Sorry for the incendiary title, but it does express my feelings pretty well. When analyzing DC circuits, students are often encouraged to use the two Kirchhoff’s laws:
- [sometimes called the node law] all current flowing into a node must flow out
- [sometimes called the loop law] the change in voltage around any closed loop is zero
Using those two, you can analyze any DC circuit that has batteries and resistors in it (and of course other elements if you appropriately define the impedance). However, while the first of the two is a no-brainer for most students, the loop law gives students all kinds of fits, mostly due to trying to imagine walking a charged particle all the way around a loop, keeping track of any voltage changes. If you go through a battery the normal way, that’s a gain. If you go through a resistor the way all other (positive – damn you Ben Franklin) charges go, that’s a voltage loss. Those two aren’t really that big of a deal. However, in any circuit that really needs this approach, you often have to imaging taking a charge through a resistor the other way, against the flow of current. You say it gains voltage during that. Students seem to have a hard time with that. Except the rule-followers, those students don’t seem to have much trouble with this. Ask them “why,” though, and you’ll regret it.
Here’s the other problem with this whole approach: at the end of the analysis, you’re usually staring at a bunch of equations with a bunch of unknowns. Often the equations outnumber the unknowns, but we try to show the students that if you use all the node equations or all the loop equations, you’re being a little redundant. Whatever. The problem is that as soon as it’s more than two unknowns, you’re really testing their math abilities much more so than their physics abilities.
So what to do instead? Here’s what I do: I give them a
complex complicated circuit with a bunch of batteries and resistors and then I do the unthinkable: I give them one of the currents for free. That’s right, I tell them one of the answers. Then I ask them to figure out the rest of the currents by building on that information. Here’s an example. I give them this circuit:
and then I tell them that the current flowing up through the center 1 ohm resistor is 6 amps. From there they try to reason out the rest, usually in this order:
- The voltage at the top is 10 volts because you lose 6 of the 16 volts of the central battery going up to the top.
- The voltage drop on the left leg is also 10, so the voltage drop across the top left resistor is 2 volts. Therefore the current through that one is 2 amp downward.
- That means that there’s 4 amps flowing down the right leg.
- The top right resistor burns 4 volts, meaning that the voltage drop across the parallel portion is 10-4=6 volts.
- The current through the right leg of the parallel section must be 6 amps to account for step (4)
- The current in the left leg of the parallel section must then by 2 amps upward so that the node law works at the top.
None of that is solving 4 equations for 4 unknowns. Rather, it’s demonstrating an understanding of ohms law and batteries all at a local level.
I like this approach, and I use it on quizzes/exams a lot. I certainly don’t want to test their linear algebra ability in those situations. Now, in the old days I used to give them a complex circuit and ask for enough loop and node equations that would, in principle, enable them to solve everything. I like this new way better.
But what about the student who asks how I got that first unknown in the first place? I’ve been thinking about that a lot lately and I think I’ve come up with a cool way to talk about that. Instead of showing them the n equations and n unknowns approach (assuming that they really understand the underpinnings of some powerful linear algebra), now I think I’ll have them tackle problems like that by just simply guessing one of the currents. Then they can follow similar steps as above until they run into a discrepancy. For example, imagine if you thought that the middle current above was 1 amp upward:
- then the voltage at the top would be 15V
- the voltage difference across the upper left resistor would be 7 V so you’d have 7 amps flowing down there.
- That would mean the right portion would have 6 amps flowing up.
- That would put the voltage at the top of the parallel section at 15+6=21 V
- That would make the current in the lower right portion 21 amps down
- That would make the current in the other portion of the parallel section 27 amps up
- BUT THAT WOULD MAKE THE VOLTAGE DROP OF THAT LEG WRONG
So you see we’d have a discrepancy. What I’ve been playing with today is to see how that discrepancy (in this case, a voltage discrepancy) varies with the initial guess of one amp flowing up in the middle portion. It turns out, unsurprisingly given the linear nature of all of this physics, that it’s a straight line if you plot the discrepancy versus the current guess. Then what you’re looking for is the current that would give you zero discrepancy. That’s the same as asking for the x-intercept of that straight line.
Now, I think students could do that, especially since never would they do linear algebra. However, I would still give them freebies in quiz/exam situations because I think they just need to do this exercise I mentioned once to see how I got the freebie. Note that to fully solve a circuit this way you have to do the 7 steps above twice (to get two points for the straight line) and then a third time with the right value. That’s a lot of work, but every step is explainable.
Your thoughts? Here’s some starters for you:
- I like this. I too hate the Kirchhoff loop law, but my issue was always . . .
- What’s wrong with the loop law?! It’s awesome, especially because . . .
- Why are you even mentioning the loop law when we all know that circuits with induction show that it’s wrong?
- Don’t you think the “linear discrepancy theory” is just as hard to explain as imagining a charge going against the current?
- Why do you think our positive charge convention is the fault of my beloved Ben Franklin?
- What happens when you teach this and your colleagues teaching lab assume the students know the loop law?
- What happens when the discrepancy is with a current instead of a voltage?