Wednesday, May 14, 2014

Lab Book 2014_05_14 YBCO levitation, Glass Dewar Mechanics, and Relativistic Projectile Trajectories

Lab Book 2014_05_14     Hamilton Carter

SummaryTested the larger of the two YBCO superconductors by trying out three different levitation demonstrations.  Ran into a hitch when preparing to vacuum test the glass Dewar.  The glass stopcock valve to the vacuum jacket is stuck shut!  Unbeknownst to me, you can replace glass valves!  Did a bit of theory work looking at hyperbolic trajectories of relativistic particles and how they compare to the trajectories of projectiles with air resistance.

Lab Work:  YBCO Rough Characterization
The YBCO superconductor was levitation tested.  Although it’s not a quantitative measure, it seems to be levitating magnets as well as it ever did.  I ran a series of three demonstrations showing the different types of field cooling/levitation.  In the first experiment, the superconductor was cooled without the magnet present.  Since the sample was very well constructed using a crystal melt method, there aren’t a whole lot of imperfections for magnetic flux to pentetrate into.  The magnet levitates as it would over a type I superconductor that rejects all flux.  In other words, it won’t levitate stably and falls off the opposing field created by the superconductor to one side of the cup or the other.

In the second demo, the superconductor was cooled with the magnet already in place, suspended by a thread a small distance above the sample.  The magnetic flux lines penetrate the superconductor before it enters its superconducting state and are trapped there.  When the superconductor has been cooled and the suspension string loosened, the  magnet levitates.  The superconductor also remembers where the flux lines were located after the magnet is removed.  When the magnet is dropped back down onto the superconductor, its landing is cushioned and it is directed back to its previous levitated location.

In the third case, the magnet is placed directly on the superconductor.  Once again, the lines of magnetic field are frozen in.  This time however, do to the proximity of the magnet to the superconductor, the superconductor won’t let the magnet be pulled apart from it.  Effectively, the superconductor is behaving like a second magnet.

Lab Work: Leak Detector Hookup and Jammed Dewar Stopcock
The leak detector was hooked up to the rubber hose that will evacuate the vacuum jacket on the glass Dewar.

We immediately ran into an issue, however.  The glass stopcock that seals the vacuum jacket was stuck shut.

At the advice of the glass shop we tried to heat the stopcock with a heat gun to loosen it.  The effort was unsuccessful.  I didn’t realize it, but glass stopcocks can actually be replaced.  The glass blower removed our stopcock and will replace it on Friday.

Theory Work: Relativistic Projectiles and their Classically Dragged Counterparts
In addition to having everything you ever wanted to know about classical trajectories of projectiles under the influence of a uniform gravitational field, McAllen includes a section on projectiles with air resistance.  In particular he includes the following figure and formula. 

The right hand side of equation 2 is the innards of an anti-Gudermannian and only needs to be wrapped in a logarithm to be the anti-Gudermannian itself.  Interestingly, the left hand side of the equation is the logarithmic derivative of the particles velocity with respect to its angle with the ground as it moves along its trajectory. The diagram is very reminiscent of the range of a relativistic projectile vs. the angle of launch.  Here’s a graph of the relativistic range as defined by MacColl:

You might notice that this formula also contains an anti-Gudermannian, however in this guise it’s written as the arctanh of a sine function.  Notice that as the factor u in the formula above goes to infinity, it will simply cancel out of the expression. 

Here’s the interesting part of the problem.   The u factor determines how close the projectile is to the speed of light and in fact does approach infinity as the projectile approaches the speed of light.  When the particle hits of the speed of light two things happen.  First, the factor of u cancels out and the range formula is a cosine function times an anti-Gudermannian.  Second, the the optimal launch angle for maximum range is equal to 56.46 degrees.  This is the same angle that will maximize the path length of the trajectory in the classical problem.

We’ve known for a while that the special relativistic effects are all tangential in this problem.  It’s very interesting, yet not unexpected, that the air resistance, (which is also a tangential effect), problem has a  few similarities.

Theory Work: Rotating Frames and Airplanes
Looking into Frenet Serret equations and Fermi-Walker transport.  Leinaas and Korsbaken are using a FS frame and not necessarily a FW frame.  This is mentioned at one point.  Letaw is more careful about that.  Svaiter and Letaw are more concerned about detectors that rotate, (spin).

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