...we need at least currents of 10A to see something noticeable. Do such high currents even flow in an antenna???
I'm just answering this point for now. Most antennas have a resistive impedance R of 50 ohms. A basic ham radio installation is 100 W, and often followed by an amp in the KW range. The current can be calculated quite simply. P=R*I² so I=sqrt(P/R). For 100 W : I = 1.4 A For 1 KW : I = 4.47 A This current is the one that really flows at the antenna connection point. For 10A, a power of 5 KW would be needed, this becomes quasiprofessional equipment. But the current in an antenna of nonnegligible size compared to the wavelength, and even compared to the quarterwave, is not constant along the conductor. If the antenna is 1/4 wavelength long, the maximum current supplied by the transmitter will drop to zero at the end of the antenna. The current distribution in a dipole antenna, as a function of its length: https://www.youtube.com/watch?v=edyFGAT_87oTo have a higher current without increasing the power, one would have to design an antenna with a lower impedance. With a 10 ohm antenna and 1 KW, one would have 10A, at least in a part of the antenna. Since the impedance of commercial transmitters is 50 ohms, a ferrite transformer or a tuner box will be needed to do the impedance matching. This may also answer your question about ferrites: yes, they can handle this kind of high RF current, with a proportional magnetic flux, but you have to choose the right size and type of ferrite for the frequencies used. This is a difficult project.

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