Zen and the art of predicting voltage across series resistors

I have a circuit in which a 14 V battery is connected to a 15 ohm and a 25 ohm resistor in series.  What's the current through and the voltage across each resistor?

In my honors-level classes, I teach a mathematical solution using the VIR chart.  They calculate the equivalent resistance of 40 ohms; use ohm's law on the total circuit to get a current of 0.35 A; recognize that series resistors each take that same 0.35 A current; then multiply across the rows of the chart with ohm's law to get 5.3 V and 8.7 V across the resistors.

In conceptual, though, we don't use a calculator, and I want to minimize (not eliminate) calculation, anyway.  So we approach this problem slightly differently.  

Take a look at the worksheet we've used in laboratory to learn how to deal with series resistors.  Students must justify their answers to each question thoroughly.

1. Which resistor carries a larger current through it?

We start with a not-so-subtle reminder of the rule that each resistor carries the same current. 

2. Which resistor takes a larger voltage across it?

Now, I insist on an equation justification with ohm's law.  V = IR; I is the same for each because series resistors carry the same current.  By the equation, then, the larger 25 ohm resistor takes the larger voltage.

3. What is the equivalent resistance of the circuit?

Fact: the equivalent resistance of series resistors is the sum of the individual resistances.  Just add 'em up to get 40 ohms.

4. Calculate the current through the circuit.

Only now do we do a calculation.  We've learned that ohm's law applies for the total voltage and resistance in a circuit.  So, we simply say that I = V/R, with V = 14 V and R = 40 ohms.  This makes the current 14/40 amps. Yes, I allow that as the answer -- we aren't using calculators, and I don't want to mess with issues of significant figures and decimals.  Look, go ahead and haul me before the Klingon Death Tribunal for my sins.  I'm looking forward several years.  When they see circuits in a senior honors class, they'll be able to call this 400 milliamps.  For now, I'm happy that they know which values to plug in to ohm's law.

5. Estimate the voltage across and current through each resistor in the chart below.  You may use fractions for current, but not for voltage.  Answers without units earn no credit.  

Here's the Zen.  Not calculating current through each resistor -- that's the same 14/40 amps through each.  The Zen is the estimate of voltage.  I'm NOT teaching them to multiply the resistance of each resistor by the current.  Nor am I teaching them to proportionalize the voltage according to the resistance.  Nope.  I'm just saying "estimate."  

All I am looking for is an answer that fits the facts they've already stated:  the voltage across each must add to 14 V, and the 25 ohm resistor must take greater voltage.  Some students will guess 2 V and 12 V; some will guess 6 V and 8 V.  I don't care.

6.  Now, set up the circuit, and MEASURE the voltage across each resistor, and across the battery.  Record your results here.

And now the Zen must be reconciled with reality.  They make the voltage measurements, and see how the voltage is distributed:  in this case, about 5 V and 9 V.  Some students got the estimate right -- they get candy.  I praise everyone else for a "good guess;" we look to see whether their guess was high or low for the 25 ohm resistor.  

I don't teach anything, still; instead, I hand out a new sheet, with different resistors.  They fill it in again, all the way from the beginning, steps 1-6.

As you may or may not suspect, by the second or third time most students are getting pretty dang close to the right voltages.  Some folks discover the proportionality rule for themselves.  Others just recognize that "close" resistors demand "close" voltages, and "far apart" resistors demand disparate voltages.  

To me, this process is teaching good physics.  I've taught the calculations for series resistors for ages, and I've been repeatedly frustrated by students who can make calculations well but can't answer simple conceptual questions like "which takes the bigger voltage."  And as well by students who frustrate themselves because they predicted 8.25 V but only measured 8.22 V.

By the second day of filling out these sheets and measuring voltages, these freshmen are getting almost bored with the process.  That's the sign I'm looking for.  When I start to see faces saying "Gawd, not again with the voltage question..." that's when I know it's time to move on.  I give the faster guys a sheet with three, not two, resistors; then, after a multiple choice quiz, we move on to resistors in parallel.  

I'll teach parallel resistors the exact same way.

GCJ

Finding an appropriate digital multimeter

Extech MN36 digital multimeter

I'm introducing my general physics class to electric circuits with an experiment.  I generally find that the idea of voltage and current doesn't really sink in until students have had a chance to measure these quantities in circuits with which they can play.  (I also find that there is little correlation between my best problem solvers and my best circuit hooker-uppers.  Circuits experiments give an opportunity for success to some otherwise weak students.)  So this week, my class will do the simplest circuit experiment one could ever devise:  keeping a constant battery voltage, graph the current through a variable resistor as a function of the resistance. 

