5 positivly charged particles on sphere, min energy configuration, rel

[Spinnor]

Assume 5 charged particles (charge 1) constrained to live on the surface of a sphere are in a configuration that minimizes electrostatic potential energy. Are there configurations that are stable but that are not the minimum energy configuration?

A simple computer program could quickly(?) examine many random configurations and slowly map out the potential energy surface E(θ_1,phi_1,θ_2,phi_2,θ_3,phi_3,θ_4,phi_4,θ_5,phi_5)?

There must be more elegant(less computer time) ways to find the minimum energy configuration?

Has this problem been solved?

Edit, E above is a function of only 8 variables, we can always let one particle be at the north pole?Thanks for any help!

 

[Simon Bridge]

It's called "the Thomson Problem".
http://en.wikipedia.org/wiki/Thomson_problem

 

[Spinnor]

It's called "the Thomson Problem".
http://en.wikipedia.org/wiki/Thomson_problem

Triangular dipyramid Nice and symmetric, I should have seen that. Things probably get more interesting with larger numbers. From the link above check out a fun Java app at,

http://thomson.phy.syr.edu/thomsonapplet.php

Check out the screen shot and add a charged particle and watch the charges rearrange to I assume the lowest energy configuration. Still would like to know if there are stable relative minimum.

Edit, I think you have to hit the Auto button on the app to get the configuration to minimize energy?

Thanks for your help!

 

[Simon Bridge]

Nice find!

 

[sardar]

I can provide a response to this content by saying that yes, there are configurations that are stable but not the minimum energy configuration for 5 positively charged particles on a sphere. This is because the minimum energy configuration is determined by the Coulomb's Law, which states that the electrostatic potential energy between two charged particles is inversely proportional to the distance between them. Therefore, the minimum energy configuration would be one where the particles are evenly distributed on the surface of the sphere, with equal distances between each particle.

However, it is possible for stable configurations to exist where the particles are not evenly distributed, but still maintain a stable equilibrium due to the repulsion between the positively charged particles. These configurations may have a higher potential energy compared to the minimum energy configuration, but they can still be stable.

In terms of finding the minimum energy configuration, there are indeed more elegant ways to do so rather than using a computer program to examine random configurations. One approach is to use analytical methods, such as solving the equations of motion for the particles on the surface of the sphere. This would provide a mathematical solution for the minimum energy configuration.

As for whether this problem has been solved, it is a well-known problem in the field of classical mechanics and has been studied extensively. There are various analytical and numerical methods that have been used to find the minimum energy configuration for a given number of particles on a sphere. However, the problem becomes more complex when considering more particles or when taking into account other factors such as the size and charge of the particles. Therefore, it is an ongoing area of research in the scientific community.