Ron wrote:

Question (repeated here for convenience):
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Assume a receiving antenna is in the center of a sphere and the received
signal is coming in equal amounts from all points on the surface of the
sphere. Which receiving antenna would capture more power, an omni or a
high gain beam? There are no noise and no losses.
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First, thanks for all the comments. They have helped me better
understand the answer. I am leaning toward the belief that the omni
(isotropic) antenna would capture more power and, as odd as it may seem,
would have more gain than a high gain beam (or any other directional
antenna for that matter). Here is my thinking:

This is a very unusual RF field. Usually the field is assumed to be
planar with coherent rays - then antennas behave as expected. But this
field originates uniformly from all points on the surface of a sphere.

Uniformly inward, outward, or both?

It does not spread but converges at the focal point of the sphere.

By focal point of the sphere do you mean the center of the sphere? How
big of a sphere are we talking about, and where is the antenna in
relation to the sphere?

An isotropic antenna placed at the focal point would collect all of the
rays whereas a directional antenna at would not.

Probably.

Therefore, in this particular situation, the isotropic would have higher
gain and capture more power than any directional antenna.

Not according to the accepted use of the term 'gain' in connection with
antennas.

Please correct me if I am wrong.

Ron, W4TQT

In the instance you describe, the antenna with gain will pick up less
signal than an antenna without gain. The gain antenna will be able to
sense signal arriving from only a fraction of the sphere, whereas the
isotropic antenna responds to signals arriving from the entire 4-pi
sphere. Therefore, the antenna with less gain produces the greater
signal level. But this should often be the case when a directional
antenna is pointed away from most of the signal. The omni, on the other
hand, is 'pointed toward' this particular signal in all directions.

Out of curiosity, what kind of signal source are you interested in?

ac6xg