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I came across this paper which describes minimum energy configuration for N equal point charges in a circular enclosure which seem to form shell like structure. For lower values of N local optimum is point charges spaced equidistant from each other on the circumference. As N increases one out of N is pushed inwards and shell began to form.

I was wondering what would happen if we introduce one more electron of charge 2e in the system i.e. (N-1) charges will have charge +e while one has charge +2e. What would be minimum energy configuration in such scenario (for lower values of N say) or possible local optimum(at least)? Will charges be equidistant from each other (I don't think so)?

Edit:

One local optimum that I am aware of is inserting a +2e charge in the center. However, placing all N charges around the perimeter results in a lower equilibrium energy (I have computationally confirmed this for N up to 4).

I am looking for optimum configuration when all N charges are placed on the circumference.

Minimum energy configuration for different values of M

Avinash Kumar Ranjan
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  • I would expect that the "special" charge will be put in the center in order to preserve rotational symmetry, and I think someone more knowledgeable about this area could prove so rather easily. – YiFan Tey Sep 26 '22 at 07:17
  • Putting the "special" charge at center is one of the possibility and it is one of local optimum. But I was wondering what sort of geometry arises when it is instead on the circumference. What will be local optimum in such scenario? – Avinash Kumar Ranjan Sep 26 '22 at 07:19
  • Hmm, okay. If you want to only consider the case with the special charge on the circumference, you might want to make it clearer in the question body. – YiFan Tey Sep 26 '22 at 07:21

1 Answers1

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For $i\in\{1,\dots,N\}$, let $q_i$ be the charge of point $i$, with $q_1=2$ and $q_i=1$ for $i\not= 1$. Let decision variables $(x_i,y_i)$ represent the coordinates of point $i$. The problem is to minimize $$\sum_{i<j} \frac{q_i q_j}{\sqrt{(x_i-x_j)^2+(y_i-y_j)^2}}$$ subject to $x_i^2 + y_i^2 \le 1$ for all $i$. Without loss of optimality, you can optionally fix $y_1=0$ and impose $x_1 \ge 0$.

Up to $N=10$, it looks like it is optimal to have all charges on the circumference but not equidistant: enter image description here

\begin{matrix} i &q_i &x_i &y_i &\theta_i &\theta_i - \theta_{i-1} \\ \hline 1 &2 &1 &0 &0 \\ 2 &1 &0.71322 &0.70094 &0.77672 &0.77672 \\ 3 &1 &0.19341 &0.98112 &1.37616 &0.59944 \\ 4 &1 &-0.38571 &0.92262 &1.96677 &0.59061 \\ 5 &1 &-0.83264 &0.55381 &2.55465 &0.58788 \\ 6 &1 &-1 &0 &3.14159 &0.58694 \\ 7 &1 &-0.83264 &-0.55381 &3.72853 &0.58694 \\ 8 &1 &-0.38571 &-0.92262 &4.31642 &0.58788 \\ 9 &1 &0.19341 &-0.98112 &4.90703 &0.59061 \\ 10 &1 &0.71322 &-0.70094 &5.50647 &0.59944 \\ \end{matrix} For $N=11$, it is better to have a 1e charge in the interior but not exactly at the center: enter image description here

RobPratt
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  • May I know how exactly did you arrive at the configuration for N=10? – Avinash Kumar Ranjan Sep 29 '22 at 05:35
  • I used the NLP solver (with the multistart option) in SAS. – RobPratt Sep 29 '22 at 15:38
  • Thanks a lot RobPratt! – Avinash Kumar Ranjan Sep 29 '22 at 21:02
  • @RobPratt Because some symmetry prevails in the diagram for the case $n=10$, I thought the angular distance (or the distance) between any two neighbouring unit (+1) charged particles is the same. In order to convince myself, I measured the mentioned angles. But, I found that they are not even approximately equal. Can you please give me the $x-$ and $y-$ coordinates of the 4 unit-charged particles on the top half of the circle, if you have them? – YNK Oct 03 '22 at 11:22
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    @YNK I added the coordinates to my answer. – RobPratt Oct 03 '22 at 15:14
  • @RobPratt Thank you very much! – YNK Oct 03 '22 at 17:29