Question regarding Coulomb's Law

sgstudent

1. The problem statement, all variables and given/known data
Coulomb's Law tells us that a force will be acted upon charges which is dependent on the distance between the two charges. So if i have a negative charge on a rod and it is placed beside a neutral metal rod, how will the charges and the whole object experience force?

2. Relevant equations

F=kQ1Q2/r^2

3. The attempt at a solution
I'm not sure how do i apply this to a neutral piece of metal. So initially as all the charges are at their fixed positions such that the metal is neutral, there will not be any net force on the whole as r is the same when i apply the law on both the positive nucleus and negative electrons. However, within the object the electrons being mobile will experience a net force pulling them forward. But for the nucleus I'm quite confused as it should not experience a net force due to it being in a fixed position. So how should the force be like acting on the electrons, nucleus and on the object as a whole when the negative rod is beside it?

Thanks for the help

mfb

Well, all nuclei have a positive charge, so if you add their forces, they will not cancel. But this is not a useful way to calculate the force (if you consider the contributions for nuclei and electrons separately, both values will be huge). Look at net charges in the different parts of the metal rod instead: The part closer to your charge will have the opposite charge (and be attracted), the other part will have the same charge (and be repelled). The net force is attractive, as the distance is different for both parts.

sgstudent

Hi thanks for the reply

But what about before it becomes polarized? Because the law explains the force acting on charged objects so just when it becomes charged how does the force act on the electron and positive nucleus? Or is the law only applicable when the charges become stable? I'm quite confused about the forces before the polarization occurs. Thanks for the help :)

mfb

If you could magically create the setup, that happens on the order of nanoseconds (assuming your setup has a size in the range of ~10cm to 1m). Every motion significantly slower than that (e.g. you, preparing the setup) will lead to equilibrium conditions all the time.
If you move and accelerate the parts so quick that an equilibrium cannot be reached, F=kQ1Q2/r^2 is no longer true. But that requires relativistic speeds for your objects.

sgstudent

Hi thanks for the reply mfb
Oh so Coulomb's Law only can be applied when everything has reached equilibrium?

However, that aside we know that there would be an attractive force on the electrons and an repulsive force on the nucleus. I'm also having trouble understanding how the forces add up in the object during this period. Under Coulomb's Law, the entire charged object experiences the force. However during this intermediate stage, I can't seem to picture how the forces are like. Is it possible to explain this part? Thanks for the help :)

mfb

Equilibrium, or slow changes (slow compared to the timescale of nanoseconds).

I don't understand which setup you have in mind here for the question.

sgstudent

Hi the set up would be the same same one with a charged object placed beside a neutral piece of metal. How will the forces be acted on the electron nucleus and as a whole? I can't seem to picture this. Thanks for the help :)

mfb

Positive charge?

Nuclei are repelled, electrons are attracted - but those forces are small compared to the forces inside the metal rod, so the influence is small. As a result, the electron density close to the positive charge increases a bit (relative to a metal rod without charges nearby), giving a net negative charge there. At the same time, the electron density far away decreases a bit and gives a net positive charge.
This gives a net attractive force between metal rod and your positive charge.
It is the same with a negative charge, just with reversed electron densities. The force stays attracting.

sgstudent

Hi thanks for replying the force acts on the electrons which causes then to move, while since the nucleus is stationary the nucleus will have another force stopping it? I'm quite confused about this part because if there is another force on the nucleus, then where would It's reaction force be.

Also, why would the force be so small during this intermediate stage? And big only when it has reached equilibrium (and use Coulomb's Law for it)?

Thanks for the help :-)

mfb

Forces from other electrons and nuclei fix its position in the material (as a very good approximation).

I still don't get what you mean as "intermediate stage". This stage does not exist in any realistic setup.
The net force is zero if there is no net charge (density) anywhere on the rod.

ehild

Imagine that the metal rod is fixed to the table so it can not move. You put a negatively charged object near it, say, a plastic rod.

The charges can not move in the plastic rod, but the electrons in the metal can. As for the positive ions of the metal, they are fixed to each other and the whole thing is fixed to the table.

The positive ions experience attraction, the negative electrons feel repulsion. You can imagine that there is an electron in the vicinity of all ions, so every part of the metal is electrically neutral, as in the first top picture. Then the attractive force on an atom is the same as the repulsive force on its electron, the net force on the metal rod is zero.

The repulsive force accelerates the electrons and they move away as far as possible, towards the other end of the metal rod. The ions can not move, they stay where they were. See the picture below: After a very short time, there are ions left without their electrons at the left end of the rod. They experience the same force as before. The electrons are accumulated at the other end of the rod, farther from the charged rod: They experience smaller forces. The rod consists of its ions and electrons. The forces exerted to the ions and electrons add and it is a net attractive force on the rod .

ehild

Attached Thumbnails

 

sgstudent

Oh so during process where the electrons move away from the positive ions, actually there should be a net attractive force already? Since now the distance between the positive ions and the negative plastic rod is smaller as compared to the electrons and the plastic rod?

ehild

Just at the very instant that the electron leaves the leftmost atom, there is a net electric force of the rod. If you keep the rod firm, it can not move. Only you need to exert a bit bigger force.

Try some experiment: cut small pieces from thin foil of aluminium. Rub a plastic comb ( I use my own hair to that :) ) and put the comb close to the alufoil pieces. They will jump onto the comb (After a short time, they jump off from it, why?)

ehild

sgstudent

Hi :)

I think it's because after all the charge has been exchanged, the net charge on both objects will be negative as the electrons will redistribute themselves?

But again this might not be the case as plastic is an insulator of electricity so it should not be able to give out those electrons to the polarized metal? If this is so then they should remain attracted and continue touching?

Thanks :)

ehild

Even the plastic can give some charge to the metal. There are free charged particles --ions or electrons- in all materials and they conduct electricity more or less. It only takes longer time.

Have you did the experiment? Repeat it with small pieces of thin paper.

ehild

sgstudent

ohh so even when a tiny amount electrons gets transferred to the aluminium, the aluminium becomes negatively charged as a whole. As the charge difference becomes smaller, the electrons in the metal will also be repelled less and as such due to their own internal repulsion start to move closer to the plastic rod. So after a while, by the same formula the repulsion will become stronger than the attraction.

I did it with the paper. It got attract to the comb but dropped off almost immediately. I suppose it is a result of the same redistribution? So actually even insulators can redistribute that charge quite quickly?

Thanks for the help :)

ehild

Well, there can be some thin water layer on the surfaces, and it can conduct electricity quite well because of the ions dissolved in it. In my experiments, the paper pieces sometimes stay attached to the comb quite long, but the air is usually dry here. It can be different at your place :)


ehild