Electrical Potential Energy of a System

AI Thread Summary
The discussion revolves around determining the smallest number of electrons, N, for which the second configuration (with one electron at the center of a ring) has lower electrical potential energy than the first configuration (with all electrons on the ring). Participants are exploring the energy equations for both configurations, noting that configuration 1's energy is greater than configuration 2's. There is a consensus that the number of electrons in one configuration must be odd while the other must be even. The conversation highlights the complexity of deriving the relevant equations and the need for a simpler approach to solve the problem. The thread invites further insights and solutions from participants.
withoutwax
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1. Homework Statement
Suppose N electron can be placed in either of two configurations. In configuration 1, they are all placed on the circumference of a narrow ring of radius R and uniformly distributed so that the difference between adjacent electron is the same everywhere. In configuration 2, N-1 electrons are uniformly distributed on the ring and one electron is placed in the center of the ring. What is the smallest value of N for which the second configuration is less energetic than the first?


2. The attempt at a solution

\sumU of configuration 1 > \sumU of configuration 2

*deduced that number of electron in one of the system must be odd, and another is even.

So, by drawing some circles with different number of electrons to understand the pattern of the summation.

But then i failed to get any relevant equations.

I think there must be some much easier way to solve it.
 
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withoutwax said:
Suppose N electron can be placed in either of two configurations.

In configuration 1, they are all placed on the circumference of a narrow ring of radius R and uniformly distributed so that the difference between adjacent electron is the same everywhere.

In configuration 2, N-1 electrons are uniformly distributed on the ring and one electron is placed in the center of the ring. What is the smallest value of N for which the second configuration is less energetic than the first?

Hi withoutwax ! Welcome to PF! :smile:

How far have you got?

Can you do an equation for the energy of configuration 1? :smile:
 
That's the problem I'm facing now..it require sum high maths solution...
 
withoutwax said:
That's the problem I'm facing now..it require sum high maths solution...

Well, can you do configuration 1 for N = 2? :smile:
 
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