

Impedance of free space is roughly ${\displaystyle 120\cdot \pi }$

Since a half wave dipole is used, its gain over an isotropic antenna (${\displaystyle {\mbox{2.15 dBi}}=1.64}$ ) should also be taken into consideration,

${\displaystyle {\mbox{E}}={\frac {{\sqrt {1.64\cdot N}}\cdot {\sqrt {120\cdot \pi }}}{2\cdot {\sqrt {\pi }}\cdot d}}\approx 7\cdot {\frac {\sqrt {N}}{d}}}$

In this equation SI units are used.

Expressing the same equation in:

kW instead of W in power,
km instead of m in distance and
mV/m instead of V/m in electric field

is equivalent to multiplying the expression on the right by ${\displaystyle {\sqrt {1000}}}$.[2] In this case,

${\displaystyle {\mbox{E}}\approx 222\cdot {\frac {\sqrt {N}}{d}}}$

## References and notes

1. ^ Reference data for radio Engineers, Howard W.Sams co,Indianapolis, 1956, 27-7
2. ^ K.H.Kaltbeitzer: Site selection, EBU Techhnical Monograph 3104,Bruxelles,1965, p 30