Why Do Capacitor Plates Have Equal and Opposite Charges?

[ZapperZ]

Most textbooks say that a capacitor whether it be a single one or one in series/parallel should have equal amounts of + and – charges on both plates and that they mostly conclude the + charges attract the same amount of – charges on the other plate without giving any reason.

Now I claim that this is supported by Gauss’ law!

When a capacitor is fully charged, there’s no electric field (no current) in the wires connecting both plates of a fully charged capacitor and there can’t be any net charge on the capacitor when enclosing the whole capacitor by a Gaussian surface.

When a capacitor isn’t fully charged, there’re 2 currents in the same direction flowing to both plates though not through the interior of the capacitor. There can’t be any net charge on the capacitor when enclosing the whole capacitor by a Gaussian surface as the whole electric flux is canceled out to 0.

Do you all agree with this argument?

I don't understand all of this. Why can't this be a simple argument based on conservation of charge?

I have an empty pail, and I fill it with water from a pond. I then lift the pail a distance h above the surface of the pond. I claim that the amount of water in the pail is equal to the amount of water missing from the pond.

What is the problem here?

Zz.

 

[feynman1]

I don't understand all of this. Why can't this be a simple argument based on conservation of charge?

I have an empty pail, and I fill it with water from a pond. I then lift the pail a distance h above the surface of the pond. I claim that the amount of water in the pail is equal to the amount of water missing from the pond.

What is the problem here?

Zz.

When there're 2 capacitors in series, merely according to charge conservation, there could be +q, -2q, +2q, -q on the plates (left to right).

 

[ZapperZ]

When there're 2 capacitors in series, merely according to charge conservation, there could be +q, -2q, +2q, -q on the plates (left to right).

No they can't! Re-read your book!

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capac.html

Zz.

 

[feynman1]

No they can't! Re-read your book!

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capac.html

Zz.

2 plates of the same conductor shouldn't conserve charge. They are independent conductors.

 

[Delta2]

No they can't! Re-read your book!

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capac.html

Zz.

I think it can. We have to assume that the current is everywhere the same along the branch containing the capacitor in series (which does not follow directly from charge conservation in the case of capacitors), aka well known assumption of typical circuit theory , see my post #16.

 

[ZapperZ]

2 plates of the same conductor shouldn't conserve charge. They are independent conductors.

I think it can. We have to assume that the current is everywhere the same along the branch containing the capacitor in series (which does not follow directly from charge conservation in the case of capacitors), aka well known assumption of typical circuit theory , see my post #16.

Are we talking about the SAME thing here? The link I stated is under static condition, and the analogy that I gave is when the pail is full. There is no current flow! All the capacitors in series must have the same charge or there is a non-conservation of charge somewhere.

This is where I choose to start and see whether this is fully understood FIRST.

Zz.

 

[Delta2]

Are we talking about the SAME thing here? The link I stated is under static condition, and the analogy that I gave is when the pail is full. There is no current flow! All the capacitors in series must have the same charge or there is a non-conservation of charge somewhere.

This is where I choose to start and see whether this is fully understood FIRST.

Zz.

I really don't understand why the configuration of #37 is prohibited by charge conservation (total charge is also 0 for this configuration), it might as well be the case, it depends how you charge the capacitors.

 

[ZapperZ]

I really don't understand why the configuration of #37 is prohibited by charge conservation (total charge is also 0 for this configuration), it might as well be the case, it depends how you charge the capacitors.

There are two separate issues here:

1. You are claiming that if a set of capacitor is in series and the ends are attached to a battery, that each of the capacitor can have DIFFERENT amount of charges? (see the figure in the Hyperphysics link that I gave if you are unsure of what I mean by a capacitor in series, because THAT is exactly what I'm referring to). So let me be clear that this is what you are saying.

2. The example I gave in my first post has nothing to do with capacitor in series or parallel. It is simply a standard scenario of a capacitor being charge by a battery, i.e. a simple closed circuit. or an RC circuit if you will. I need to know if this is fully understood as a case of charge conservation FIRST. Because if it isn't, then I had to dig more elementary case of electrostatic charging! I have no idea where this is going, but at some point, I have to established a common knowledge that everyone can agree to! Otherwise, and this appears to be the case here now, we are talking about different things doing different events!

A simple capacitor being charged by a battery. At equilibrium, charge on one is equal to charge on the other, but opposite in sign. Again, as my example with water in the bucket, what is the issue here?

Zz.

 

[Delta2]

There are two separate issues here:

1. You are claiming that if a set of capacitor is in series and the ends are attached to a battery, that each of the capacitor can have DIFFERENT amount of charges? (see the figure in the Hyperphysics link that I gave if you are unsure of what I mean by a capacitor in series, because THAT is exactly what I'm referring to). So let me be clear that this is what you are saying.

