Capacitors fully charged, Captain.

(I'm going to try to post about once a week from now on. We'll see.)

On Wednesday we did a fun lab that involved charging capacitors in an RC circuit. I only expect to hear about charging the capacitor banks in Star Trek and other such shows, so this was a neat experience.

The point of the lab was to observe the exponential decay of a discharging capacitor. A capacitor is a circuit element that takes the energy of a voltage source, such as a battery, and stores it for later use. The energy comes from moving charges, which are clumped together on capacitor plates. But like charges repel, so something needs to keep the charges in place. That something is another nearby capacitor plate where opposite charges are also being clumped.

If the space between the two plates is non-conducting, then the charges will feel an attraction to the other side but won't be able to do anything about, and this attraction balances the repulsion on their side. A capacitor has reached its, um, capacity when the voltage source is unable to overcome the repulsive force of the charges already on the plate. Each time you add an electron to the capacitor, it gets harder to add another one, because there's even more negative charge repelling the next electron. So every capacitor has a limit to the amount of charge per volt it can hold that is measured in farads.

Once the capacitor is fully charged, the energy is said to be stored in the electric field between the two capacitor plates. An analogous situation is a crane suspending a large hunk of steel in the air. The crane's engine supplies the energy necessary to lift the mass up, and the energy is then stored in the earth's gravitational field as potential energy. If the crane lets go of the steel, that potential energy is converted into kinetic energy and the steel slams into the ground. Similarly, if the capacitor is discharged, the electric potential transforms into kinetic energy, shooting the electrons out and establishing a current.

(In a further extension of this analogy, if a crane tries to lift something too heavy for too long, the cable will snap, damaging the whole setup. If too much voltage is applied to a capacitor, then the insulating material between the plates momentarily transforms into a conductor and charge shoots across, ruining the capacitor. This is lightning. When charge builds up during a storm, the air acts as an insulator between the clouds and the ground. Once the voltage gets too high, the air ionizes and creates a path for the charged electrons.)

What differentiates this from gravity is that the force pushing the charges off the plate is proportional to the number of charges on the plate in that instant. For the same reason that it gets harder and harder to add more charges onto a plate, the charges initially leave the plate very quickly. But then, as they leave, there are fewer repulsive charges in place, and the remaining ones leave more slowly.

When the rate at which some amount of stuff changes is proportional to the amount of stuff present, you have an exponential process. Radioactive decay is exponential because it depends on the amount of an isotope present in a sample. Population growth is ideally (that is, under perfect conditions, not as in I want it to be that way) exponential because the more people there are, the more children will be born.

The most recognizable feature of an exponential process is that the amount of time it takes for the process to double (or halve, or reach any specified multiple) is constant. This is where we get the idea of half-life. If some radioactive material has a half-life of a thousand years, and you've got 50 grams of it, then after a thousand years you will have 25 grams, but it takes another thousand years to get down to 12.5 grams.

Which leads us to this pretty graph:

As you can see, our capacitor discharged itself in a very cooperative exponential fashion. But it didn't cooperate fully. Every exponential decay has an associated "time constant" which is the amount of time that has to pass before the sample has reached 1/e of its original size. In an RC circuit, the time constant turns out to be the product of the circuit's resistance and capacitance. We had a 22 megaohm resistor and a 1 microfarad capacitor, which gave us a time constant of 22 seconds.

If you analyze the data we collected, however, our half-life turns out to be about 24 seconds. Now, there's a relationship between half-life and time constant, but right away it should be obvious that something is wrong. It absolutely has to take longer for the voltage to decay to 1/e (.368) its starting value than 1/2 its starting value. So if we accept that our calculated half-life is correct (which I'm willing to do, because it's such a pretty graph), then the time constant must be higher by about 13 seconds. For the time constant to be higher, there must be more total capacitance or resistance in the circuit.

But we have a fairly simple circuit that looks like this:

(We are fortunate that our circuit diagrams are not graded on artistic merit.)

When we close the switch, the battery charges the capacitor but is hampered by the resistor. The DMM is a digital multimeter which measures the voltage across the circuit. When we open the switch, the battery is no longer a part of the circuit, the electromotive force keeping the capacitor charged goes away, and the voltage discharges across the circuit. We measured this with the DMM.

Now, the wires have resistance, but not on the order of 12 or 13 megaohms. And it would be nearly impossible to add any capacitance to the system because capacitors in series have a lower net capacitance than that of each individual capacitor. Capacitors in parallel can add up, but we don't have anything in parallel when the switch is open. The only explanation is that the DMM has some internal resistance. As it turns out, the DMM is supposed to have an internal resistance of between 10 and 20 megaohms. So our 12 missing megaohms fit perfectly into that range. w00t.

As a final note, there was another section of the lab dealing with fast decay on the order of milliseconds. Because we can't measure that with our eyes, we used the oscilloscope to plot voltage versus time. As I explained before, the scope's display is a grid of squares. The width of each square corresponds to a length of time, the height to a voltage. How much time and voltage are calibrated with knobs. Anywho, we were measuring half-life, and we saw that the voltage decayed to half its original value across 1.5 squares. Our time knob was set to .5 milliseconds.

One of my partners will only take raw data, so he wrote down 1.5 * .5 milliseconds and left it at that, because he didn't want to make any math errors during the lab. My other partner whipped out his calculator and began typing 1.5*5x10^-4 into it. I looked at them both rather strangely and said "Guys, you just divide by two. It's .75 milliseconds." They looked at me like I was crazy and kept writing/calculating. Sigh.