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, $T_{}$. 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

$\displaystyle \ensuremath{B_{}}(\ensuremath{\theta_{},\phi_{}},\ensuremath{\nu}) = \frac{2\,\ensuremath{k}\ensuremath{T_{}}}{\ensuremath{\lambda}^2},$ (78)

and the observed spectral power is

$\displaystyle \ensuremath{dw}(\ensuremath{\theta_{0},\phi_{0}},\ensuremath{\nu}...
...math{\nu}) \,\frac{2\,\ensuremath{k}\ensuremath{T_{}}}{\ensuremath{\lambda}^2}.$ (79)

If we plug a matched resitor at the antenna output, it will deliver a noise power of

$\displaystyle \ensuremath{dw}(\ensuremath{\theta_{0},\phi_{0}},\ensuremath{\nu}) = \ensuremath{k}\,\ensuremath{T_{}},$ (80)

because its temperature is $T_{}$. 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.