Can anyone explain this about Directivity relation to effective area?

[Alan0354]

I am learning Antenna.

http://ptuece.loremate.com/awp/node/2

In question 15, It tries to relate directivity with effective area by using two antenna A1 and A2. I can follow the derivation to the last part where we let \[P_1\;=\;P_2\]. Then it said according to reciprocity, \[P_{1-2}\;=\;P_{2-1}\]. Can anyone explain why this is true even if the two antennas have different directivity and effective area?

Thanks

Alan

 

[Khashia]

It is explained lower in the article.i.e in the reciprocity theorem according to which

"If Antenna A1 is a transmitting antenna and antenna A2 is the receiving one;
the ratio of transmitted to receive Power Pt/Pr will not change if
A1 becomes the receiving antenna and A2 becomes the transmitting one"

Here P1-2 is the receive power when A1 is transmit and A2 is receive antenna and
P2-1 is the receive power when A2 is transmit and A1 is receive antenna

P1 = transmit power by antenna A1
P2 = transmit Power by antenna A2

then according to reciprocity theorem:
P1/P1-2 (Pt/Pr if A1 transmits) = P2/P2-1 (Pt/Pr if A2 transmits)

Now imposing condition P1 = P2;

results in P1-2 = P2-1