A2 Physics - Capacitors

This is being done on a boat, and as such without internet. Thus I am typing this up in OpenOffice, and without the Blogger Draft's curious ability to know exactly what I want to do with my images. So let's do this thing. Today I will be talking about, oh, lets go for capacitors. I do not like capacitors. So, all the more reason to learn about them, I guess. I shall however be recording my state of boredom through judicious use of reaction images.

This is going to be much more factual than historical, no internet and all, so all I'm working with is my textbook. So let's start with an idea of what a capacitor is. It stores charge. Nothing much else. Get two metal plates near each other, boom, you gots yourself a capacitor. Connect it to a battery blam, you've got a charged capacitor, the two plates actually having equal and opposite charges. This I because the same amount of electrons leave the plates as they do jump on. It's like a water flume. One kid comes out the bottom, the guy at the top pushes another one in. Eventually the first kid gets back to the top of the flume to be pushed down by the guard guy, who I guess in this metaphor represents the battery.

Ok, so, charging one of these bad boys. This can be done by hooking it up to a direct current right? So what we are going to do is stick in a variable resistor, an microammeter, and a voltmeter. Exciting stuff right? Hold on, it gets even better. The voltmeter could be a data logger, which sends data to a computer, about the p.d.. Or you could use a regular voltmeter and use a stopwatch to record the p.d. at regular intervals. Pussy. So now we have some badass information in the form of amperes, p.d., and time. You know what we can do with these? Fucking everything. We want to know how much the capacitor has charged by? Q=It, bitch. We have I, we have t, so you know what that means? Yeah you do you BAMF of a physicist you. We have motherfucking charge Q.

But shit son, there's more. We still got a V hanging around. That's where shit gets interesting. You see capacitors have their own damn quantity called capacitance. This is defined as the charge stored per unit pd. You know what that means? You got it. C=Q/V. We have p.d. V and thanks to our braniac skills above we have charge Q. You know what's coming. Go ahead, tell me, Feel good about it. You got it, we have capacitance C.

And oh shit, you know what? These capacitors are used in tonnes of everyday items, including smoothing circuits, back-up power supplies, timing circuits, pulse producing circuits, tuning circuits and filter circuits. Shit man, this last paragraph is like an electronics student's wet dream.

Ok, so we have the basics. Capacitors are like temporary batteries. And you know what batteries store? Energy. So you know what capacitors store? Energy. Excitement abound. Specifically Electrical potential energy. So, picture this. You have a bunch of electrons on a plate. You know, one of the plates the capacitor is made of. So the plate has a charge, yeah? Then you try to put another electron on the plate. This requires work do be done on the electron/charge. So you put that energy into putting that electron on the plate, and this becomes electrical potential energy. For a comparison, think about gravitational potential energy. You put energy into lifting an object onto a table, thereby giving the object E_GP. Do you get it? Because I kinda do now. This actually feels kinda good.

But how much energy? Consider this in graph form. You mathsy people out there, you know what a graph gives you? Two things, other than the values it already represents. Yeah, the gradient and the area underneath. So now we are going to imagine a graph of charge Q against p.d V. Q is on the x axis and Q and V are directly proportional. You know what that kind of graph looks like I hope. Got it? Good because I'm not going to be mocking up one just for you. Besides, you were already doing so well. Feel free to draw your own if needed. Lets do this. Ok, think in graph terms, we have a capacitor with charge q and we want to add Δq to that value. This is represented on the graph by a vertical strip over Q. This area is the energy required to force that charge onto the plate. (Think of a particularly heavy kid in that water flume).

Excuse me, the original Pokemon theme tune came on iTunes. Just need to rock out.

That was awesome. Where were we? Right. Consider the strip on the graph. Now consider lots of those strips leading to 0Q. The energy still is represented by the area under the graph, but now its hella easy to work out. You know who to work out the area of a triangle? And shit brother, look what we have in that graph. Half base times height that shit, and what do you get? The equation for energy is what. E=½QV. Magic.

Christ I'm bored. And just as we get to the important bit. Charging and discharging these bad boys through a fixed resistor. OkOk, this tenuous metaphor represents the discharging of a capacitor, where the insults and general ill demeanor represent the resistor. So where as before the electron kids were continuous as long as the battery guard kept pushing, the resistor guard just doesn't care. As lots of kids that have built up are now spilling down the flume and possibly over the side of the rails. It's very messy. But the amount of kids falling down slows as there become less of them, and word goes around that this guard may have a thing against kids. Ok, it's not a perfect metaphor.

The point is that the decrease in kids happens exponentially. If we have Q₀ kids at t=0, and then at t=1 we have 0.9 Q₀ then at t=2 we with have 0.9*0.9Q₀, etc etc.

Ok, just a couple more pages. Why is it exponential? Pretty much because some rearranged equations tell us so. Skipping some stuff, some how you end up with ΔQ/Q = -Δt/RC. This tells us that a fractional drop in charge ΔQ/Q is the same in any short interval of Δt during discharge. Rearrange that bastard a bit and you get ΔQ/Δt = -Q/RC. Δx/Δy? Its a differential equation mother fucker! But physicists leave maths to the mathemagicians. We just do some trickery and end up with Q=Q₀ e^-t/RC. Voila. We have created the time constant, RC. Let's just do a final check of my boredom.

Wonderful.