Mass of a capacitor plate

[jartsa]

There's a force F between oppositely charged parallel plates. If the plates are free to move, they accelerate towards each other with acceleration a.

If the plates are large and close to each other, we can reduce the mass of the system by moving the plates closer to each other, while keeping the force between the plates unchanged. Does this mass reduction operation result in increased acceleration of plates when the plates are let free?

If the acceleration of a plate is indeed increased, can we then conclude that the mass of that plate was decreased?

 

[jbriggs444]

If the plates are large and close to each other, we can reduce the mass of the system by moving the plates closer to each other, while keeping the force between the plates unchanged. Does this mass reduction operation result in increased acceleration of plates when the plates are let free?

If the acceleration of a plate is indeed increased, can we then conclude that the mass of that plate was decreased?

[Ignoring edge effects], the electric field produced by a charged plate is constant regardless of distance. Accordingly, the acceleration of a test charge in the field will be a constant regardless of distance. Accordingly, each plate should accelerate toward the other at a fixed rate that does not depend on separation.

If I understand where you are going with this...

One would conclude that the mass of each plate is fixed regardless of the configuration of the capacitor system of which it is a part. Yet we know that the capacitor system carries more energy when the capacitors have a larger separation and less energy when they have a smaller separation. If one requires that the extra mass be located somewhere, where is it?

It's in the field.

 

[jartsa]

[Ignoring edge effects], the electric field produced by a charged plate is constant regardless of distance. Accordingly, the acceleration of a test charge in the field will be a constant regardless of distance. Accordingly, each plate should accelerate toward the other at a fixed rate that does not depend on separation.

If I understand where you are going with this...

One would conclude that the mass of each plate is fixed regardless of the configuration of the capacitor system of which it is a part. Yet we know that the capacitor system carries more energy when the capacitors have a larger separation and less energy when they have a smaller separation. If one requires that the extra mass be located somewhere, where is it?

It's in the field.

What I was getting at is very simple: The plates lost mass and therefore the plates accelerate more.

Now the case of one plate and one test charge. Well first I intuitively think that the acceleration of the test charge should be the same whenever the force is the same. But if the mass of the test charge changes, then the acceleration of the test charge changes. But it's not impossible that only the mass of the plate changes, while the mass of the test charge stays the same, because the plate is a plate and the test charge is a small ball, or something like that. I mean there's no symmetry like in the case of two plates.

 

[jbriggs444]

What I was getting at is very simple: The plates lost mass and therefore the plates accelerate more.

Do they accelerate more?

 

[jartsa]

Do they accelerate more?

Well if you ask me, I guess they do accelerate more.Hey how about this argument why the plates should accelerate more when the capacitor palates are initially closer to each other:

Let's say that after the plates have accelerated towards each other for a distance d, both plates are perfectly elastically deflected to the same direction, so that we now have a moving capacitor.

This moving capacitor should have a lower speed when there is more electrical energy stored i the capacitor. IOW when the initial distance between the plates is larger, the final speed should be lower.

But you are saying that the speed of the plates right before the deflection does not depend on the mass of the capacitor. Now If we can assume that the speed of the plates right after the deflection is the same as right before the deflection, then according to you the mass of the capacitor had no effect on the speed that the capacitor reached. This seems to be a problem.

I hope the above is understandable.

 

[jbriggs444]

But you are saying that the speed of the plates right before the deflection does not depend on the mass of the capacitor. Now If we can assume that the speed of the plates right after the deflection is the same as right before the deflection, then according to you the mass of the capacitor had no effect on the speed that the capacitor reached. This seems to be a problem.

I was suggesting that the speed of the plates depends on the mass of the plates. Your contention appears to be that the speed of each plate depends on the mass of that plate, inclusive of its share of the mass due to the capacitor charge.

You are suggesting a thought experiment in which a charged plate is accelerated without producing any electromagnetic radiation. Is that not a flaw in the thought experiment?

 

[Comeback City]

If the plates are large and close to each other, we can reduce the mass of the system by moving the plates closer to each other,

How do you expect to reduce the mass simply by moving them closer to each other?