Yes this is the issue except that I had not in mind a DC battery but rather the AC case with very high frequency such that the wavelength of current in the circuit is in comparison with the dimension of the branch containing the capacitors in series.

2. The example I gave in my first post has nothing to do with capacitor in series or parallel. It is simply a standard scenario of a capacitor being charge by a battery, i.e. a simple closed circuit. or an RC circuit if you will. I need to know if this is fully understood as a case of charge conservation FIRST. Because if it isn't, then I had to dig more elementary case of electrostatic charging! I have no idea where this is going, but at some point, I have to established a common knowledge that everyone can agree to! Otherwise, and this appears to be the case here now, we are talking about different things doing different events!

A simple capacitor being charged by a battery. At equilibrium, charge on one is equal to charge on the other, but opposite in sign. Again, as my example with water in the bucket, what is the issue here?

Zz.

This example I believe is fine.

 

[ZapperZ]

Yes this is the issue except that I had not in mind a DC battery but rather the AC case with very high frequency such that the wavelength of current in the circuit is in comparison with the dimension of the branch containing the capacitors in series.

So, can you explain to me at what point from the OP's original post did it somehow morphed into this situation? Have we all agreed upon the simplest case first before going into this rather unusual state?

Zz.

 

[Delta2]

So, can you explain to me at what point from the OP's original post did it somehow morphed into this situation? Have we all agreed upon the simplest case first before going into this rather unusual state?

Zz.

Well you decided to bring in the charge conservation principle, which works for your example but not for the example of capacitors in series in the AC case or when ,in generally ,they are being charged in an irregular way (not via a battery).
My thoughts on this regarding conservation of charge and KCL (or that the current along the same branch is everywhere the same):
Conservation of charge is equivalent to the continuity equation:
$$\nabla\cdot \vec{J}=-\frac{\partial \rho}{\partial t}$$
But this is not enough to infer KCL. We have to assume that $$\nabla\cdot \vec{J}=0$$ everywhere along the circuit and from this (by integrating both sides over a closed surface S that encloses the junction point and using divergence theorem) we can infer KCL.

 

[ZapperZ]

Well you decided to bring in the charge conservation principle, which works for your example but not for the example of capacitors in series in the AC case or when ,in generally ,they are being charged in an irregular way (not via a battery).
My thoughts on this regarding conservation of charge and KCL (or that the current along the same branch is everywhere the same):
Conservation of charge is equivalent to the continuity equation:
$$\nabla\cdot \vec{J}=-\frac{\partial \rho}{\partial t}$$
But this is not enough to infer KCL. We have to assume that $$\nabla\cdot \vec{J}=0$$ everywhere along the circuit and from this (by integrating both sides over a closed surface S that encloses the junction point and using divergence theorem) we can infer KCL.

That's fine, but that wasn't my question. I wanted to know, starting from the OP's question in the first post, how that morphed into this. How is this relevant within the scope of what I perceived the OP was asking?

Zz.

 

[Delta2]

That's fine, but that wasn't my question. I wanted to know, starting from the OP's question in the first post, how that morphed into this. How is this relevant within the scope of what I perceived the OP was asking?

Zz.

I don't know its just a forum's thread, conversation can divert (a little or a lot) from the original topic.

 

[ZapperZ]

I don't know its just a forum's thread, conversation can divert (a little or a lot) from the original topic.

Once again, that's fine. It occurs a lot in this forum. However, when you jumped all over my post using a situation that is beyond the scope of what I think the OP is asking, then you are being unfair.

I can easily cite many discussions on here in which, if we apply a more general or unusual situations, the standard and common explanation simply will not work or incomplete. Every time there's a discussion on the photoelectric effect, for example, and the claim that increasing intensity of the light source that has photon energy below the work function will not cause any electron emission, I will let that pass by even though I have personally done many experiments where this is clearly not true. Why? Because within the scope of the question at that level, this is an added complication that is totally unnecessary and irrelevant.

At this point, I have no idea what the OP knows or have understood, because the situation just got way too complicated and confusing. I don't see a clearly-established baseline.

Zz.

 

[Delta2]

Once again, that's fine. It occurs a lot in this forum. However, when you jumped all over my post using a situation that is beyond the scope of what I think the OP is asking, then you are being unfair.

Sorry for being unfair I blame @feynman1 he has the tendency of diverging from the thread's original topic in the threads he makes.

At this point, I have no idea what the OP knows or have understood, because the situation just got way too complicated and confusing. I don't see a clearly-established baseline.

Zz.

