Wednesday, October 29, 2014

Here's today's special relativistic EM question.  Can the Thomas precession be shown to be a special case of the perihelion advance of relativistic elliptical orbits?  Any ideas?  Here's what's going on:

We've been deriving the special relativistic  orbit of a charged particles around another fixed charged particle.  At the end of the day, you wind up with a perihelion advance which is a fancy way to say that major axis of the elliptical orbit won't stay put.  It swivels around, (orbits), the charged particle as well.  The advance angle of the major axis winds up being\\

$\delta\phi = 2\pi\left[\left(1 - \dfrac{\kappa^2}{l^2}\right)^{-1/2} - 1\right]$

Which is very, very, similar to the Thomas angle for the spin precession, or gyroscopic precession along a circular orbit at special relativistic speeds:\\

$\delta\phi = 2\pi\left[cosh\left(w\right) - 1\right]$

$= 2\pi\left[\left(1 - \dfrac{v^2}{c^2}\right)^{-1/2} - 1\right]$

In the expression for the perihelion advance, $\kappa = eQ$ where $e$ is the charge of the orbiting particle and $Q$ is the charge of the fixed particle.  $l$ is the angular momentum and is written as $l = mr^2\left(\dfrac{d\phi}{d\tau}\right)$

Just a little more massaging of the above before I'm done for the day.  For a circular orbit, $\dfrac{d\phi}{d\tau} = \omega$ is constant and we can write $l = m\omega r^2$.  If we plug this in as to the Perihelion advance equation we wind up with

$\dfrac{\kappa^2}{l^2} = \dfrac{Q^2 e^2}{m^2v^2r^2}$ where $v = \omega r$

Here's where I get into trouble for playing fast and loose with things that migh actually be rapidities instead of velocities like $v$ above.  However, if you carry the simplificatins a bit further out, I believe you wind up with something that looks like an units of potential energy over momentum squared.

$\dfrac{Q^2 e^2}{m^2v^2r^2} = \dfrac{F}{m} \dfrac{1}{mv^2} \sim \dfrac{a}{E}$


Notes du jour:

Tuesday, October 28, 2014

EM Notes Part I: The visual bit of relativistic EM fields pointing at the observer

This is kind of cool from yesterday's EM notes.  Our professor pointed out that if you calculate the field from a relativistically moving electric charge, you'll always find that it's pointed straight at the point of observation.  Anyone have any idea why?  The argument could certainly be made that if you measure the field from a static charge that it will also be pointing straight at you.  Then, there's also the realization that the Lorentz transformation only affects the E and B fields in a frame that are perpendicular to the frame's tangential velocity.  I'm not sure that's either here or there since the point of observation can be anywhere.  Here's the associated diagram for the curious.


Friday, October 24, 2014

Day o' Videos: Presentation and Flying Superconductors

The lab book today was a bit sparse and a bit dry.  This is a bit odd considering I got to play lab yesterday...  You'll see.

First, here's an archival video of the presentation I did last Sunday for the TX APS section meeting here in College Station.  I fumble a few times, but the content is all there.  If you have any questions, they are very, very welcome!

The second video has some kind of cool stuff in it.  Not stuff that went the way I had hoped mind you, but cool nonetheless.  Here's the deal; we'd hoped to make a spiffy little superconductor visibly quenching video.  The idea was to suspend a superconductor as a pendulum in a magnetic field.  It was hoped that as the field increased, the superconductor would swing away from the pole of the magnet, (it did), and that as the field increased more, the superconductor would quench and fall from it's suspended state, (it didn't).  Our melt-texture growth superconductor from CAN just won't quench in 12.8 kG fields.  We already knew that.  Our next hope was to go with a low quality superconductor which should have quenched at a lower field strength.  I didn't consider the fact that quality also affects the force available for superconductor levitaiton, and so the lower quality superconductor did absolutely nothing.

In any event, it was educational and fun to watch the high quality superconductor as the field was ramped up.  Here's what you'll see.  As the field between the magnet poles is increased, the cylindrical superconductor which must have frozen in a bit of residual field when it was cooled will first orient it's magnetic moment perpendicular to the field to minimize the torque it feels.  As the field strength is increased, the superconductor will then begin to move itself out of the magnet altogether eventually swinging off the screen.  You'll also catch the odd snippet of extraneous lab conversations.




Picture of the Day:
New Mexico Mountains



Thursday, October 23, 2014

Pickup Coils, Faraday's Law and Back in the Lab! Lab Book 2014_10_23

 As always, look to the bottom of the post for background on what's going on.

Finally, enough of theory and presentations!  I got back to the lab today!  Here’s the apparatus I built/used.

NOTE:   As always, look to the bottom of the post for background on what's going on.




No, the oscilloscope is not sticking its tongue out, that’s a floppy disc.  Remember those?  

The small solenoid is what’s deemed a pickup coil. 


It’s the first prototype, of the coil that will be used to measure the actual currents and magnetic fields produced by the can crusher magnet.  It’s exactly what it looks like, six complete turns made from a jumper wire.  The Styrofoam cup is to avoid abusing the small magnet block too much when it’s dropped.  

