A generic demonstration of this statement uses a thought experiment. Let's
assume that the antenna is put in a box at a given physical
temperature,
. This box can be considered an emitting black-body. If the
frequency is appropriately chosen, the Rayleigh-Jeans regime is
fulfilled. This implies that the source brightness is
 |
(78) |
and the observed spectral power is
 |
(79) |
If we plug a matched resitor at the antenna output, it will deliver a
noise power of
 |
(80) |
because its temperature is
. As it is matched to the antenna, this
power is also the spectral power measured by the antenna. Hence, equating
the last two equations, we yield the searched relationship.