Lec 6

Judge of a man by his questions rather than by his answers.

Voltaire

Links for this lecture: https://www.youtube.com/watch?v=yVkdfJ9PkRQ

Lecture overview for today:

We will cover three topics and then you will complete a worksheet. The topics are: asking pertinent questions from a youtube video prompt, repeatability of signals from a photodetector, and adding harmonic functions. Answers to this worksheet are discussed after the worksheet is completed.

Lab overview for today:

In the lab you will also complete a worksheet on the following book and lab exercises. This is to be done as a group. You can use your laptop for this worksheet.

Lab Handout on “How to Lie with Statistics”: chapters 8-10 -Ask students to list two questions that should be asked when given data (who says so, how does he know, what's missing, did somebody change the subject, does it make sense?) -What major points were covered in each of chapters 8-10 of the "How to lie with statistics" book? -What do you think you might remember from these chapters a year from now? -Look at the spurious correlation link (just below the "How to lie with statistics" book link). Give one way in which the author adjusted the figures to better show a correlation? Ans: No indication of repeatability of results (error bars). Vertical Scales have been adjusted to make the graphs look similar.

Topic 1: Practice asking pertinent questions

Watch the pendulum link here https://www.youtube.com/watch?v=yVkdfJ9PkRQ

Before thinking of questions, think first about a model which describes what you see!

In class worksheet question:

write 3 questions about the video each of which must be in a different category. Those of you who think this is trivial ought to try and come up with questions in the analogy and creative categories.

There is no hierarchy in this questioning scheme. Any category can lead to important insights.

Photodetector:

In class worksheet question:

Sketch the photodetector output with 10 percent noise. On the same plot, sketch this output before and after a planet passed in front of a star if the planet generated a 10% drop in photon flux. How could you have the detector measure a 1% change in photon flux? Explain!

How are photons detected? Photoelectric effect or photodiode. Some electrons in a metal are free to move. If you hit a copper hammer on a table. What happens? Why don’t the electrons spew out the end of the hammer in a spark? Inertia moves the electrons to one end leaving positive charges at the other (this has been measured). It is hard to remove the electrons due to the left over positive charge. Yet in thermionic emission, used in the e/m lab, there are some electrons in the thermal distribution of speeds which have kinetic energy greater than the work function and can escape the metal surface.

This is a simple model of the work function in the photoelectric effect. One photon liberates one electron which is made to flow across a resistor in a phototube or photomultiplier.

What are those fluctuations in photon rate at a detector? Sketch a circuit of a reversed biased photodiode charging a cap (with C=Q/V or V=Q/C) or passing through a resistor across which the voltage is measured. Let the capacitor charge for 1 nanosecond before the output voltage is measured. It is then reset by switches which short the capacitor. This is repeated for another nanosecond.

What does the data look like? WHAT HAPPENS IF YOU REPEAT THE MEASUREMENT? IT YIELDS DIFFERENT VOLTAGES.

How do you display this data? -sketch a histogram of number of reading along the y-axis and voltage along the other. This yields a mean of 1.0 V with stnd dev of 0.1 V. -sketch a time series of voltage vs time which yields a series of dots which can be connected by a line to yield a display like on an oscilloscope.

Draw this histogram for the 1 nsec window assuming a Gaussian distribution. What is the minimum % change in intensity that could be measured? The percent noise is 0.01V/.1V = 10%. Draw a voltage vs time series rather than a histogram to show what this looks like. Show that a 10% signal would be buried in the noise.

Phasors:

In class worksheet question: What fundamental physical principle is illustrated in this graph of voltage vs time across the three circuit components?

How do you add harmonic wavefunctions A cos(kx-wt)+B cos(kx-wt+del). Use trig identities. Sketch a harmonic wave incident normally on a grating. Note that the hemispherical waves coming from each slit approximately from plane waves far from the grating. These waves have the same frequency and wavelength (why?) but are offset in phase.

What if you add 100,000 wave as with a grating? -use euler's theorem: take real part. Show this by adding the real part of two complex exponentials in the real-imaginary plane. -how do you add graphically two traveling waves? Show each with the sum as a phasor of the hypotenuse. -sketch an LR circuit. -sketch the graph of voltage vs time for the resistor, inductor and source.

Answers to worksheet questions:

Topic 1: Photodectector:=

The noise is 10% of the voltage due to the sun. We need to see a 1% noise to sunlight ratio to see the planet.

What could you do to see this small variation in the sun’s output? Charge the cap for 100 nsec or 100 times longer. The histogram of voltages is now 10 V mean and stn dev 100x 0.01/Sqrt[100] = 0.1 with a percent fluctuation of 0.1/10 = .01

( aside: When put across a resistor we get a DC voltage Io R = R e N/sec (where e is the charge on an electron in Coulombs) with fluctuations R e sqrt[N]/sec yielding a percent error in the DC voltage = 1/sqrt[N]. This is the percent absorption that can be measured.) Current in a circuit is (N electrons per sec) N q +- sqrt[N] q Percent fluctuation is sqrt[N] q/N q = 1/sqrt[N] The per second is IMPORTANT. Say you measured in 1/10000 of a second. You would have had 10000 less particles and therefore a higher percent fluctuation. Why would you measure in a shorter time interval? The signal may be your voice and it amplitude modulates at say from 200 to 2000 times a second. This requires a bandwidth of a kHz and therefore a shorter time given by delta nu delta t = 1 This is the same issue with the voltmeter set to measure 1 or 100 PLC's. During the time interval for 1 PLC there is more sqrt[N] noise than with 100 PLC's.