The ‘scope pictured can capture a single waveform.  Here, it’s slowed way down to make a sweep over the course of several seconds.  It’s being used to look at the signal from the coil as the magnet is dropped through it.  Each set of spikes you see is created by dropping the boxy looking magnet through the solenoid.  Faraday’s law does the rest.


This is a very basic test run in preparation for measuring the changing magnetic field that will be generated by the pulsed magnetic field that’s to be used in the experiment.  Here’s a trace of a single magnet drop



Coil diameter  13.5 1/16ths
Number of turns 6
3178



Background
Hirsch's theory of hole superconductivity proposes a new BCS-compatible model of Cooper pair formation when superconducting materials phase transition from their normal to their superconducting state[1].  One of the experimentally verifiable predictions of his theory is that when a superconductor rapidly transitions, (quenches), back to its normal state, it will emit x-rays, (colloquially referred to here as H-rays because it's Hirsch's theory).

A superconductor can be rapidly transitioned back to its normal state by placing it in a strong magnetic field.  My experiment will look for H-rays emitted by both a Pb and a YBCO superconductor when it is quenched by a strong magnetic field.
This series of articles chronicles both the experimental lab work and the theory work that’s going into completing the experiment.

The lab book entries in this series detail the preparation and execution of this experiment… mostly.  I also have a few theory projects involving special relativity and quantum field theory.  Occasionally, they appear in these pages.

Call for Input
If you have any ideas, questions, or comments, they're very welcome!

References
1.  Hirsch, J. E., “Pair production and ionizing radiation from superconductors”, http://arxiv.org/abs/cond-mat/0508529

Tuesday, September 30, 2014

Finding Quenching Field Magnitude Using Levitation Force: Lab Book 2014_09_29


Summary:  Working more on using the superconductor to detect its own quenching field.  The initial setup is shown below.  The quenching test is described in the following.  A YBCO superconductor is placed between the poles of a very uniform magnet and then cooled into its superconducting state.  The field frozen into the sample at the state transition opposes the fringing fields on the magnet.  However, had the magnetic field been strong enough to quench the superconductor, the results would have been the pendulum swinging freely beyond the pole pieces' diameter until it encountered a field less than its critical field at which point, it would have re-entered the superconducting state and frozen in those field lines, suspending itself.  There's another realization of this process that will be tested today.  The pendulum is again suspended in a uniform field and the field is slowly increased.  It is suspected the sample will be deflected until the quenching field is reached at which point, the pendulum will fall back to its equilibrium vertical position.  By measuring the angle of the pendulum, the levitation field could also be determined.




The superconductor is placed in a Styrofoam cup supported on a wood plank wedged between the two poles of the magnet.   The magnet gap was set at 2 and 9/16 inches.  This could be much smaller for the sample used here, I just need to find a smaller reservoir.

There are two movies.  The first contains the cooler alarm going off.  After the alarm went off, the magnet current supply was slowly ramped down, and water was added to the reservoir after the cooler was switched off.  The cooler alarm did not start again after it was turned back on, nor when the magnet supply was ramped up to 49 amps. 

The second movie detailed the superconductor not moving while the reservoir slipped out from underneath it. 
We’re measuring the magnetic field with a F. W Bell 5180 Hall Effect Gauss meter.
7.32 – 7.35 kG at a 2 and 9/16 inch gap.

12.8 kG at the gap setting, 1 and 1/8 inch gap setting.
A small Dewar was carved from blue Styrofoam to fit in the smaller gap space, see the first picture below.  The Dewar was suspended as a pendulum between the poles of the magnet as shown in the second picture below.  Dental floss was used to support the Dewar pendulum from the upper yoke of the electromagnet.

The quenching test was as follows:
A YBCO superconductor is placed between the poles of a very uniform magnet and then cooled into its superconducting state.  The field frozen into the sample at the state transition opposes the fringing fields on the magnet.  However, had the magnetic field been strong enough to quench the superconductor, the results would have been the pendulum swinging freely beyond the pole pieces' diameter until it encountered a field less than its critical field at which point, it would have re-entered the superconducting state and frozen in those field lines, suspending itself.  There's another realization of this process that will be tested today.  The pendulum is again suspended in a uniform field and the field is slowly increased.  It is suspected the sample will be deflected until the quenching field is reached at which point, the pendulum will fall back to its equilibrium vertical position.  By measuring the angle of the pendulum, the levitation field could also be determined.




The fringe field produced at the edge of the electromagnet pole, mentioned above, is shown in the diagram below from Lawrence's cyclotron patent application.  Note that the 'magnetic lines of force' become less uniform as the edge of the pole piece is approached.



The small YBCO sample did not quench at this gap and field setting.
Note in the video that at 35 amps during the ramp down, the sample seems to be drawn to the pole piece .  At 10 amps, the lower right corner of the Dewar relaxes.
Link to quenching test video