My summary from this thread: in the DC case it holds that the charges in capacitors plate are opposite and equal (and we can explain this with various ways, like with gauss's law and ideal wires (and ideal capacitor) , or with conservation of charge). But it doesn't necessarily hold in the AC case, not when the frequency becomes too high.

 

[ZapperZ]

Sorry for being unfair I blame @feynman1 he has the tendency of diverging from the thread's original topic in the threads he makes.

You're slowly catching up, grasshopper. :)

Zz.

 

[feynman1]

I don't know its just a forum's thread, conversation can divert (a little or a lot) from the original topic.

No, I never diverted.

 

[feynman1]

where the so called +2Q and -2Q could be +3Q and -3Q etc, if there were no Gauss' law (under the umbrella of DC+ideal wires+ideal capacitors (no electric field leaking out) premise).

 

[Delta2]

No, I never diverted.

I think you did, you start with one capacitor and then you speak about capacitors in series.
Also you did divert in that other thread about current not being a vector but current density being a vector.

Its ok, it happens a lot in these forums you are not the only one.

 

[Delta2]

where the so called +2Q and -2Q could be +3Q and -3Q etc, if there were no Gauss' law.

I think that you are also using the assumption that the electric field outside the capacitor is zero (together with Gauss's law). But this is not always a valid assumption, especially in the time dependent case with high frequency -low wavelength current. It can be valid if you assume ideal capacitor and ideal connecting wires as you say, but this is not the case in real world situations.

 

[feynman1]

I think that you are also using the assumption that the electric field outside the capacitor is zero (together with Gauss's law). But this is not always a valid assumption, especially in the time dependent case with high frequency -low wavelength current. It can be valid if you assume ideal capacitor and ideal connecting wires as you say, but this is not the case in real world situations.

i changed the recent post with a pic to restrict our discussion to DC+ideal wires+ideal capacitors (no electric field leaking out).

 

[Delta2]

i changed the recent post with a pic to restrict our discussion to DC+ideal wires.

I think under DC and under ideal wires +ideal capacitor (No E-field escapes outside the capacitor) you are right that conservation of charge alone doesn't prevent the configuration that you show in post #53. It is gauss's law that prevents it (or alternatively the circuit theory assumption that the current along the same branch is everywhere the same, which is valid for the DC-case).

 

[vanhees71]

Yes this is the issue except that I had not in mind a DC battery but rather the AC case with very high frequency such that the wavelength of current in the circuit is in comparison with the dimension of the branch containing the capacitors in series.

Then you cannot treat the problem in quasistationary approximation and it's far from the DC case discussed in this thread since for this you need the full Maxwell equations and their retarded solutions. This would be an entirely different topic and thus should be discussed in a separate new thread.

 

[vanhees71]

i changed the recent post with a pic to restrict our discussion to DC+ideal wires+ideal capacitors (no electric field leaking out).

With only "ideal wires" cicruit theory becomes singular either. You should have some finite resistance to not overcomplicate things.

 

[Delta2]

With only "ideal wires" cicruit theory becomes singular either. You should have some finite resistance to not overcomplicate things.

Ok let's assume finite resistances, and charge the capacitors with a DC voltage source, how can conservation of charge only prevent the configuration of charges shown in post #53?

 

[vanhees71]

As discussed several times, then you'd have an electric field outside the capacitors and a current would flow in the wire connecting the capacitors, which by assumption (electrostatic situation) should not be the case.

 

[Delta2]

As discussed several times, then you'd have an electric field outside the capacitors and a current would flow in the wire connecting the capacitors, which by assumption (electrostatic situation) should not be the case.

I agree, but we are using gauss's law here don't we? how else can we infer there would an e-field outside the capacitors (because there would be net charge enclosed).

 

[vanhees71]

Of course, we use Maxwell equations all the time when dealing with electrodynamics. What else should we use?

 

[rude man]

I don't have the temperament to read all the posts, but:

it is perfectly possible for a capacitor to have differing amounts of charge on its plates. You just put separate charge magnitudes on each plate the way you would normally charge a plate - by a charged object held against each plate.

Take two identical plates, charge each of them separately with different charge q1 and - q2, $q2 \neq q1$. Then bring them close to each other.

The sides of plates facing each other will have the same magnitude of charge, just like in a capacitor carged by current. This as you point out can be shown by Gauss's theorem, or by the fact that $\nabla \cdot \bf E = ~0$.

The outside faces will have differing charge densities:

The total picture is:
outside faces charge = (q1 - q2)/2
inside faces charge = (q1 + q2)/2 and -(q1 + q2)/2

If the capacitor is charged the normal way i.e. by current, q2 = q1 and the inside faces will have all the accumulated charge while the outside faces will have zero charge.

