Why does the linear antenna radiate at k^2?

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SUMMARY

The discussion centers on the radiation characteristics of linear antennas, specifically dipoles, as described in Jackson's "Classical Electrodynamics" (3rd edition). It clarifies that while the total power radiated by a dipole is generally proportional to k^4 (equation 9.24), the effective radiation for a linear antenna is proportional to k^2 (equation 9.29) due to the dipole moment's dependence on angular frequency ω in the denominator (equation 9.27). This relationship arises because the terms involving the dipole moment replace two of the k factors in the power equation.

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qnach
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I am reading Jackson's book on classical electrodynamics.
It said in page 412 to 415 (3rd edition) that the total power radiated by a
dipole is proportional to k^4 (equation 9.24)
quadrupole is proportional to k^6 (equation 9.49)
But why does the linear antenna at page 412, which is a dipole, radiate at k^2 (equation 9.29)?
 
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The dipole moment, p, has ω in the denominator (equation 9.27). Thus the c2|p|2 terms in 9.24 replace two of the k's.
 
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so the (real) power for a dipole should be proportional to k^2 instead of k^4 ?
 
Yes
 

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