## Premature Jocularity

You may recognize the above term from old episodes of *SportsCenter*; it’s a term they used when covering athletes who celebrated some achievement before it was, technically, complete, and in doing so either endangered it or actually kept it from happening. Canonical examples include that soccer goaltender who was too busy doing the “I am the man” dance to notice that the ball he’d just deflected was, in fact, rolling into the goal anyway, and that one snowboarder in the 2006 Winter Olympics who was leading a race by such a huge margin that she decided to do a showboaty stunt on the last jump – and promptly stuffed it and had lost quite resoundingly by the time she extracted herself from the snowbank.

I don’t like to risk premature jocularity, because a) it’s obnoxious and b) it leads to a particularly agonizing sort of embarrassment when it all goes wrong as a direct result of showing off. Besides, it constitutes a very specific type of tempting fate. I’m as superstitious as the next guy, and the last thing I need is to jinx myself by some round declaration that I’ve got something surrounded.

But all the same, I have to say… last night I sat down and, having accumulated a bit of a backlog of online-math-class homework sets owing to illness, fall break, and a bit of losing track of time, undertook to catch up. I approached this task with a measure of dread, because in addition to having fallen a bit behind on the homework, I also haven’t managed to make it to lecture for a couple of weeks. That’s not actually a problem, administratively, because I’m technically in the online class and am not expected to show up for the live lectures at all, but as previously discussed, I think I do a lot better when I make it to them. But between one thing and another, I haven’t been able to attend since the exam, which covered through Chapter 4.

This meant that I had the homework sets for all the sections of Chapter 5 we’ve covered to do – six of them, plus a quiz on 5.1 through 5.4 – and no live instructor face time for any of it. Just the “see an example” button and the little videos and animations the online course tools provide.

And… it wasn’t that hard.

Mind you, it *took* a long time – something like four and a half hours to get it all done – and I felt pretty crispy at the end, because I didn’t actually intend to do *all six sets and the quiz* in a single day. I figured I’d do two of them yesterday, two and the quiz today, and the remaining two tomorrow afternoon, between ECE seminar and (touch wood) the observatory opening. Instead it was like the math homework equivalent of one of those occasions where you sit down to have a cookie and discover yourself, an indeterminate time later, covered in crumbs and clutching an empty cookie bag. (Or does that only happen to me?)

It took a long time and it left me slightly reeling, but… and again I have to stop and glance furtively around – it really didn’t seem that hard. It was all to do with logarithms and exponential equations, and the relationships between them, and I’m sure I could still do with some reviewing on the purely rote parts of the process (which bits of an exponential go where in a log, for instance – can be worked out from context in a lot of cases, but it’s presumably easier to just learn by rote which number goes where), but overall it wasn’t nearly as agonizing as, say, rational functions.

In a perverse way, I almost hate it when I start to feel like I’m getting something. It always places me in danger of pausing too long to admire what I’ve just learned and then finding myself running behind the school bus as it plows remorselessly on to the *next* stop on whatever route the class is taking. I think my tattered old brain would like college better if it ran on about a ten-year timescale. Still, I get what I’m expected to get this week, and that feels pretty good.

Mind you, I’m still not sure where *e* comes into it over in ECE 101 – I know it’s involved in the calculation of a capacitor’s voltage over time, but why that particular bizarrely irrational number should come into it I’m not all that clear on… but one epiphany at a time. At least I know what to *do* with it in that context.

There is a reason, but you need to get to calculus before it’ll be a reason that makes intuitive sense. I think the thing to know at the point you’re at, is that

eturns up “naturally” whenever the rate at which something changes is directly proportional to the magnitude of that thing.Interesting. Well, the voltage across a capacitor certainly does that. I just finished a homework problem the whole purpose of which was to make it apparent that the rate at which voltage increases across a capacitor in a circuit with an unchanged time constant (resistance times capacitance) depends directly on the voltage of the source. If the cap and the resistor are the same, the cap takes the same amount of time to reach full voltage, regardless of what that full voltage is. Which is kind of cool.

Think of it as being kinda like Zeno’s paradoxes. A half full capacitor is harder to put stuff into than an empty one. So every time you fill half the remaining capacity, it takes more and more oomph. And while Achilles and the Hare ends because Zeno’s example was missing a piece, you can’t ever fill a capacitor up 100.00000… %. The (1 – e^-t/RC) term shows how that happens. t has to go to infinity for the whole thing to equal exactly 1.

Well, yes, I should’ve said “the cap’s voltage increases by the same proportion per time constant,” since technically ALL capacitors reach full voltage at the same time, that being infinity.

For example, in the one I just did, the voltage in one example was greater than the one in the other example by a factor of ten. This had the interesting (and perhaps not entirely surprising) effect of bumping up all the vc(t) slices at the first five time constants by a factor of ten as well.

I see that the next one in the set multiplies the resistance by 10 instead. I expect this will have the same “values at each time constant are the same” effect, but increase the length of the time slices. Which makes sense.

I find the whole thing pleasingly elegant.

Yeah, the relationship between “pure math” and all the low-level “physics” stuff is actually very cool, and calculus and basic electrical engineering is one of the places where it’s easiest to see how all these subjects (often taught as completely separate at the high school level) start fitting together.

I hope the homework binge is a good omen, and that practice makes everything go more smoothly (faster and more accurate).