[Phys-L] Fw: non-polarized capacitor

[Jeffrey Schnick <JSchnick@Anselm.Edu>]

________________________________________
From: Jeffrey Schnick <JSchnick@Anselm.Edu>
Sent: Thursday, February 25, 2021 9:56 PM
To: John Denker
Subject: Re: [Phys-L] non-polarized capacitor

> From: John Denker <jsd@av8n.com>
> Sent: Wednesday, February 24, 2021 2:38 PM
> To: Jeffrey Schnick
> Subject: Re: [Phys-L] non-polarized capacitor

> > In the case of the electrolytic capacitors,
> [...]
> >  charge is leaking out of the
> > central conductor.  This makes the central conductor acquire a
> > negative net charge.

> I disagree with your model of leakage.
> The current doesn't come out of nowhere; it flows across the
> capacitor gap.  So when it is charging one plate of the capacitor,
> it is discharging the other, so the net effect is pure gorge, not
> an unbalanced charge, not a free charge. The leakage does *not*
> violate Kirchhoff's so-called laws.

> Just now I added some discussion and a diagram that may help.
> It shows ideal capacitors plus an explicit model for the leakage.
> This may make it easier to see that the leakage sub-circuit is
> just gorging and disgorging capacitors in the usual way.
>  https://www.av8n.com/physics/non-polarized-capacitor.htm#point-flow-through

I like your model.  I think I would have used the symbol for a non polar capacitor for each of the capacitors but you clearly state that they are ideal capacitors so that's a minor quibble.  I still fail to see how, for the case of a conductor in a circuit containing check valves that allow charge charge to flow out of the conductor but not into it, when the circuit (starting with all conductors, including capacitor plates, neutral) undergoes a cycle in which some charge (in a positive charge carrier model) does flow out of the conductor, how one can conclude that the conductor remains neutral.

We agree that charge doesn't come out of nowhere.  We agree that, during a cycle, the capacitors are doing nothing more that gorging and disgorging.  (We disagree about whether or not this implies that the net charge of the central conductor never changes.)  We agree that the net charge of any wire in the circuit is at all times negligible.  We disagree on whether the central conductor, the one consisting of the bottom plate of the upper capacitor, the top plate of the lower capacitor, and the wire connecting those two plates, remains neutral.  (In your model, we can include, in the definition of the central conductor, the vertical wire connecting the two diodes to each other and the horizontal wire connecting that wire to the vertical wire connecting the just-mentioned two plates together as well.)  I think we agree that what I call the central conductor is what you call the central node.  Thus, we disagree on whether or not Kirchhoff's Current Law, a.k.a. the junction rule is violated.  I think it is violated.

I think it might help for me to use your model to run through part of a cycle in a way that makes it clear that the capacitors are doing nothing more than gorging and disgorging and shows where the charge that has left the central conductor resides, at a few stages of the cycle.  (We know in advance it resides on the top plate of the upper capacitor and/or on the bottom plate of the lower capacitor because those are the only other places in the circuit where charge can hang out.)

In your model, starting with everything neutral, we connect an ideal battery designed to maintain a constant DC voltage V_max between its terminals to the two left terminals of the circuit with the lower voltage terminal of the battery connected to the grounded terminal of the circuit and the higher voltage terminal of the battery connected to the wire labeled V in your diagram.  Before we talk about what happens, let's establish some definitions.

Let the gorge of a capacitor be the charge on the top plate of that capacitor minus the charge on the bottom plate of that capacitor, all divided by 2.
Let the voltage of a capacitor, a.k.a. the voltage across a capacitor be the voltage of the top plate of that capacitor minus the voltage of the bottom plate of that capacitor.
Let the constant G be that amount of gorge that a capacitor must have in order for its voltage to be V_max.
Let the constant Q = G be the amount of charge on the top plate of a neutral capacitor having gorge G.
Note that V_max, G, and Q are all positive constants.