 

[jbriggs444]

How do you expect to reduce the mass simply by moving them closer to each other?

The point would be mass-energy equivalence. The larger the distance between the capacitors, the smaller the capacitance and, accordingly, the larger the potential difference between the two charged plates. Same net charge and larger potential difference means more energy stored.

The fact that it takes work to separate the plates against their attraction also demonstrates that more energy is stored in the system when they are separated further.

 

[timmdeeg]

which then doe

The point would be mass-energy equivalence. The larger the distance between the capacitors, the smaller the capacitance and, accordingly, the larger the potential difference between the two charged plates. Same net charge and larger potential difference means more energy stored.

But isn't this different to the case of the stretched spring where the invariant mass of the spring increases because sloppy speaking the work to stretch the spring is stored in the spring itself. In my opinion this however isn't comparable to the case of the two charged plates because here the work is "stored" in the increased potential difference and not in the invariant mass of the plates which then remains unchanged. Kindly correct, if I'm wrong.

 

[pervect]

Didn't we have another thread on this? I recall typing an answer about how relativistic mass is just energy, and that energy is stored in the electric field between the plates of a capacitor.

But maybe I didn't hit send.

 

[jartsa]

Let's put two neutral and non-conducting plates on two scales. Then we add N electrons to one plate and remove N electrons from the other plate. For each electron removed we adjust the mass of the plate upwards by mass of one free electron. For each electron added we adjust the mass of the plate downwards by mass of a free electron.

We all perhaps agree that the readings on the scales are going up. The gravitational masses of the things on the scales are increasing, so the inertial masses of the things on the scales are increasing.

Now if we start bringing the two scales closer to each other, we can see that the gravitational and inertial masses of the things on the scales are decreasing. When the plates are touching, the masses are the same as originally.

Now let us imagine putting those scales on a frictionless surface, and then observing their acceleration towards each other.

Can we assume that larger readings on scales are correlated with smaller acceleration of scales? If yes, then capacitor plates accelerate at different rates at different distances, like the plates had different masses at different distances.

 

[jartsa]

I was suggesting that the speed of the plates depends on the mass of the plates. Your contention appears to be that the speed of each plate depends on the mass of that plate, inclusive of its share of the mass due to the capacitor charge.

Yes.

You are suggesting a thought experiment in which a charged plate is accelerated without producing any electromagnetic radiation. Is that not a flaw in the thought experiment?

Yes.

 

[jartsa]

Didn't we have another thread on this? I recall typing an answer about how relativistic mass is just energy, and that energy is stored in the electric field between the plates of a capacitor.

But maybe I didn't hit send.

I don't disagree with that.

In what situations does the energy between the plates resist the acceleration of the plates, causing the plates behave like they had extra mass?

Or in what situations does the energy between the plates not resist the acceleration of the plates, causing the plates to not behave like they had extra mass?

 

[Dale]

What I was getting at is very simple: The plates lost mass and therefore the plates accelerate more.

I don't think this is the case. The system lost mass, but that doesn't mean that the plates have lost mass.

 

[timmdeeg]

I don't think this is the case. The system lost mass, but that doesn't mean that the plates have lost mass.

Thanks, the latter is what I've tried to say. I think we talk about rest energy vs. (invariant) mass.

 

[jbriggs444]

Thanks, the latter is what I've tried to say. I think we talk about rest energy vs. (invariant) mass.

I consider "rest energy" and "invariant mass" to be almost synonymous. The only difference would be the conventional choice of units (mass for one, energy for the other) and the fact that "rest energy" is not a good phrase to use when invariant mass is zero.

The question I see being posed is about exactly where in a composite system one can find the mass (or lack thereof) due to the potential energy.of the system.

 

[jartsa]

I don't think this is the case. The system lost mass, but that doesn't mean that the plates have lost mass.

So the plates have not lost mass, but the system has lost mass.

Does that mean that:

1. Plate A has not lost mass and plate B has not lost mass, but system that consists of plate A and plate B has lost mass.
2. Plate A has not lost mass and plate B has not lost mass, but the system that consists of plate A, plate B and an electric field has lost mass.

