Amplification Circuit for Generating High-Current Sinusoidal Signal to Feed a Coil
Force is proportional to current and Inductance.

For 100Hz ( 628rad/s ) your R/L ratio must be T&lt;=1ms .  

So using Litz wire with the most turns and diameter and core permeability is what you need with an inductance and resistance calculator for the best Litz wire you can get. (most strands)

Matching the ESR of the batteries to the DCR of the Coil will maximize your power transfer but also load you batteries to the max. which must be minimized by prudent low duty cycle.

Now if you know how to measure battery ESR, and capacitance in kiloFarads , you can use the same conjugate impedance  for the coil for the theoretical max. power transfer.  Everything is a compromise when you realize how big that is, then you have to cascade coils or think outside the box.