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?

I am super jealous of you right now. Tomorrow I am teaching in a 6th grade science class and my biggest worry is which kid is going to be sent out of the room first.

I love the quiz/exam type of question that you have been using. I too use these ones and like how it asks them to apply the rules but avoid the grinding. However, I’m not sure I really understand your concern with the hypothetical student that asks how you got the one current. They certainly don’t ask we measured the work or energy that we provide them as givens in all of the mechanics questions we ask them to solve. Plus current is something that they completely understand that there is a special kind of measurement device whose purpose is to measure that thing.

Your example of using guess and check twice to solve the circuit strikes me as being on par with having to solve the linear equations in the first place in terms of being tedious.

That’s a good point about the ammeter approach. I think I was just nervous that I’m not fully preparing them to understand how to do the whole circuit if I concentrate on this freebie approach. Of course the book goes through the usual approach, so maybe I shouldn’t be so nervous.

I also agree with the tediousness of the discrepancy approach, but I’m ok with that as long as each step is clear (unlike gaussian elimination for lots of linear equations).

On Tue, Oct 7, 2014 at 11:20 PM, SuperFly Physics wrote:

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I have to say, I really appreciate your approach to teaching Kirchhoff’s Laws. I was learning about these laws in my physics labs last week and seeing how they apply to actual circuits, but unfortunately, solving the system of equations ended up completely subverting the point of the lab. Each of the four groups in the lab spent at least 30 minutes trying to successfully solve the equations, and it was incredibly frustrating for many people, myself included (I do not have a background in linear algebra).

thanks for the comment, Kevin! There is, of coures, an argument that having lab and lecture approach things differently can be good for learning, but tedium, especially tedium that gets in the way of learning, needs to be watched out for.

Hi Andy, I think what you’ve described in your post is brilliant. I wouldn’t say I ever hated Kirchoff’s loop rule, but I have tended to simply avoid giving my students complex circuits, feeling that I don’t want to bog them down in trivial arithmetic that they would almost certainly mess up. I have asked them in the past to evaluate another students’ solution (hopefully implementing the loop and node rules). That is a fun exercise. But this is a really nice way to get them to use the node and loop rules in a practical way. I’m going to use this idea with my students, if you don’t mind.

Of course I don’t mind! I’m glad to hear it could be useful to you. I’m intrigued by the idea of having them critique others’ solutions. I could see that working in so many different contexts.

On Thu, Oct 9, 2014 at 5:07 AM, SuperFly Physics wrote:

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I can definitely appreciate the value of this method of teaching Kirchhoff’s loop law. The way I had to go through Kirchhoff’s loop law equations in lab to come up with the eventual answers took hours. It was frustrating to mess up a single negative and get the wrong answer. Due to the lengthy math involved, most of the concept was actually lost. At the end of lab I knew that if I was asked to replicate my work it would have been nearly impossible, especially in a test period. I think giving students a small amount of information, to reduce the unknowns, and for them to fundamentally understand the law itself is more important. I have personally learned more about circuits’ behavior and trends via this method. The linear algebra way, may make more sense to those who like to conceptualize with math, but that is not my learning style. Obviously there is value to traditional ways of teaching, but I agree too much time is spent on math instead of the physics. The guessing at a circuit’s unknown amperes sounds like an interesting method. It seems like a nice middle ground between the math way and the concept way. Using the math and concept together to prove if you are right. I think it could become frustrating though. It is discouraging to go through all of the work and be wrong. The discrepancy plot sounds very interesting, but I would have to actually use it to give any more thoughts.

Thanks for the comment Hailey! I like how you’re able to articulate (can you tell that’s one of my favorite words?) how the different approaches affect your learning. A colleague and I were discussing the notion of whether a learning experience that yields learning needs to be remembered fondly by a student. I argued that it should. In other words, of course learning matters, but if the students memory of the experience is negative, that affects learning other things that need that idea.

I like your idea as a way of emphasizing the physics without getting bogged down in the math. This is an especially great idea for a way to present a multi-loop circuit on an exam.

One thing I do love about the math for Kirchoff’s laws is that you can pick current directions when you are setting up your equations and even if you pick wrong (for example, you guess that current is flowing down through your center resistor) you still get correct results, you just have to interpret the signs of your current values. There are several places in physics where you can guess a direction when setting up an equation and as long as you are consistent the math will work out. I find that beautiful and always want to point that out to students.

Yeah, I really like that notion of picking the wrong direction at the beginning but dealing with it at the end. I don’t like the texts that really try to get students to guess correctly at the beginning. Instead I tell them to just always choose down and/or to the right and just deal with the negative signs at the end.

On Mon, Oct 13, 2014 at 6:41 PM, SuperFly Physics wrote:

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very nice

very nice

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Thanks, dear. It’s a nice post. I like the way you have described it. It’s really appreciable. But if you provide calculation of current and applications of Kirchhoff’s law . Then it will be a greate post.

great*