Charged Sphere Pump

Exercise

You have 3 perfectly conducting spheres of equal size. Two spheres are charged, one with +10 and the other with -10 (units will be left out for this exercise), and the third is neutral. There is no ground. The goal of the game is to get as much positive charge onto the negative sphere, and as much negative charge onto the positive sphere as possible by successively touching the spheres. You may touch any sphere to any other, any number of times. In other words, what order do you need to touch spheres so that the negative sphere gets as positive as possible, and the positive sphere gets as negative as possible?

How much maximum charge can be transferred this way? How much charge could be transferred this way with two uncharged spheres, instead of one, to work with? How much charge could theoretically be transferred this way if you had an infinite amount of uncharged spheres to work with?

 

Facts

It turns out you can get the positive sphere to be negative and the negative sphere to be positive, but they will not have the same value.

With 3 total spheres, the +10 sphere can change to -2.5 and the -10 sphere can change to +1.25. Illustrated solution below in the Thoughts and Opinions section.

With 4 total spheres, two of them being neutral, the +10 sphere can change to -3.75 and the -10 sphere can change to +2.1875.

Continuing the pattern, with an infinite number of uncharged spheres to work with, you can get the +10 sphere to become -5 and the -10 sphere to become +3.333…

 

Thoughts and Opinions

This exercise is dedicated to the class of 2023, especially Matt and Mike. Thank you for your hard work helping put this one together.

The concept behind this puzzle is the transfer of charge between two conducting objects when they touch (a process named “conduction”). Before charge was experimented with in wires, batteries, and light bulbs, scientists may have experimented with the ability to transfer charge in other ways. Illustrated below is the solution to the first case of 3 total spheres. Notice they need to be the same size, or the charge transfer will not split the charge evenly. One could engineer a much more efficient charge pump by simply having 4 spheres of different size.

Operations A, B, and C are marked on the diagram above to show the order spheres need to be touched to maximally transfer charge. It is natural for the reader to ask why you might need to move positive charge from one end to the other. This exercise, after all, requires a lot of imagination.

I will not include an illustration for the solution with two neutral spheres, as it turns out illustrating every iteration up to infinite spheres is quite a lot of work to do. It can be satisfying work to prove the case of 4 or even 5 spheres. For infinite spheres, a new technique is required.

I would like to say I am clever enough to find the general rule, but in fact that answer was found by a clever young man named Matt. I named it the “zipper method” and that too I will leave as an exercise for you to find out. The minimum number of operations required turns out to be 2N – 3, where N is the total number of spheres. Instead, I personally took the easy way out and created a computer program to successively solve the case of 4, 5, 6 … 1,000 spheres and deduced the answer from the pattern.

Here are some insights from our work: (1) It is possible to make the negative sphere positive and the positive sphere negative (2) It is not possible to completely transfer all negative charge and all positive charge (3) There will be “waste charge” left over on the neutral spheres (4) The initially charged spheres start out equal in magnitude (10 each, positive and negative), but will never be equal by the end of a “zipper method” process.

Feel free to email me with your thoughts and opinions and any further insights on this matter at andrew@ahogan.org.