Finding Electric Potential Energy of N Electrons On A Circle

[knsnim]

Homework Statement

It's a problem on Halliday's Fundamentals of Physics's 24th chapter.
This problem gives us N-electrons on a circle with radius R.
The electrons are placed on the same distances so these electron positioning has a
circular symmetry.
And it also gives us another positioning of N-1 electrons like above on the same circle,
but it has an additional electron on its center. In other word, it also has N electrons in it.

Then, the problem is asking us the least N value which makes the latter's potential energy smaller than the former's.

Homework Equations

Potential energy of a pair of two electric charges :

U= k q_1 q_2 / r

The Attempt at a Solution

I found an equation of by using sine function.

k/2 e^2 sum_(k=1)^(N/2) N/(sin((k pi)/N))
(Sum from k=1 to [N/2], [N/2] means the greatest natural number smaller than N/2.)

But this equation does not help me in any way. So I tried using wolfram alpha to compute
this equation but I quited because this problem is from fundamental physics!
I thought that physics for 1-graders will not call for complex math like this.
So I'm thinking of simpler form of calculation but I have no idea.
Can you help me, supporters, please?

(Sorry for my weird English. haha...)

 

[anathema]

Thank you for sharing your problem with us. It seems like you are on the right track by using the equation for potential energy of a pair of electric charges. However, I would suggest that you approach the problem by considering the potential energy of each individual electron in the two configurations.

In the first configuration, where all N electrons are placed on the circle with equal distances, the potential energy of each electron would be the sum of the potential energies of all the other electrons. This can be expressed as:

U_1 = N-1 ∑ i=1 (k q_1 q_i / r_i)

where i represents each individual electron and r_i is the distance between the electron and the center of the circle.

In the second configuration, with N-1 electrons placed in a circular symmetry and one electron in the center, the potential energy of each electron would be the sum of the potential energies of all the other electrons plus the potential energy of the central electron. This can be expressed as:

U_2 = (N-1) ∑ i=1 (k q_2 q_i / r_i) + k q_2 q_c / R

where q_c represents the charge of the central electron and R is the radius of the circle.

To determine the minimum N value for which U_2 < U_1, you can set the two equations equal to each other and solve for N. This will give you the minimum number of electrons needed for the potential energy to be smaller in the second configuration compared to the first.

I hope this helps and good luck with your problem!