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If we surround an energy emitting radio antenna by large number of energy absorbing radio antennas, do the surrounding antennas have a large effect on the central antenna?

The emitting antenna emits circular waves, and the absorbing antennas form a large circle around the emitting antenna, at distance of many wavelengths.

Perhaps the absorbing antennas emit EM-waves that all interfere constructively at the position where the emitting antenna is?

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Two antennas interfere with each other's operation if they are close to each other. In practice, close means to be within the Fraunhofer distance outside of which they can be treated essentially independently. Now make a spherical surface and tessellate it with elementary dipoles placed about a $\lambda/2$ distance. Match each dipole properly and they will be nearly independent of each other. Now place at the center of the sphere a properly matched radiating antenna and let the radius be larger then its Fraunhofer distance, that is the spherical surface be electrically far away. Then almost all, practically all, energy you radiate away will be absorbed by the dipoles on the surface without interfering with the emission, as you were asking in your question.

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