Finding potential energy difference of two electronic system

[withoutwax]

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

$$\sum$$U of configuration 1 > $$\sum$$U 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.

 

[anathema]

thank you for posing this interesting problem. I would approach this problem by first considering the basic principles of energy and charge distribution in a system of electrons. We know that electrons repel each other due to their negative charges, and they tend to distribute themselves in a way that minimizes their potential energy. In the first configuration, all the electrons are equally spaced along the circumference of the ring, resulting in a symmetrical distribution and a lower potential energy compared to the second configuration where one electron is in the center.

To solve for the smallest value of N for which the second configuration is less energetic, we need to compare the potential energies of the two configurations. We can use the formula for the potential energy of a system of point charges, U = kQ1Q2/r, where k is the Coulomb constant, Q1 and Q2 are the charges of the two interacting particles, and r is the distance between them.

In configuration 1, the potential energy is the sum of the interactions between all the electrons, which can be calculated by considering the distance between adjacent electrons on the ring. In configuration 2, the potential energy is the sum of the interactions between the central electron and all the other electrons on the ring.

Using this information, we can set up an equation to compare the potential energies of the two configurations and solve for the smallest value of N. This approach avoids the need for trial and error and allows us to find a specific value for N.

I hope this helps in your quest to find the solution to this problem. Good luck!

 

[Moetasim]

I would approach this problem by using the principle of conservation of energy. In this case, we can consider the potential energy of the electrons in each configuration as the energy being conserved. We can also assume that the electrons are non-interacting and have no kinetic energy.

To find the potential energy difference between the two configurations, we can use the formula for the potential energy of a system of point charges, which is given by U = kq1q2/r, where k is the Coulomb's constant, q1 and q2 are the charges of the two particles, and r is the distance between them.

In configuration 1, we have N electrons placed on the circumference of the ring, with each adjacent electron having the same distance between them. This means that the potential energy for each pair of adjacent electrons will be the same, and we can simply sum up the potential energy for all the pairs to get the total potential energy of the system.

In configuration 2, we have N-1 electrons placed on the circumference of the ring, with one electron at the center. In this case, the potential energy for the central electron with each of the N-1 electrons on the circumference will be different, as the distance between them will vary. However, the potential energy for each pair of adjacent electrons on the circumference will still be the same. Again, we can sum up the potential energy for all the pairs to get the total potential energy of the system.

Now, to find the smallest value of N for which configuration 2 is less energetic than configuration 1, we simply need to equate the two potential energies and solve for N. This can be done by considering the distance between the central electron and each of the N-1 electrons on the circumference in configuration 2, and comparing it with the distance between adjacent electrons in configuration 1.

In conclusion, by using the principle of conservation of energy and the formula for potential energy of a system of point charges, we can find the potential energy difference between the two configurations and determine the smallest value of N for which configuration 2 is less energetic than configuration 1. This approach avoids the need for drawing circles and trying to find a pattern, and provides a more systematic and scientific solution to the problem.