= Antenna gain-effective-area relation
{title2=$A_{\rm eff}(\Omega)=\lambda^2G(\Omega)/(4\pi)$}
For a reciprocal <antenna> with matched polarization and load, the available <antenna effective area> is proportional to the directional <antenna gain>. For a lossless <antenna>, a <thermal equilibrium> argument fixes the constant: one polarization of <Rayleigh-Jeans law> radiance gives incident <power> $(k_BT/\lambda^2)\int A_{\rm eff}\,d\Omega\,d\nu$, and matched <thermal electrical noise> gives $k_BT\,d\nu$. Thus $\int A_{\rm eff}\,d\Omega=\lambda^2$. Since $\int G\,d\Omega=4\pi$, the displayed relation follows. Reciprocal losses multiply both area and gain by the same efficiency.
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