# Field strength in free space

Field strength in free space is a term in telecommunications. It is the 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]

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

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

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

The following relation is obtained by equating the two,

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

or by rearranging the terms

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

## Numerical values

Impedance of free space is $120 \cdot \pi$

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

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

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
is equivalent to multiplying the expression on the right by $\sqrt{1000}$.[2] In this case,
$\mbox{E} \approx 222\cdot\frac{\sqrt{N}}{d}$