I’m on sabbatical working on an text (Introductory Computational Physics using Python) and, to take a break from writing, I am also writing interesting questions to better prepare our physics students for the physics GRE, and well, because it’s fun to think about physics.

So, here’s a question I wrote (thinking that the answer wouldn’t be too hard to derive—ha!) that turns out to be more interesting than I thought. First, the question as stated, with the geometry shown in the figure at the right.


Geometry for the rotating ring. The ring rotates about a the z-axis and the magnetic field is into the page along the -x direction.

Geometry for the rotating ring. The ring rotates about a the z-axis
and the magnetic field is into the page along the -x direction.

Suppose that a thin copper wire of diameter a is bent into a circular ring of radius r >> a. The ring is rotating without friction about a diameter and that this diameter is perpendicular to a magnetic field

    \[\vec{B} = -B_0 \hat{x} = - 0.01\, \hat{x}\;\mathrm{Tesla}\]

If the intial angular velocity of the copper ring is \omega_0, calculate the time it will take for it to drop to \frac{1}{e} of this initial value, using the assumption that the dissipated energy goes into Joule heat. You may use the fact that the resistivity of copper is 18\times 10^{-9}\,\Omega\cdotm and its density is 8900 kg/m^3.

How does your answer depend on the radius of the ring, r and it’s cylindrical wire radius a?


This problem has a lot of depth and subtlety. Sit down with a pen and paper and derive the initially non-linear differential equation for the angular velocity (by considering kinetic energy dissipation), make an appropriate approximation and presto, your equation is simple to solve.

The above hint is the only analytical way I can see to solve this problem…approximately, anyway. And, if you’re paying attention, you might wonder why I asked you to consider kinetic energy dissipation and not total energy dissipation.  Why not include the magnetic dipole potential energy term -\vec{\mu}\cdot\vec{B} ?  I think that the answer to this is going to be non-trivial and might depend on fully understanding the January 1999 paper in the American Journal of Physics entitled: Magnetic dipole orientation energetics by G. H. Goedecke and Roy C. Wood.

In any case, one can work out the “exact” dynamics by equating the magnetic torque on the induced dipole moment to the rate of change of angular momentum and avoid energy considerations entirely, obtain a non-linear differential equation and find the resulting dynamics. I turned to Python and SciPy to write a quick snippet to solve this in the iPython Notebook. More when I post a solution.

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Here’s a derivation of the minimum detectable magnetic field fluctuation we will be able to detect as a function of our measured angular deflection. There are some limitations to this derivation:

  1. I’ll assume that we’ve perfectly aligned the torsional zero with the external magnetic field first; i.e. at t=0, the magnetic dipole moment is precisely aligned with the torsional zero.
  2. I’ll assume that the magnetic field fluctuation is completely perpendicular to the residual initial magnetic field B_0.

Within the limits of these two assumptions, this derivation is exact. So, here we go—the figure below shows the geometry of our situation.

Initially, the magnetic dipole moment \vec{\mu} is aligned with the torsional zero, and a residual magnetic field B_0 exists; then a perpendicular field component \Delta B is applied. The dipole moment experiences a magnetic torque \vec{\mu}\times B which tries to align with the net field B, but this torque is thwarted in its effort by the restoring torque from the fiber -\kappa \delta. Hence, the equilibrium position is defined by

    \[\mu B \sin\theta = \kappa \delta\]

but, since \Delta B is perpendicular to B_0, we see that

    \[\sin\phi = \frac{\Delta B}{B},\]

and by inspection, \theta = \phi - \delta, so that we have

    \[\mu B \sin(\phi - \delta) = \kappa \delta,\]

and therefore

    \[\mu B \sin(\arcsin(\frac{\Delta B}{B}) - \delta ) = \kappa \delta.\]

By standard angle addition identities for the sine, we then have

    \[\mu B (\frac{\Delta B}{B} \cos\delta - \frac{B_0}{B} \sin\delta  ) = \kappa \delta\]

simplifying and solving for \Delta B, we have

    \[\Delta B = \frac{\frac{\kappa}{\mu} \delta + B_0 \sin\delta}{\cos\delta}\]

Clearly, for a given measured shift in orientation \delta, we obtain the smallest \Delta B when the residual field B_0 is as small as possible.

