Charge in RLC circuit when starting charges are 2Q and Q

[alan123hk]

You can in principle have such an unusual initial condition and why not asking for the dynamics of this unusual "circuit", but I think it's not as easy to solve as it looks at the first glance. I'm not so sure, whether one can use the standard AC circuit theory in terms of the parameters R, L, and C; particularly the latter, which has a well-defined meaning for the case that the capacitor is connected to a (DC) voltage source, which implies that one plate carries a charge Q and the other a charge -Q, and then you use the quasistatic approximation for the AC case too, which is legitimate if the typical wavelength of the em. waves involved is very large compared to the extension of the circuit.

This is indeed a thorny issue. I think whether we can use AC circuits to handle its dynamics will depend on how we define the circuit model. Because the capacitor has an unusual initial condition [2Q,0Q], although there will be a potential difference between the two conductors of this capacitor, part of the charge stored on the two conductors must not belong to this capacitor.

If we assume that this experiment is carried out in an empty cosmic space, and there are no objects in the endless space around this circuit, including people who try to do this experiment will use the remote control to carry out this experiment, then the electric field lines generated by the extra charge mentioned above will extend to the endless cosmic space and never return, and its distance is likely to be much greater than the EM wave wavelength of the original AC LCR circuit, which will make the effectiveness of the use of the AC circuit questionable.

 

[sophiecentaur]

If we were to examine the behaviour of an RLC circuit on Earth (one terminal of the C being earthed) and then charge the Earth with an "extra" +Coulomb, would our answers be any different? I think superposition would work here, as long as the surface field of the Earth could be considered uniform over a reasonably big area.

 

[alan123hk]

No matter where the experiment site is, people may ask similar questions. It may not be much different on the surface of the Earth or in the empty space of the universe

After further thinking about this problem, I think it is not difficult to solve it from an engineering perspective. Simply put, it is to ignore the extra charge on the capacitor that is not clearly defined or difficult to handle in the analysis (of course, the premise is that it does not constitute a non-negligible danger or impact).

The method is very simple, that is, first measure the potential difference and capacitance of the capacitor, and then we can calculate the charge that belongs to the capacitor and the extra charge that does not belong to the capacitor. When the LR circuit is connected to the capacitor, we only perform dynamic analysis based on the charge belonging to the capacitor, that is, the measured voltage and capacitance are all the information we need for dynamic analysis.

Of course, when the LR circuit is connected, the extra charge may also move between the two conductors of the capacitor, but if for special reasons, the effect of the extra charge cannot be integrated into the equivalent AC circuit by adding an additional capacitors, then except ignoring it , I can't think of any other way to deal with it now.