I'm new to this forum and turned here because I have no other help to turn to. In studying antenna theory and I came across the relationship between E0(Electric field at aperture) and Er(Electric field at distance r). It was stated as follows:

$$ |Er| = \frac { |E_0| \times A } { r \times \lambda } $$ where:

\$ Er \$ is the electric field broadside to aperture at distance \$ r \$,

\$ E_0 \$ the electric field at aperture,

\$ A \$ the effective aperture of the antenna,

\$ r \$ the distance between the two fields,

\$ \lambda \$ the of radiating filed wave length.

I have tried extensively to find proof to this either mathematically or intuitively but I am really confused, specially how lambda makes an effect on the field intensity change. My only idea is that as r increases, the field varies according to inverse square law which I believe is a very basic view than what is expected.

Any help will be appreciated, or links to sources where I can research myself.

  • 1
    \$\begingroup\$ What have you found so far, get out your EM book and start hashing equations together. Do some units analysis. \$\endgroup\$
    – Voltage Spike
    Jan 8, 2017 at 3:24
  • \$\begingroup\$ I am currently following Antennas 2nd edition by John D. Kraus and I find the derivation quite difficult to understand. I was hoping to find a simpler explanation somewhere. Unit analysis does verify the equation by doesn't really explain why. \$\endgroup\$ Jan 8, 2017 at 3:43
  • \$\begingroup\$ Hazarding a guess here! Antenna aperture is a function of wavelength. So lambda needs to be included (antennas have less gain as wavelength increases). On Ham.SE: Why is antenna aperture a function of wavelength? \$\endgroup\$ Sep 7, 2018 at 16:15

1 Answer 1


Once a proper EM wave is formed by an antenna, the power density (watts per square metre) reduces with distance squared. Hopefully this can be seen but if not, think about the conceptual isotropic antenna - it takes in a signal power from the feed wires and emits that power equally in all directions. This is just like a light bulb emitting approximately equally in all directions and, if you measured the light power passing through a fixed area at some distance you would find the light power measured falls with distance squared.

Given that the power density is simply the electric field (volts/m) multiplied by the magnetic field (amps per metre) either of these terms reduces linearly with distance so, it's not surprising to see this formula.

The "A" bit refers to the effective aperture of any antenna. If an antenna is to liberate "watts" from "watts per sq metre" then it must have an effective area so that it can "net" the power. For any antenna this is: -

enter image description here

Where lambda is wavelength and G is antenna gain. An isotropic antenna has G set to 1. Also note that an isotropic antenna is just a mathematical concept for comparing performances of different antennas.


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