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Data about our Helmholtz Coil pair:

  1. Diameter = 1.016 \pm 0.008 m
  2. R_{\mathrm{coils}} = 0.939 \pm 0.001\;\Omega
  3. R_{\mathrm{coils\, in\, parallel\, with \,leads}} = 0.516 \pm 0.001\;\Omega
  4. R_{\mathrm{coils\, in\, series\, with\, leads}} = 2.067 \pm 0.001\;\Omega
  5. diameter of coil wire = 0.075 \pm 0.001 in = 1.905\pm 0.025 mm
  6. Predicted number of turns = 49.9\pm 1.4 turns

These (home made) coils which I’ve inherited are clearly  marked as having 43 turns, so I’ve got quite a discrepancy here. I’m inclined to believe my measurements, but will have to check by putting a known current and measuring the magnetic field. A good project for a student measurement.

I’m adding a sensing resistor (1.420 \pm 0.001\; \Omega ) to the above in order to measure the current using the LabJack A/D. This resistance value includes the lead wires to the LabJack.

Addendum: (12 April 2012): I think that I have overestimated the resistance; will have to re-do measurements! Think this is not supposed to happen in a laboratory notebook? Unfortunately, it happens all the time, but hopefully we catch our mistakes.


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Here’s a 12 hour data run showing the noise pickup on a 1.0 m diameter helmholtz coil in my lab.

I wanted to see how much noise exists in the my laboratory. Looks to my eye to be  about \Delta V \approx 0.5  mV peak-peak fluctuation. No obvious temperature correlation (at least crudely, but I’ll check it out for real). Next step will be to measure the fluctuations while a “constant-current” power supply is driving the coils.

Of course, I suspect that the noise has a large 60 Hz content, but I could only get the LabJack interface to sample at about 20 Hz. I was using the LabJack python interface API, so it may be possible to achieve a higher rate if I look into lower level calls. Fortunately, this high frequency noise will not affect our pendulum, as it’s period is significantly lower (on the order of 0.6 Hz).

The plot was made with qtiPlot, as Matplotlib could not plot the data array, which consisted of about 1 million data points. The error that python returned to me indicated that the error occurred in the Agg backend, but I haven’t tried other backends, and haven’t checked if this is a numpy issue.

Voltage Fluctuations through Helmholtz coil when connected to LabJack input.

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Oxygen, phytoplankton, and Physics

On 2012-Mar-21, in Biology, by paul

Recenlty, I listened to a fascinating RadioLab podcast about the battle that rages between phytoplankton in the ocean (called coccolithophores) and the viruses that like to use the plankton as hosts to reproduce. Here’s a picture from the RadioLab site:

An electron microscope picture of coccolithophores (from the RadioLab site

That’s a pretty amazing exterior—it’s designed to protect against viral invasion; but according to the scientist William Wilson, it’s not entirely successful. In any case, the podcast (which you should now listen too) goes on to detail the battle between the coccolithophores and the viruses and the many contortions that occur during the battle. The IMPRESSION I was left with was that there is a constant stream of evolutionary adaptations that occurr (read: mutations) that make the battle continue. The interesting things to me are:
a) a full 50% of the oxygen we breathe come from the phytoplankton that live in the ocean,
b) the battle between the coccolithophores (and presumably other plankton) creates a varying (sinusoidally? with what period? random? or not?) output of Oxygen into the atmosphere.

I’m really interested in (b). What does the time/space dependence of this battle (i.e the phytoplankton blooms/die-off) look like? What are the governing parameters that effect this battle? Ocean temp? seasonal solar gain? presence of other types of plankton? Is there anything to be gained by trying to simulate this? How would one model this realistically?

William Wilson — if you’re ever passing through Portland, I’d love to hear more about these critters over a pint of beer.

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In praise of Steve Innes

On 2011-Oct-19, in Death, physics, Progress Report, by paul

Yesterday, my machinist (okay, I like to think of him as MY machinist!) Steve Innes was going to come over to my laboratory to talk about the last design work for our experiment, and pick up the stock needed to finish construction. I was about to sketch the last
drawings he needed to finish, and I received an email announcing that Steve died this past weekend. Suddenly. Unexpectedly. Gone.