Alternative 2 is maybe the better one. It's better because it includes more stuff.Now the word plate. Does that word refer to:

1. An object with a net charge, but not the electric field around it
2. A system consisting of an object with a net charge and the electric field around it

Now this is a tough choice. Alternative 2 means that folding a charged plate increases its mass. Alternative 1 means that the real mass of a proton is something else than what can be found in textbooks, or it means that we must treat charged particles and charged plates differently when we consider their masses.

EDIT: No sorry. It's not a problem if the mass of a plate is changng, when the plate itself is changing, like when a charged plate if folded. Right?

 

[Dale]

Plate A has not lost mass and plate B has not lost mass, but the system that consists of plate A, plate B and an electric field has lost mass.

This is the system that I was thinking of. You are, of course, free to define the system however you want. I would include the field because you cannot move the plates without changing the field, but it is up to you.

An object with a net charge, but not the electric field around it

That is what I was intending. The word "plate" refers to the object and the "system" includes the plate and the field.

 

[jartsa]

Let's put two neutral and non-conducting plates on two scales. Then we add N electrons to one plate and remove N electrons from the other plate. For each electron removed we adjust the mass of the plate upwards by mass of one free electron. For each electron added we adjust the mass of the plate downwards by mass of a free electron.

We all perhaps agree that the readings on the scales are going up. The gravitational masses of the things on the scales are increasing, so the inertial masses of the things on the scales are increasing.

Now if we start bringing the two scales closer to each other, we can see that the gravitational and inertial masses of the things on the scales are decreasing. When the plates are touching, the masses are the same as originally.

Now let us imagine putting those scales on a frictionless surface, and then observing their acceleration towards each other.

Can we assume that larger readings on scales are correlated with smaller acceleration of scales? If yes, then capacitor plates accelerate at different rates at different distances, like the plates had different masses at different distances.

Is the above not a proof that a plate of charged capacitor has extra mass?

Here's a short version:

A single plate gains mass when being charged. If bringing two charged plates close to each other does not change the mass of either plate, then each one of the plates of a charged capacitor has extra mass.

 

[jbriggs444]

A single plate gains mass when being charged. If bringing two charged plates close to each other does not change the mass of either plate, then each one of the plates of a charged capacitor has extra mass.

That assumes both that mass is an additive quantity and that fields do not contribute to mass.

 

[jartsa]

That assumes both that mass is an additive quantity and that fields do not contribute to mass.

I did not assume those things.

A single, lonely plate is charged. Its mass increases, right?

Moving the plate to a place where electric potential is different either causes the mass of the plate to change, or it does not cause the mass of the plate to change. Either the mass of the plate changes, or it stays extra large.

Either the mass of a charged capacitor plate is extra large, or it changes when the plate is moved.

 

[jbriggs444]

I did not assume those things.

Yes, you did.

A single, lonely plate is charged. Its mass increases, right?

A field is created as the plate is charged. The field has energy. The field is not the same thing as the plate.

 

[Dale]

Is the above not a proof that a plate of charged capacitor has extra mass?

No. I feel like we have explained why not already and you are just repeating things.

 

[jartsa]

Yes, you did.

A field is created as the plate is charged. The field has energy. The field is not the same thing as the plate.

Is it perhaps so that when a plate is charged, its mass stays the same, and an electric field, which has mass, appears around the plate?

I thought that a plate can be thought as a system, whose mass can change, if the configuration of the system changes.

 

[Dale]

I thought that a plate can be thought as a system, whose mass can change, if the configuration of the system changes.

A system whose configuration changes can have a change of mass. But trying to localize the mass within the system is not typically unique, and can usually be contradicted as shown above.

 

[Herman Trivilino]

If bringing two charged plates close to each other does not change the mass of either plate, then each one of the plates of a charged capacitor has extra mass.

Changing the distance between the plates changes the potential energy of the two-plate system. This is equivalent to changing the mass of the two-plate system. This is an example of the true meaning of the Einstein mass-energy equivalence. But as @jbriggs444 points out, the mass of the two-plate system does not equal the sum of the masses of the two plates. This is an example of what Einstein meant when he told us that mass-energy equivalence means that mass cannot be used as a measure of the quantity of matter.