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16 January 2017

Mail Time: Rigorous definitions of circuit properties in AP Physics 1

Buckeye native Matthew writes in with a question about circuits in AP Physics 1.  He’s referring to my summary post of the topics on the exam

Greg - Happy New Year!. As I am outlining the second semester of the year I am having difficulty finding information about

Non-rigorous definitions of voltage, current, resistance
Rigorous definitions of voltage, current, resistance

Any information you could provide me about finding the differences between non-rigorous or rigorous would be appreciated. 

I am thinking (hoping) that I already address this and just have not been exposed to the terms non-rigorous and rigorous when it comes to the definitions?!?!

Matthew, great timing -- I just worked on this difference with my AP class last week.  

"Rigorous" and "non-rigorous" definitions are my own personal terms, not anything to do with materials published by the College Board.

I start circuits on the very first day with the non-rigorous definitions:

Non-rigorous definitions of voltage, current, resistance
Voltage is provided by a battery.  Voltage is measured in units of volts.
Resistance is provided by a resistor, a lamp, or any electronic device.  The units of resistance are ohms (W).
Current relates to the amount of charge flowing through a resistor.  The units of current are amps.
Ohm’s law states that voltage is equal to current multiplied by resistance:  V = IR.

With just these facts, I can have students graph current and voltage to verify or discover the relationships in ohm's law; I can have students measure brightness of a bulb as a function of voltage and resistance to discover the power equation. And then we can do basic semi-quantitative questions with single resistor circuits, like "I replace a 10 ohm resistor with a 20 ohm resistor, by what factor has the current in the circuit changed?"  

Then we move on to circuits with series and parallel resistors, then to combinations of resistors, then to light bulbs, then to circuits with switches, using ammeters and voltmeters. I like to give circuit TIPERs, but make the students set up the situations experimentally to verify their prediction.

During these first couple of weeks, I never mention Kirchoff's laws -- rather, we have rules about current and voltage for parallel and series resistors which are a poor person's statement of Kirchoff.  (“Voltage across series resistors is different for each, but adds to the total.”)

Finally, once we've done all of this... everyone has a personal, intuitive understanding of what current and voltage are.  That understanding has been built on experience through problem solving, lab work, right and wrong answers.*  In eduspeak, this personal, intuitive understanding is referred to as an "operational definition."

*Never through analogy, though.  If students create their own analogies, great.  But direct experience without analogy has proven far more effective at building knowledge and avoiding misconceptions than any analogy I've ever tried.  Voltage and current aren't truly LIKE anything else.  

So, with that personal understanding built, it's time to introduce the rigorous definitions:

Rigorous definitions of voltage, current, resistance
Voltage is energy per charge.
Current is charge per time.
Power is energy per time.
Potential difference is a synonym for voltage.

Remember, your students aren't likely to come into the course with an operational definition of charge; and gaining the experience necessary to develop what charge truly means requires, I think, a full-on AP Physics 2/C treatment.  And "energy" is still a bit fuzzy in students' minds.  (These rigorous definitions can actually help students develop their operational definition of energy and charge, since they're so solid on voltage and current.)

I therefore tell the students to translate from rigorous language into our non-rigorous definitions.  When they see a problem like "rank these bulbs based on how much energy is gained by an electron passing through" they recognize that as asking about energy per charge; that just means "rank by voltage," which my class is well trained to do.

The last bit about circuits we do is to use Kirchoff's laws, and to make voltage vs. position-in-circuit graphs.  Here I use the terms "electric potential" and "electric potential difference" with impunity.  But by this time voltage is such an ingrained concept that the class has little difficulty anymore.


  1. I recently switched to teaching circuits only with Kirchoff's laws and treating the rules for parallel/series as the afterthought. So it is a bit opposite of what you outlined here. I found this really helped my students with conceptual understanding of what circuits are as an application of conservation concepts. How do you avoid them simply thinking circuits are just a "game" with rules to memorize?

  2. Great question, David. Firstly, I keep everything grounded in experiment -- we're not playing mathematical games to get the right answer, we're making predictions that are verified by experiment. They develop their understanding of Kirchoff's loop rule as much by measuring voltages in a series circuit as by calculating voltages in a table.

    That said, I don't mind the class *beginning the unit* memorizing rules and applying them blindly. By this point in the year I've built to a pretty good understanding of how to learn physics, of the primacy of process over result. They know that just getting answers is not a strategy leading to long-term success. (Or short-term success -- they know by now that an unjustified answer earns no credit, ever.)

    Internalizing basic rules of parallel and series resistors is really no different than internalizing Kirchoff's laws. You're going big picture to little picture, going from conservation laws to Kirchoff to specific application of Kirchoff. I'm going little to big, from specific applications of rules to Kirchoff's rules to conservation. My students get there, as evidenced by performance on AP- and TIPERS-style questions that test understanding, without the opportunity to game the question. No doubt your students get to the same place.