I’m still not really adjusted to this reality of his death, and I’m pretty upset. I really liked Steve–he loved what he did, and I really admired
his grasp of mechanical design and his ability to translate this ability into tangible functional and BEAUTIFULLY machined equipment.
His work was absolutely top notch. Invariably when I set out to design some apparatus, I would run into either a design logjam—either I wouldn’t know how to get where I wanted to go, or there would be too many ways to get there and I wouldn’t know what the best method was. I’d call Steve, he’d come over, and we’d have a design discussion at the whiteboard. Problem solved. Steve had a design intuition that was invaluable; every experiment I did had his hand in the design, and it’s no exaggeration to say that I would NEVER have received tenure at the University of Southern Maine without his help.

I owe my career to him, and I’ll never be able to thank him, and that’s very upsetting, and sad, sad, sad.

So, to Steve’s wife, and children: please know that Steve will be much missed by me, everyone in the Physics Department, the Planetarium, and countless others at USM—and that he has contributed enormously to everything we do. I’m very, very sorry for your sudden and shocking loss.

And, to all experimental physicists out there—thank your machinists for the enormous contribution they make to your research. Do it now.

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A mock-up of our pendulum hanging in our enclosure. The magnets are gold plated Neodymium magnets (1" diameter), and they sandwich the fiber between to identical aluminum disks. The final pendulum will have 4 magnets and the outer magnets will have 50mm diameter front surfaced mirrors super-glued in place; the aluminum disks are designed so that the pendulum will have equal moments of inertia about each principle axis.

Our machinist (Steve Innes) has finished the main enclosure. Now what’s left is for me to give him plans for the optical system. I’ll be designing two different optical systems. The first will mount on the rotating plate visible in the photograph and will be used for calibrating the pendulum (finding out the torsion constant and the moment of inertia — simultaneously; but that’s a story for a future post). This internal system will be removed after the calibration is complete, and replaced with an optical system on the exterior plate (removed so that you can see the interior).
Now, I’m awaiting a new quadrant photodiode, as I fried the old one by back-biasing the diode accidentally. If there’s good news in this, it is that I found a source that sells the same detector for a cheaper price — Pacific Silicon Sensor.

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Aluminum oxide layer

On 2011-Jul-16, in experimental, Materials, physics, by paul

According to this source: http://www.finishesltd.com/anodizing.html , the naturally occurring oxide layer on aluminum is about 500 nm thick. Much more than I would have guessed.

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Our ace machinist, Steve Innes, sent me this picture of the main apparatus enclosure in the center of which will hang our torsion pendulum. Steve has yet to mount the 1 meter long “chimney” onto the cube, and has to make the interior optical lever platform which will allow us to calibrate and monitor the fluctuations of our pendulum.

Here’s a question. Since the pendulum will be surrounded by aluminum (save a small opening at the bottom through which the wiring will emerge, one would expect pretty good shielding. However, aluminum oxidizes readily, and I guess then we have a layer (how thick? 1 atom? 10’s of atoms? more?) of aluminum oxide with dielectric constant of between 9-12.
(I gave a range of values, because the proper way to express the dielectric constant with a crystalline structure is with a susceptibility tensor, and aluminum oxide has diagonal elements which are not all equal.)

Now, the question: Should I be concerned about the aluminum oxide layer? Should I paint the interior of the cube with something conductive—like graphite in alcohol?

I’ll have to think about this. I don’t know the answer, but if the pendulum itself was a perfect conductor (which is likely a good approximation) and was electrically connected to the body of the aluminum enclosure, I think I know the answer, or at least how to get there.

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Many scientists use or have used LabView (by National Instruments) for data acquisition and experimental control. I used to too, but the cost of the hardware and the software is rather formidable (unless you have a large research budget), so I am attempting to run our torsion pendulum magnetometer using a python-capable A/D interface by a company called LabJack. $350 buys one a pretty capable interface that is easily controlled by Python. I have the U6Pro, which supposedly allows one to capture roughly 20 bit resolution (since the voltage range is \pm 10 volts, this gives a theoretical resolution of about 19 \mu V, which is pretty good.

Here’s a simple python script (see bottom of post) to:
a) read data from analog input channel 0 for a user specified time interval,

b) save the output time and voltage readings,

c) calculate the fluctuations in the time and voltage readings

(i.e. \delta t = t_n - t_{n-1}, and similarly for the voltage),

d) write out the time, voltage, and fluctuation data to 4 ascii files, and

e) make plots (via Matplotlib) of Voltage vs Time, and histograms

of the time and voltage fluctuations.

This is half baked code, and I didn’t even bother to label axes :-(.