Why can a capacitor get discharged through a (long) cable?

[greypilgrim]

Hi.
If a plate capacitor is charged with some charge $Q$, then there's a voltage $U=Q/C$ between the plates. The electric field between the plates is $E=U/d$.
Voltage is path independent, if I connect the plates with a cable of length $l\gg d$ the voltage across the cable is the same. The electric field along the cable will probably not be constant, but the mean field $U/l$ is much smaller than the field between the capacitor plates. So how can the force directing the charges through the cable be larger than the force from the other plate holding them back?

 

[Jonathan Scott]

The force isn't larger, but the cable is a conductor (so charge can move freely) whereas the region between the plates is an insulator (so charge cannot move across it regardless of force).

 

[CWatters]

Hi.
If a plate capacitor is charged with some charge $Q$, then there's a voltage $U=Q/C$ between the plates. The electric field between the plates is $E=U/d$.
Voltage is path independent, if I connect the plates with a cable of length $l\gg d$ the voltage across the cable is the same.

If the cable connects/shorts the plates together then the voltage "across" the cable is not the same as that of the capacitor before the cable is connected. +1 to what Jonathon said.

If the cable is open circuit (eg two unconnected wires) then the voltage can be the same but no current flows and it's not discharged.

 

[jartsa]

So how can the force directing the charges through the cable be larger than the force from the other plate holding them back?

A small charge, like an electron, feels a force from two plates.

(While a plate feels a force from one plate)

 

[greypilgrim]

The force isn't larger, but the cable is a conductor (so charge can move freely) whereas the region between the plates is an insulator (so charge cannot move across it regardless of force).

I don't understand this. If no cable is connected, a charge experiences an attractive force from the other plate. This force is somehow compensated by a "normal force" preventing the charge from entering the insulator. This "normal force" adapts such that it exactly cancels the other force (as the normal force of the floor exactly cancels the gravitational force of an object resting on it). If the cable now is connected and a smaller force acts in the other direction, this only means this "normal force" will be smaller. Why should it make the charge move?

If the cable connects/shorts the plates together then the voltage "across" the cable is not the same as that of the capacitor before the cable is connected.

Actually that's weird, I learned that in electrical circuits, everything connected by (perfect) conductors is on the same potential. However, this is clearly not the case here. So what's correct?

A small charge, like an electron, feels a force from two plates.

Now this really makes sense. The electron feels a stronger repulsive force from its own plate because it's closer.

 

[Jonathan Scott]

Perhaps it would help to imagine the equivalent with water and water pressure. A capacitor is like having a piece of rubber stretched across a pipe, preventing direct flow but allowing alternating flow to propagate. When it is charged, the rubber is pushed to one side. The pressure tries to make the water move away from the high pressure side in any direction (and towards the low pressure side from any direction), but it cannot move across the barrier. If the sides are connected by a long pipe, the water will flow around the pipe.

 

[greypilgrim]

I know this model, but it has its limits. One of the main problems is that it allows leaking, if the pipe with the rubber membrane is disconnected without closing the ends, the water would just flow out and the membrane relaxes. Also it's more of a macroscopic model, my question is what forces make single charges move.

 

[Jonathan Scott]

I think the water model is more accurate than you think. The pressure of the water is equivalent to the electrostatic potential. When there is nowhere for it to flow, the potential on each side is a single value. It is only when there is somewhere for it to flow that the pressure can go lower and a potential gradient forms, causing flow.
In terms of individual electrons, the potential corresponds to whether there is an unbalanced relative excess or deficit of electrons within a given region, so each electron is being affected by the excess or deficit of electrons in its immediate vicinity, and by any unevenness in that distribution, causing a local electric field.

I don't understand why you think that "leaking" is a problem with the water model. That's the same as connecting the ends to electrical Earth or similar.

The electrostatic effect of more distant parts of the apparatus is not relevant to electrons in a conductor, as the electron distribution evens itself out automatically.

 

[person_random_normal]

The electric field inside a conductor is consequence of the potential difference across the conductor only which can be given by the formula, current density equal to conductivity X the electric field inside the conductor
What I mean is, if it would have been for a case where a conducting wire is placed inside some electric field, then by virtue of shielding effect electric field inside the conductor will be zero as there has to be no current flowing inside the conductor

So if we extrapolate that logic here
We can say that the effects of the external electric field are nulified by deposition of some electric charge on the surface of the conducting wire, and only that amount of electric field is established in the wire which is given by the formula I stated above(j=Sigma X E this formula is experimental and basis for derivation of Ohm's Law)

 

[jartsa]

If we have a charged plate capacitor, and we put an electron outside the negative plate, exactly centered at the middle of the plate, then the electron will escape to infinity, right?

The question is: "How can the electron escape the pull of the positive plate", right?

Here's a picture of the setup, electron on the left, capacitor plates on the right:

- ||Here "escape to infinity" means "moves to infinity along a field line".

So, if an electron at the middle of the negative plate can get off the plate, it moves to infinity. Then a positive charge might move from the positive plate to infinity. Then an electron might move to infinity from the negative plate ... and so on.

 

[jartsa]

If we have a charged plate capacitor, and we put an electron outside the negative plate, exactly centered at the middle of the plate, then the electron will escape to infinity, right?

The question is: "How can the electron escape the pull of the positive plate", right?

Here's a picture of the setup, electron on the left, capacitor plates on the right:

- ||Here "escape to infinity" means "moves to infinity along a field line".

So, if an electron at the middle of the negative plate can get off the plate, it moves to infinity. Then a positive charge might move from the positive plate to infinity. Then an electron might move to infinity from the negative plate ... and so on.

... until there are, let's say five negative charges on the negative plate and five positive charges on the positive plate. Now an electron leaving the negative plate moves to a position where the five positive charges exert a force whose magnitude is the same as the magnitude of the force from the four negative charges. (Let's say there is some friction that stops the motion of the electron)

 

[person_random_normal]

I answered your question wrt #1 in my #9

Aren't you convinced ??