Why does the linear antenna radiate at k^2?

[qnach]

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)?

 

[the_emi_guy]

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.

 

[qnach]

so the (real) power for a dipole should be proportional to k^2 instead of k^4 ?

 

[the_emi_guy]

Yes