### Demonstration

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.