For a laser beam generating photoelectrons in a photodiode N can easily be 10^14/s. Changes in intensity can be as small as 10^-7 of the DC intensity.

Critical thinking discussion:

what are the assumptions: The sun does not have intensity fluctuations over the time you expect planets to move across its disc and it period. Only one planet crosses the sun at a time. The DC reference voltage for the amplifier and voltmeter do not vary in time at the period measured. No periodic scattered light between the star and earth enters the detector. logical fallacy: assume that the change in intensity is proportional to planet size. You really need to overlap circles for the planet and sun which may not lead to a linear change in intensity with planet size. Light scattered by the planet back to your detector as the planet moves to the dark side is not detected. different kinds of evidence: wobble of the center of mass of the sun due to the orbiting planet yields the same period.

This is a discussion not about how to quantify the errors in an expt due to noise but how to reduce the noise. Like our previous discussion averaging over a large number of samples reduces the noise.

What could you do to see this small variation in the sun’s output? Charge the cap for 0.1 sec or 100 times longer. The histogram of voltages now has a 10 V mean and stn dev 100x 0.01/Sqrt[100] = 0.1 with a percent fluctuation of 0.1/10 = .01

( aside: When put across a resistor we get a DC voltage Io R = R e N/sec (where e is the charge on an electron in Coulombs) with fluctuations R e sqrt[N]/sec yielding a percent error in the DC voltage = 1/sqrt[N]. This is the percent absorption that can be measured.) Current in a circuit is (N electrons per sec) N q +- sqrt[N] q Percent fluctuation is sqrt[N] q/N q = 1/sqrt[N] which decreases as N increases! The per second is IMPORTANT. Say you measured in 1/10000 of a second. You would have had 10000 less particles and therefore a higher percent fluctuation. Why would you want to measure in a shorter time interval? The signal may be your voice and it amplitude modulates at say from 200 to 2000 times a second. This requires a bandwidth of a kHz and therefore a shorter time given by delta nu delta t = 1 This is the same issue with the voltmeter set to measure 1 or 100 PLC's. During the time interval for 1 PLC there is more sqrt[N] noise than with 100 PLC's.

For a laser beam generating photoelectrons in a photodiode N can easily be 10^14/s. Changes in intensity can be as small as 10^-7 of the DC intensity.

Ask the class: What is the smallest percent absorption possible to measure in 1 sec?

Topic 2: Phasors:

-show how Kirchoff's law is satisfied for all times.

Critical thinking: There is a logical fallacy in a solution to Kirchhoff's law in a RLC circuit. It is why don’t you have a solution V=cos (kx - omega t)? How do you address this?

Find k = 2 Pi/lambda approx 0 since lambda nu = c and nu = 10^3 leaving lambda = 3 10^5. Note that if the circuit went from Denver to Tokyo and back you would need to have k in the solution.

Then do the phasor diagram for just an AC driven RC circuit to show better Kirchhoff’s law being satisfied. Note C=Q/V so V=Q/C. Vo cos(omega t) - R dQ/dt - Q/C = 0. Now assume an oscillating current solution dQ/dt = cos(omega t +phi). Then the voltage across the cap is sin(omega t + phi)/(omega C). The voltage across the cap is out of phase with that of the resistor. Note sin(theta+Pi/2) = cos(theta) so the voltage across the cap has a Pi/2 phase shift relative to that across the resistor (the voltage across the resistor leads that of the cap by 90 degrees but don’t want to go into more details since that’s not the main point)

Topic 3: Questioning:

-incongruous: How could the filmmakers get the masses equal sufficiently for this pattern to repeat?

-congruous: I know Newton’s laws govern this behavior but how do I calculate the period from them? I know from Newton’s laws that the period is proportional to the square root of the length/g but how does this cause the pattern shown?

-modify: what if there were more air friction? What if the lengths were random?

Causal/creative: could this be done with standing waves? Here it is done with EM waves and it could also be done with sound. https://en.wikipedia.org/wiki/Mode-locking

-emphasize the importance of informational questions: how do you know that? Why did you do that (repeat this question for each answer)? What are the assumptions and conditions for which the model is valid?

-emphasize the analogy question, “is this related to the orbits of the planets?” A peg on a rotating circle executes simple harmonic motion when viewed “end on.” If each peg were a planet rotating at a different rate, then the end-on view would look like masses attached to springs with different spring constants. Would these masses line up after some time?

-Focus on a model about which to ask questions or on how to determine a model. -What is the model? T= 2 Pi Sqrt[L/g] Simplify with 3 pendula, with periods T1, T2, and T3. They will repeat after m T1 = nT2 = p T3 where m, n, and p are integers. If T1=2, T2=3, T3=4 and the three pendula start at the same angle then after 24 seconds they all will be back in the same place. -emphasize the informational questions: how do you know that? Why did you do that (repeated)? What are the assumptions and conditions for which the model is valid? -causal creative looks for a new model. In this case you might ask how this looks if the oscillators were treated with quantum mechanics or with elastic strings .