Chapter 23 homework

Although my son has taken both the AP Physics C: E&M exam and the SAT2 Physics exam already, we haven’t finished the Matter and Interactions book yet, so we’ll keep going over the summer to finish off the last 3 chapters (23: Faraday’s Law, 24: Electromagnetic Radiation, 25: Waves and Particles).  He read the chapters before taking the exams but has not done any exercises or labs for them yet.

Chapter 23 includes inductors, so we’ll probably do some inductor labs, looking at current in response to a step change in voltage and perhaps making an LC oscillator.  I have some 220µH inductors, so we should be able to do current changes slow enough to track with the Arduino data logger and make an audio frequency oscillator.  We may try winding our own air-core inductors with different core diameters, and measure the inductance in several ways (say by fitting a time constant to the current change, with a frequency measurement of an LC oscillator, and by nulling a bridge circuit comparing to a known inductance).  We’ll probably also do some sort of eddy current demo (dropping a magnet through a copper tube, for example).

Homework exercises for Chapter 23:  23P29, 23P30, 23P32, 23P34, 23P35, 23P38, 23P40, 23P42, 23P43, 23P45, 23P46, 23P47, 23P51, 23P52.

 

Snell’s Law lab

We are way behind on physics—my son took the AP C: E&M test before we even got to Faraday’s Law.  He read through the last 3 chapters of Matter and Interactions in 3 days, rather than the 8 weeks we had originally planned, and he hasn’t done any of the exercises in those chapters yet.  Because he is planning to take the SAT 2 Physics test this Saturday, I decided that he should at least have a cursory familiarity with Snell’s Law.  Since there wasn’t time or energy for a problem set, we did a lab instead.

I had noticed when playing around with the violet (405nm) laser pointer, that the water in the fish tank fluoresced brightly.

Laser beam causing fluorescence of water in fish tank. The beam comes in from the right and is reflected off the water-air interface.
The blue is probably from bacterial cells in the water, and the red from chlorophyll from algae growing on the walls of the tank.  We get the blue fluorescence even from water right out of the tap, though not as brightly.

We made some very crude measurements of the angle of the beam coming into the water and of the beam in the water using a protractor.  (The beam coming in was invisible in the air, so measuring the incoming angle was very inaccurate.)

Despite the very inaccurate measurements, my son got a decent estimate of the index of refraction of the water. We don’t know the true index of refraction, since the water has a lot of “stuff’ in it.

Here is the gnuplot script he used for fitting the data (with some editing by me, which he did not entirely approve of):

set angle degrees

set xrange [0:90]
set yrange [0:60]

set title 'Index of Refraction in a Fishtank'
set key top left
set xlabel 'normal angle in (degrees)'
set ylabel 'normal angle out (degrees)'

refract(a_in, rfr_ind) = asin(sin(a_in)/rfr_ind)

water_rfr = 1.333
fishtank_rfr = 1 # initial guess

fit refract(x, fishtank_rfr) 'snell1.gnudat' using (90-$1):(90-$2) via fishtank_rfr

plot 'snell1.gnudat' using (90-$1):(90-$2) title 'Measured', \
    refract(x, fishtank_rfr) title sprintf('Fitted %f', fishtank_rfr), \
    refract(x, water_rfr) title sprintf('Pure Water %.3f', water_rfr)

I’m sure that with more careful measurement, we could get much less scatter around the theoretical curve, but we were tired at the end of the day and couldn’t be bothered to do the measurements right.

Chapter 22 homework

We finally finished off Chapter 21 of Matter and Interactions today, about 2 months behind my original schedule, having been repeatedly distracted.  We never did get around to measuring the magnetic field of a coil as a function of distance or current, either, though we’ll probably get back to trying that after the AP exams.

It looks like there is a chance my son will get to take the AP CS and AP Physics C: E&M exams this year, even though my first 5 attempts to find a place for him to take them failed. He needs to take the late exam for AP CS, since it conflicts with the Oregon Shakespeare Festival field trip and no one in the County offers Physics C—my attempts to get one of the high schools to offer the exam (which is at the same time as Physics B) all failed.  His consultant teacher is trying to arrange to be the proctor for him on the late AP CS and the late Physics C: E&M exams (it is now too late to register for the regular exams) through another high school in the same district.  I’m hopeful that she’ll be more successful in moving the bureaucracy than I was as an outsider.

Of course, he’ll probably never get any credit for taking the exams, since many of the schools he is applying to don’t do AP credit anyway, and he’ll have to retake physics at any of the schools he’s likely to choose.  But the exams will help validate that he has done rigorous work in physics, which should help him get into the colleges that would be a good fit for him. The AP CS exam is so low level that all it validates is that one has learned some Java syntax—but it might help with admissions offices also, as most will not be familiar with the new Art of Problem-Solving Java course.

In any case, we have to speed up a bit on the physics, despite the distractions, so here are the problems for Chapter 22 “Patterns of Fields in Space”: 22P15, 22p16, 22p18, 22P22, 22P23, 22P25, 22P29, 22P31, 22P33, 22P37.

