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.