How Do You Determine When a Chain of Charges Breaks?

[TSny]

This gives an alternating sequence converging to -ln(2), but the smallest magnitude term is the second one.

Here's a graph for fun.

 

[Saitama]

Well, maybe I do. Consider the net Coulomb force on the first particle, the first two particles, the first three particles, etc.

https://www.physicsforums.com/attachment.php?attachmentid=66869&d=1393004218

I calculate the net force on the first particle already which came out to be 8.22 N. The force is towards positive x-axis.

Won't the net force on the first two particles be simply the twice of 8.22 N?

 

[TSny]

https://www.physicsforums.com/attachment.php?attachmentid=66869&d=1393004218

Won't the net force on the first two particles be simply the twice of 8.22 N?

Can you explain how you got this? For the second particle, how many other particles are to the right of the 2nd particle and how many other particles are to the left of the 2nd particle?

 

[Saitama]

For the second particle, how many other particles are to the right of the 2nd particle and how many other particles are to the left of the 2nd particle?

To the right of 2nd charge, there are infinite charges so the force due to them is $F_0\ln(2)$ but maybe I went wrong here.

If I consider that there are $\infty-1$ charges to the right, do I subtract the force due to first charge on left of it?

 

[TSny]

To the right of 2nd charge, there are infinite charges so the force due to them is $F_0\ln(2)$

Now wait a minute. The first charge feels the force of all of the charges to it's right (essentially an infinite number). And you said that the force is 8.22 N. Good.

The second charge has essentially an infinite number of charges to it's right. How do you get that the net force of these charges on the second charge is $F_0\ln(2)$?

Besides the force from all the charges to the right of 2, there is also the force on 2 due to particle 1 on the left. How much is that force? Is it in the same direction or the opposite direction as the force due to the particles on the right?

 

[haruspex]

To the right of 2nd charge, there are infinite charges so the force due to them is $F_0\ln(2)$ but maybe I went wrong here.

If I consider that there are $\infty-1$ charges to the right, do I subtract the force due to first charge on left of it?

The ln(2) only arises when both sides of the link under study are infinite.
With 1 on the left of that link and infinity on the right (1:∞) we have π²/12.
When we introduce a second atom on the left, the force it feels from the right side is the same as for its neighbour, but missing the first term of the series.

 

[Saitama]

Now wait a minute. The first charge feels the force of all of the charges to it's right (essentially an infinite number). And you said that the force is 8.22 N. Good.

The second charge has essentially an infinite number of charges to it's right. How do you get that the net force of these charges on the second charge is $F_0\ln(2)$?

Besides the force from all the charges to the right of 2, there is also the force on 2 due to particle 1 on the left. How much is that force? Is it in the same direction or the opposite direction as the force due to the particles on the right?

The ln(2) only arises when both sides of the link under study are infinite.
With 1 on the left of that link and infinity on the right (1:∞) we have π²/12.
When we introduce a second atom on the left, the force it feels from the right side is the same as for its neighbour, but missing the first term of the series.

I did the following. I calculated the net force on the second particle due to the charges on the right and it comes out to be $F_0 \pi^2/12$. This force is towards right. The force due to first charge on the second charge is towards left and has the magnitude $F_0$. Hence, the net force on the second particle is $F_0(1-\pi^2/12)=1..7\,N$ towards left.

Similarly, the net force on the third particle is: $F_0(\pi^2/12+1/4-1)=0.724\,N$ towards right.

 

[TSny]

Looks good.

 

[Saitama]

Looks good.

Erm...

What to do with the net forces I have calculated?

Btw, looking at my post #35, I wanted to write $F_0\pi^2/12$, I don't know why I wrote $F_0\ln 2$.

 

[TSny]

What to do with the net forces I have calculated?

You're looking for the "weakest link". The first link is the force that is felt by particle 1 alone due to all of the other particles. The second link is the force felt by particles 1 and 2 together due to all of the other particles (3, 4, 5, ...). One way to obtain this is to add together the force felt by 1 alone and the force felt by 2 alone.

Btw, looking at my post #35, I wanted to write $F_0\pi^2/12$, I don't know why I wrote $F_0\ln 2$.

Ah, I feel better now.