Battery internal resistance lab and Chapter 21 homework

In our home-school physics class today, my son and I did two things: comparing answers on homework questions and a lab measuring the internal resistance of a battery.  I only had time to read Chapter 20 of  Matter and Interactions this morning, and only finished about half the problems.  I’ll have to finish them and compare answers with my son later. He’s already finished—he found these problems as easy as the ones in Chapter 19, but much more fun.  I suspect that Chapter 19 is one of those things that only a physicist can love—those of us who have more of an engineering mindset just find it tedious make-work.

The internal resistance of the battery was a simple experiment: we put known resistors across the battery pack and measured the resulting voltage.

The top circuit shows the setup we used—the bottom shows the equivalent circuit we were modeling.

I also measured the short-circuit current (briefly) with an ammeter.

The data was much noisier than I had expected, probably because the “switch” just pushed one battery away from contact, and the contact resistance between the battery holder and the batteries varied. We tried cleaning the contacts on the battery holder and on the batteries, but the noisy data were after that cleaning.  The noise is probably not due to self-heating of the resistors, as it was highest for the larger resistors.

The load line using the open-circuit voltage and short-circuit current look reasonable, but many of the measured voltages corresponded to a lower current, and fitting just the data points that exclude the short-circuit current does not produce a reasonable load line, though the estimated internal resistance is not too far off.

The battery-measurement lab took longer than I expected, because the data wasn’t as clean as I expected.  It might be worth trying again with a different battery holder, and a better switch, to see if the problems were just with the crummy contacts on the battery holder.  Variations of 0.5Ω in the resistance of the contacts would throw the measurements off by this much.

Chapter 21 of Matter and Interactions appears to be about magnetic force (though we don’t get to inductors until Chapter 23).  I find magnetism much more confusing than electricity, so I suspect we won’t be as quick with Chapters 21–23 as we were with Chapter 20. Problems for Chapter 21:  21P38, 21P39, 21P40, 21P44, 21P50, 21P60, 21P61, 21P66, 21P69, 21P71, 21P72, 21P79, 21P80, 21P90, 21P103, 21P105 (computational).

RC time constant lab

Since Chapter 20 of Matter and Interactions deals with RC time constants, and we’re behind on physics (the Spring Hill Science Fair judging took up most of my day, and the AMC-12 exam took up my son’s time before Java class), today we did a simple RC time constant lab, charging a capacitor to 5V, then discharging it through a resistor, recording the discharge with my son’s DataLogger program (the one he wrote for my circuits class).

The circuit we used for the discharge of the capacitor. Our first version used 4.7µF and 1MΩ, but we got some anomalous results and redid it with these values.

The data looked pretty good when plotted, but we had a bit of a surprise when we tried fitting exponential decays to the data:

The initial fit for the first 5 seconds after flipping the switch behaves pretty much as expected, but the discharge slows down, and has a much larger time constant after a while.

We are at a loss to explain this gradual slowing of the discharge.

I tried inserting a unity-gain buffer between the capacitor and the Arduino pin A0, to make sure that the Arduino was not disrupting the capacitor discharge.  This did not change the behavior, other than at the very low end, where we ran into the limits of the “rail-to-rail” output of the MCP6002 op amp chip, which is supposed to be able to get within 25mV of the power rails, but which seemed to be stopping at 100mV.  I think that this particular copy of the chip may be fried, though, as the other op amp on the chip was taking a huge current through the input pins. I replaced the MCP6002 op amp and got down to within 5mV of the rail—better than the spec.  The other chip was definitely fried, and I’ll discard it.  The stretching of the RC time constant as the capacitor discharges is still present, but to a much smaller extent:

The RC decay curve as seen after a unity-gain buffer made from an MCP6002 op amp. The fit for the first few seconds has the RC time constant we expect, but fitting later in the discharge gets a longer RC time constant, though not as badly as when directly connected to the Arduino.

I’m still looking for a convincing explanation for why the RC time constant is not constant in the discharge.  The power supply voltage was not changing, nor was there enough power anywhere for significant self-heating.  I’m having trouble believing in enough inductance to cause any measurable effect, especially as that would have had the biggest effect at the beginning of the discharge, not the end.

I suppose I could be seeing non-ideal conversion in the Arduino ADC (different Arduino boards were used for the two plots on this page: the first one was done by my son on a Leonardo board, the second one by me on an Arduino Uno board, so the difference in the fits may be from different  non-ideal conversions in the board, rather than from the unity-gain buffer.  Indeed, taking out the unity-gain buffer, but still using the Uno board gets very similar results:

With no unity-gain buffer, the results are not changed much, which means the buffer is not really needed (a good thing, since I don’t usually want to add an amplifier every time I use the DataLogger).

I suppose that the most likely explanation is that the gain and offset errors in the Arduino ADC may be throwing things off. They are only supposed to cause errors of 2LSB (about 10mV with a 5V reference voltage). I can get a better fit if I add a constant offset error of about 10mV, though that doesn’t quite explain all the error. It looks, though, like what I’m seeing is limitations in the Arduino ADC as a measuring device, and not interesting physics or electronics in the capacitor.

I wonder whether the circuits course should have a capacitor discharge lab early in the quarter next year, as a refresher of physics thinking about RC time constants, before we do impedance and low-pass filters.