For years, though, I could not do this experiment effectively.  The ammeters -- really multimeters -- in my classroom wouldn't measure a wide range of currents, and would blow a fuse if they measured too much current.  You can certainly trust introductory students to blow an ammeter's fuse, no matter how much preventitive instruction and warning you give.

I really wanted a meter which would measure currents as small as a few microamps, and as large as a few milliamps.  Then we can use the standard available power supplies that provide 5-15V with tens of kilohms of resistance.

Why are these values important?  Most cheap resistors are rated at about 1/4 watt... so to keep the resistor from getting hot and ruined, the voltage squared divided by the resistance for any individual resistor must be less than 0.25 watt.  Even with as much as 15 V across as small as a 1 k resistor, the power is acceptable.  That calculates to a maximum current of 15 milliamps... and my preference isn't to come too close to this maximum. 

The meters I bought 10-15 years ago weren't this sensitive to current.  Their sensitivity was in the 1 mA range.  But last summer I searched google shopping under "digital multimeter."  I found a number of reasonably priced meters that would measure currents as small as 1 μA!  I've pictured one, the Extech MN36, which was listed at $17 on Amazon.  No price guarantees, and I've never used this particular meter -- I'm just showing an example of what I found.  Look around, there are gazillions of meters, many of which will serve your needs.

When you're looking for a class set of multimeters, DON'T look at science supply stores!  Their prices are inflated, and they don't necessarily give you a good selection.  Look around at electronics retailers.  Make sure to check the specs to see that the resolution for DC current is in the 1-10 microamp range.  If you have to, buy four this year, and four more next year, etc. 

Once you have the meter, all you need is a huge pile of 5-200 kilohm resistors -- which can be found from electronics stores in bulk for 1-2 cents apiece -- and some battery holders.  You can add more expensive power supplies, breadboards, or "resistance substitution boxes" later if you want.  But don't let cost or procurement be an obstacle to simple and fun electronics experiments.

Projectile lab with a marble: use a photogate!

Greetings from the AP reading in Fort Collins, Colorado.  I'd say 3/4 of my teaching ideas have germinated in  this enclave of friendly and professional physics teachers.  Today's thought comes courtesy of David Moore, who is part of the team grading this year's fluids experiment problem. 

A common  laboratory exercise asks students to predict the landing spot for a marble projected off of a table top.  Usually the marble is rolled down a ramp from the same height every time to ensure a consistent initial horizontal speed.

Measuring that horizontal speed is tricky.  Motion detectors don't pick up objects as small as marbles very well.  I suppose video analysis would work, but that's too intricate for a general physics class, I think.  In the past, I've had the class use stopwatches.  If the marble rolls across a flat tabletop, then the distance of the flat region divided by the time to travel that region gives the marble's horizontal speed.  However, my lab groups have made consistently incorrect predictions using this method.  Just a small reaction time issue can cause the marble to miss the target by 30% or worse. 

David says he uses a photogate placed near the end of the table!  Knowing the marble's diameter and the time during which the gate is interrupted, the marble's speed can be calculated.  Even better, use two photogates near each other:  the speed is the distance between beams divided by the time between the beams' activation times.  Reaction time or stopwatch clumsiness is not an issue when photogates are involved.

Mail Time: B field of a straight wire, and waves before magnetism?

It was good to hear from Fed Duay, a two-year veteran of my AP Summer Institute at Manhattan College*

* Which, for the uninitiated, is in the Bronx.  No, I don't get it, either.

Fed said:

Hello!

I have two questions. We are doing the "B field of a straight wire lab", where we can use a compass aligned to the earth's magnetic field and trigonometry to find B's value; then we graph "B vs. 1/r" and use the slope to find the "vacuum permeability" value [or should we find the current as compare to the ammeter reading?]. However I have come across two values for the earth's field: 2x10^-5 T and 5x10^-5 T (or 20 microT and 50 microT respectively). What do you use for the earth's field value?

I actually do the experiment the other way -- I use the ammeter reading and mu naught to find the magnetic field.  I like either of the two ways you suggested.  That's one of the beautiful aspects of the graphical approach to laboratory... Depending on what you measure or what you look up, a single experiment can be done in a wide variety of ways.

As for the value of Bearth:  You're only finding the HORIZONTAL COMPONENT of the earth's magnetic field.  Along the east coast, the magnetic field points more down than north, at a "dip angle" that can be close to 70 degrees off of horiontal. 

