Balancing resistors are needed only if capacitors are under DC voltage. If only AC voltage is applied it will distribute along capacitors inversely proportional to capacitance of each capacitor. Assuming all capacitors in that chain has the same capacitace AC will be divided to equal AC voltages.
The number that actually matters for generated field is ampere turns.Is there any part of the circuit I can modify to increase the voltage across the antenna, apart from the size as that is fixed?
Increasing the ampere turns doesn't not necessarily mean more turns. It can also be achieve by thicker wire (possibly litz wire), resulting in lower ESR and higher current.Maybe I need to change the amplifier then because as I increase the number of turns, the current from the amplifier decreases.
A tuned series resonant circuit is no inductive load.Some audio amplifiers have protection against very inductive loads though it is not mentioned on the data sheet for this amplifier.
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