 

[feynman1]

I don't have the temperament to read all the posts, but:

it is perfectly possible for a capacitor to have differing amounts of charge on its plates. You just put separate charge magnitudes on each plate the way you would normally charge a plate - by a charged object held against each plate.

Take two identical plates, charge each of them separately with different charge q1 and - q2, $q2 \neq q1$. Then bring them close to each other.

The sides of plates facing each other will have the same magnitude of charge, just like in a capacitor carged by current. This as you point out can be shown by Gauss's theorem, or by the fact that $\nabla \cdot \bf E = ~0$.

The outside faces will have differing charge densities:

The total picture is:
outside faces charge = (q1 - q2)/2
inside faces charge = (q1 + q2)/2 and -(q1 + q2)/2

If the capacitor is charged the normal way i.e. by current, q2 = q1 and the inside faces will have all the accumulated charge while the outside faces will have zero charge.

Many thanks for agreeing on the argument of Gauss' law. How did you calculate inside faces charge = (q1 + q2)/2 and -(q1 + q2)/2?

 

[rude man]

Many thanks for agreeing on the argument of Gauss' law. How did you calculate inside faces charge = (q1 + q2)/2 and -(q1 + q2)/2?

Glad you asked.
OK, we have +q1 on plate 1 and -q2 on plate 2. There are 4 sides: the two outside ones and the two inside ones. Say the plates are of unit area and lined up left to right.

Let
$\sigma1 =$ left plate outside face charge density
$\sigma2 =$ left plate inside face charge density
$\sigma3 =$ right plate inside face charge density
$\sigma4 =$ right plate outside face charge density

By Gauss, $\sigma3 = - \sigma2$. I think you got that already.
Then,
$\sigma1 + \sigma2 = q1$
$\sigma 3 + \sigma4 = -q2$

The last equation is the interesting one. Imagine a unit test charge inside plate 1. This charge will see a force from each of the four face charges:
Charges 2 and 3 forces cancel.
Charge 1 and charge 4 forces also cancel (charge1 pushes the test charge to the right, charge 4 pushes the test charge to the left). So $\sigma1 - \sigma4 = 0$.
That gives you 4 equations in the four unknown $\sigma 's$.

 

[Delta2]

The last equation is the interesting one. Imagine a unit test charge inside plate 1. This charge will see a force from each of the four face charges:
Charges 2 and 3 forces cancel.
Charge 1 and charge 4 forces also cancel (charge1 pushes the test charge to the right, charge 4 pushes the test charge to the left). So $\sigma1 - \sigma4 = 0$.
That gives you 4 equations in the four unknown $\sigma 's$.

Just to add one little thing: The net E-field inside the plate is zero, so total force is zero, so because the force from 2 and 3 cancel, the forces from 1 and 4 have to cancel too, otherwise the total force and the e-field wouldn't be zero.

 

[feynman1]

Glad you asked.
OK, we have +q1 on plate 1 and -q2 on plate 2. There are 4 sides: the two outside ones and the two inside ones. Say the plates are of unit area and lined up left to right.

Let
$\sigma1 =$ left plate outside face charge density
$\sigma2 =$ left plate inside face charge density
$\sigma3 =$ right plate inside face charge density
$\sigma4 =$ right plate outside face charge density

By Gauss, $\sigma3 = - \sigma2$. I think you got that already.
Then,
$\sigma1 + \sigma2 = q1$
$\sigma 3 + \sigma4 = -q2$

The last equation is the interesting one. Imagine a unit test charge inside plate 1. This charge will see a force from each of the four face charges:
Charges 2 and 3 forces cancel.
Charge 1 and charge 4 forces also cancel (charge1 pushes the test charge to the right, charge 4 pushes the test charge to the left). So $\sigma1 - \sigma4 = 0$.
That gives you 4 equations in the four unknown $\sigma 's$.

Many thanks. I agree with your 4 equations. The last 1 simply ensures both plates be conductors.

So in what circumstances will outside faces charge not vanish? I think when the wires connecting the capacitor are ideal, they should vanish whatsoever, unsteady charging/discharging included.

 

[Delta2]

So in what circumstances will outside faces charge not vanish? I think when the wires connecting the capacitor are ideal, they should vanish whatsoever, unsteady charging/discharging included.

I think they vanish only in the static case and in the quasi-static approximation(DC and low frequency-high wavelength AC). In the full dynamic case I believe they don't necessarily vanish (neither that the inside charges are opposite and equal)

 

[feynman1]

I think they vanish only in the static case and in the quasi-static approximation(DC and low frequency-high wavelength AC). In the full dynamic case I believe they don't necessarily vanish (neither that the inside charges are opposite and equal)

By dynamic, do you mean AC circuits or LRC ones? Even in these, they still vanish.