The site http://www.ngdc.noaa.gov/geomag/magfield.shtml will tell you the local magnetic field, including all components. I find at Woodberry Forest the northward magnetic field component is 2.0 x 10^-5 T.

(Fed continues with his second question:)

Also, I see that last year you covered part of waves before finishing magnetism; I am guessing that you needed to do the "standing wave lab on a string" before finishing EM. Is this the reason or is there something else I should be aware of?

Nothing other than personal preference is in play here.  The intricacies of electromagnetic waves aren't included on the AP physics B exam.  Certainly students are expected to know the EM spectrum, the visible wavelengths, which colors of light have higher frequencies, and so on; but the fact that electric and magnetic fields oscillate in accordance with Maxwell's equations is irrelevant at the physics B level.  Standing waves are in no way a prerequisite to magnetism.

I tried sticking in the wave section before magnetism because that breaks up the toughest parts of the AP course. Electricity and magnetism kick my students' butts, especially coming in the dead of winter when they're busy and in bad moods, anyway. I put waves in between, because waves have some cool demonstrations, are easily vizualizable, and are (comparatively) easy .

(You got a question you want answered in Mail Time?  Either post a comment, or email me at greg_jacobs@woodberry.org.  Those who include an astute and witty criticism of the Cincinnati Bengals or Reds impending disastrous seasons are most likely to see their questions answered.)

Introduction to circuits: laboratory exercise

I've already posted about how frustrating electrostatics can be, both to teach and to learn.  Why?  Because I've never found a way to do quantitative, or even qualitative, demonstrations.  Sure, you can hang a balloon from a wall or use an electroscope, but these aren't nearly as satisfying as, say, measuring the pressure at the bottom of a flask to be exactly the 103 kPa you predicted.

I teach circuits immediately after electrostatics.  It's pretty straightforward to do quantitative demonstrations with circuits -- all you need is a voltmeter.  (Or, if you want to be fancy, Vernier voltage and current probes.)  Thing is, I don't do more than a couple quick quantitative demos.  Instead, I run a laboratory exercise over the course of several days.

Especially in the wake of electrostatics, I want the class to get their hands dirty experimentally.  I want them to see, once again, that physics is not a math class, even though the vector addition from electric fields may have given that impression. 

So:  on day one of circuits, I give a crash course of definitions and memorization material.  I write on the board how to deal with series and parallel resistors, including the equivalent resistance formulas and definitions (current through series resistors is the same for each; voltage across parallel resistors is the same for each).  I give one brief example of how to use Ohm's law to deal with a set of two series resistors.

The next day I stop with the theory.  Instead, I show briefly how to use a breadboard to connect resistors to a power supply.  I bring out a large set of labeled resistors, all between 5k and 200k, and a set of voltmeters.  Each student gets a nine-page packet which they are to fill out.  I must sign off on each page before they proceed to the next.  Completing the packet will take about four class periods -- and it's worth every minute.

Here's the first page:

This page takes a while for most folks.  In fact, it might take the entire first class period.  The conversion to microamps should be trivial, but often is not; getting the correct circuit connected on the breadboard is not simple. 

The good news is, the class tends to help each other.  Once one person makes a connection, then that information spreads throughout the class quickly.  I want them to learn from each other as well as from me. 

Once I'm satisfied that a student has both predicted and measured voltage across each resistor, I initial the page and tell the student to move on to the next page.  What's on the next page, you ask?  It's the exact same set of questions... but with a different circuit diagram.  At the end of the post are pictures of the circuits on pages 2-9.

Students work at their own pace.  I ask everyone to get through the first seven circuits over four class periods.  The last two are essentially "extra credit" for the quick workers.  Those who finish early are given time to work on problem set problems, or to help the others in the class. 

Now, I'm pretty picky about the answers to the questions.  If a student says, "The voltage is bigger across the bigger resistor because V=IR," I don't accept that.  "Because V=IR"  is not a reasonable explanation for anything.  For the series resistors, students must indicate that current is the same for each, meaning that bigger R leads to bigger V in Ohm's law.  For the combination circuit, they must show me that they are simplifiying the circuit to a set of series resistors. 

Does this approach work?  Is it worth the four class periods?  I certainly think so.  (If I didn't, I wouldn't do it, right?  :-)  )  I find that my class has performed better on circuit problems on tests and quizzes since I started this exercise about five years ago.  More importantly, when someone does miss a question, we all have a common experience that I can refer to.  "Remember on the circuit lab?  Did the biggest resistor ALWAYS have the biggest voltage?  Remember when the resistors were in parallel?  Oh, yeah..."

GCJ

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