Can anyone please clarify the following for me?
Two part question:
If I take a solenoid and feed a positive DC offset sinewave (ie. 6V max 2V min) into one end and an inverted copy with a similar negative offset (ie -2V max -6V min) into the opposite, what, in terms of electrodynamics, occurs within the coil?
If I now split the coil in half and ground the center point, what happens now?
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Do you realize that voltage is relative? You can add any voltage to one point as long as you add it to any other and the "physics" will not change.
So, your first case is identical to setting one side to 0 volts and the other side to (2sin(wt) + 4) + (-4 - 2sin(wt)) = 0. So both ends of the coil are at the same voltage. I could have set one end to Zeta(wt)^w + cos(wsinw(wt))^t and wouldn't have changed anything.
In the second case when you ground the center you end up with two coils. You can separate them mentally or even physically since you know(or forced) the center(or even some other point) to be a certain value. It is exactly the same when we know(force or decide) that something is "ground".
Note that this is distinctly different from the first case. In the first case we have one coil and the second case we have two... but each one similar to the others.
Anyways, if you think about it a bit you shouldn't have any issues. Use resistors instead of coils. The only difference with coils is that the voltage drop across the coil may not be linear due to self and mutual electromagnetic effects and fringe effects.