In arriving at steady state, here is what happens:

A transient flow of charge flow occurs from the bottom plate of the upper capacitor, down through the lower diode, up through the battery, and onto the top plate of the upper capacitor.  The charge flows until the voltage across the capacitor is equal to the battery voltage V_max.  At this point the gorge on the upper capacitor is G, the charge on the top plate of the upper capacitor is Q and the charge on the bottom plate of the upper capacitor is -Q.

Boom!  There it is.  The central conductor was neutral before.  Now it has a net charge -Q.  That means (in the positive charge carrier model) than an amount of charge Q has left the central conductor.  Where is that charge now? It is on the top plate of the upper capacitor.

If we now vary the source voltage, as long as we keep it in the range -V_max to V_max, the amount of charge on the central conductor will remain at -Q.  Let's check out a couple more values of the source voltage.  First, lets replace the battery with a piece of wire, thus making the source voltage 0.  A transient flow of charge occurs from the top plate of the upper capacitor, down through the new piece of wire to the bottom plate of the lower capacitor.  At the same time, a transient flow of charge occurs from the top plate of the lower capacitor to the bottom plate of the upper capacitor.  Each time a charge carrier departs the top plate of the upper capacitor, a charge carrier arrives at the bottom plate of the same capacitor.  The upper capacitor is disgorging.  Each time a charge carrier arrives at the bottom plate of the lower capacitor a charge carrier departs the upper plate of the same capacitor.  The lower capacitor is gorging or disgorging, depending on what you want to call it when the gorge is becoming more negative.  This goes on until the voltage of the upper capacitor has dropped to V_max/2 and the voltage of the lower capacitor is -V_max/2.  The algebraic signs are an artifact of how I defined a capacitor voltage but there is no need to be confused, the two capacitors are in parallel with each others with the ends connected to the new wire both being at the higher voltage and the other ends (both of which are part of the central conductor) being at the lower voltage.  In the end, the gorge of the upper capacitor is G/2 and the gorge of the lower capacitor is -G/2.  The charge of the bottom plate of the upper capacitor is -Q/2 and the charge of the top plate of the lower capacitor is -Q/2.  Thus the charge on the central conductor is still -Q.  Q/2 of the charge that originally left the central conductor is now on the top plate of the upper capacitor and Q/2 of the charge that originally left the central conductor is now on the bottom plate of the lower capacitor.

Now let's replace the new piece of wire with the battery hooked up with the polarity opposite the way we had it the first time we hooked it up.  This makes the source voltage -V_max.  A transient flow of charge occurs from the top plate of the upper capacitor, down through the battery to the bottom plate of the lower capacitor.  At the same time, a transient flow of charge occurs from the top plate of the lower capacitor to the bottom plate of the upper capacitor.  Each time a charge carrier departs the top plate of the upper capacitor, a charge carrier arrives at the bottom plate of the same capacitor.  The upper capacitor is disgorging.  Each time a charge carrier arrives at the bottom plate of the lower capacitor a charge carrier departs the upper plate of the same capacitor.  The lower capacitor is gorging or disgorging, depending on what you want to call it when the gorge is becoming more negative.  This goes on until the voltage of the upper capacitor has dropped to 0 and the voltage of the lower capacitor is -V_max.  In the end, the gorge of the upper capacitor is 0 and the gorge of the lower capacitor is -G.  The charge of the bottom plate of the upper capacitor is 0 and the charge of the top plate of the lower capacitor is -Q.  Thus the charge on the central conductor is still -Q.  All of the charge that originally left the central conductor is now on the bottom plate of the lower capacitor.

In this system: Charge is conserved.  Each capacitor is always neutral.  Kirchhoff's Voltage Law is never violated but Kirchhoff's Current Law is, there is a net flow of charge out of the central conductor.

Once the charge on the central conductor is -Q, the diodes in your model no longer play a role.

I think the next refinement to your model would be to put a resistor in series with each of your diodes, in which case the charge of the central conductor would not reach -Q the first time the source voltage reached V_max or -V_max, but, because the diodes in the model and the dielectric in the original electrolytic capacitors act like check valves that only allow charge to flow OUT of the central conductor, the net charge of the central conductor would monotonically change from 0 to -Q and once it got to -Q, the charge of the central conductor would remain at -Q and the system would behave as a non polar capacitor.