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18 July 2013

"Because V=IR"

My first laboratory exercise with circuits shows the students a circuit with a battery and two resistors in series.  Part (a) asks which resistor takes the larger current through it -- that's easy, because they see the fact of physics that says "current through series resistors is the same through each and equal to the total."

Part (b) is slightly more complicated.  It asks which resistor takes the larger voltage across it.  Everyone recognizes that the 70 ohm resistor is the one with larger voltage.  But how should that statement be justified?

Incomplete justification:  "The larger resistor must take the larger voltage."  Says who?  All justifications in physics must begin with a fact, an equation, or a calculation.  We have no fact of physics in our textbook or our fact sheet that says larger resistors take larger voltages.  So try again.

Incomplete justification:  "The larger resistor takes the larger voltage because V=IR."  Not good enough.  Everything in circuits can, in some sense, be justified with the phrase "Because V=IR."  Just citing an equation isn't enough -- I need to see how the equation is used to justify the answer.

Incomplete justification:  "The larger resistor takes the larger voltage because in V=IR, when resistance increases so does voltage."  Almost there... but missing an important element.  This equation contains more than just resistance and voltage.  Mathematically, a slightly larger with a much smaller I would produce a smaller V!  

When justifying an answer with an equation, always start with the variable(s) that don't change:

Complete justification:  "The larger resistor takes the larger voltage because in V=IR, each resistor takes the same I because they are series resistors.  Then mathematically a larger R gives a larger V."  I need to see the statement about constant current for both resistors before I accept the justification.

The protest:  "But Mr. Jacobs, I already told you that the current was the same for both resistors in part (a)."  You did, and I appreciate it.  I nevertheless need you to say it again in part (b), or at least to refer me back to part (a).  Without the explicit statement about constant current, Ohm's Law doesn't lead to the conclusion you think it does.*

*  In the very rare situation when the student gets huffy with me, I remind him firmly that I'm asking him to write just four extra words; if he wants to ask the headmaster to replace the physics teacher because I require four more words than he personally wants to write, he's welcome.  Until he has that conversation with the headmaster, he may stop whining and do the work like I asked him to.

Why I'm picky here:  In general, the key to justifying an answer directly from an equation without calculation relies on identify the constant variables.  So I'm emphasizing the technique that will help students be successful beyond this single problem, or even this single exercise.

Beyond that, I've established how this sort of justification should look throughout this circuit exercise. When the students come to the combination parallel-series circuit, they see quickly that current is not the same for each resistor; so the reasoning "R2 takes the largest voltage because it is the larges resistor" is never even written down for me to criticize.  The class may groan at my pedantry, but they gain the benefits.  There's a method to my madness, even when my students don't see the method right away.

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