# Relation between Power density and temperature in an antenna

Considering an antenna placed inside a blackbody enclosure at temperature T, the power received per unit bandwidth is:

$latex \omega = kT$

where k is Boltzmann constant.

This relationship derives from considering a constant brightness $latex B$ in all directions, therefore Rayleigh Jeans law tells:

$latex B = \dfrac{2kT}{\lambda^2}$

Power per unit bandwidth is obtained by integrating brightness over antenna beam

$latex \omega = \frac{1}{2} A_e \int \int B \left( \theta , \phi \right) P_n \left( \theta , \phi \right) d \Omega $

therefore

$latex \omega = \dfrac{kT}{\lambda^2}A_e\Omega_A $

where:

- $latex A_e$ is antenna effective aperture
- $latex \Omega_A$ is antenna beam area

$latex \lambda^2 = A_e\Omega_A $ another post should talk about this

finally:

$latex \omega = kT $

which is the same noise power of a resistor.

source : Kraus Radio Astronomy pag 107