**WONDERFUL Question!!!** First, there can certainly appear to be something weird about counter- wound bifilar coils--where does the magnetic field go??? The answer is that in such a case there is *NO* magnetic field created, so we don't need to worry about where a magnetic field goes. Consider the capacitive analog: a cap with each end connected to the same potential. Now, the capacitor stores electric energy in an electric field. But because each end of the cap is at the same potential, no electric field is created and so the capacitor does not store energy. It's exactly the same situation in the counter-wound bifilar coils: the magnetic field created by the current flowing in each wire is exactly cancelled by the current flowing in the opposite direction in the other wire, so no magnetic field is created, no energy is stored (important point!), and so you don't have to worry about where the energy goes, because no energy is stored. I hope that this answers your question (a). b--you are exactly correct. Since there is no magnetic field created, there is no inductance and, strictly speaking, there is no resonant frequency in a tank circuit. [Another way to say this is that the resonant frequency is infinite]. c--since there is no inductance, you should get zero effect in an RF antenna. A finer point here is that such coils are, of course, never perfect, so a real-world coil is sometimes used in tuning the RF antenna or used to block unwanted frequencies. I hope this helps!
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