My first blog post discussed the following interview question I received many years ago.
You are handed a 1 F capacitor charged to 10 V and two boxes containing uncharged capacitors: one box contains an uncharged 1 F capacitor and the other contains 1 million 1 µF capacitors. You can connect the capacitors from one box, one at a time, across the charged capacitor. Your job is to determine which box will allow you to discharge the voltage on the charged capacitor the most?
I received a comment recently from a reader who had looked at the problem and brought up an interesting aspect of the problem that I had not mentioned – energy does not appear to be conserved. My solution used charge conservation, so I skirted the energy conservation issue. Here is how the question came to me.
Meant to mention that these two problems together beg the question of how much energy is stored in the million charged 1uFs and, added to 6.7668 joules remaining on the 1F, how does the total compare to the original 50 joules. This is a tougher problem because each small cap is charged to a different – though calculable – value. I haven’t tackled that question yet; maybe you have.
Many problem solutions depend on some property remaining true throughout some transformation. A property that remain true throughout a transformation is called an invariant. There were clearly two candidate invariants in this problem: energy conservation and charge conservation. It turns out that the bookkeeping for the charge conservation is easier because this problem presents no opportunity to lose charge. Unfortunately, the bookkeeping for energy conservation is more difficult because energy can be lost in the interconnection between the capacitors. I will illustrate how using energy conservation for this problem presents an issue – how do I determine the losses that occur in the interconnection?
Let's begin my computing the energy on the 1 F capacitor before and after the million 1 microfarad capacitors have been attached. Since I am computationally lazy, I will write a Mathcad program to compute the result.
From this calculation we can see that the 1F capacitor loses about 43.2 Joules of energy. We now need to compute how much energy did the million one microfarad capacitors pick up, which is done below.
The million microfarad capacitors only ended up with 21.6 J of energy. Therefore, we are missing 42.3 J - 21.6 J = 21.7 J of energy.
Energy must be conserved, so where did the energy go? It was lost during the charge transfer through a combination of heat and electromagnetic radiation. This is why I used charge conservation for my invariant. I did not have a way to determine the energy loss that occurs during the connection.