There is a point that could be cleared up a little in the derivation of the Friis transmission formula. The power received is Pr=IΔΩr, where ΔΩr=Ar/r2 is the solid angle formed by a receiving antenna with effective aperture Ar. Pt/4π is the intensity for an isotropic radiator which radiates uniformly in all directions. The derivation with the additional step is,
The Nyquist noise is due to the random motion of the electrons in the resistor. It is similar to Brownian motion. The formula is only true if the frequency is not too large. See Kittel, Elementary Statistical Physics for more information on this.
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