### Lenz's Law, Induction, and Levitation

+Bruce Elliott asked an excellent superconductor levitation question, so, no lab stuff, journal articles, or homework problems today, just superconductors.  A few days ago, I posted the following video of eddy current levitation to Google + with this explanation...

Staying with the eddy current theme, what if you played the game a different way?  The video below shows a coil of wire driven by wall current, (60 Hz here in the USA).  The alternating current creates a rapidly changing magnetic field.  The eddy currents in the aluminum plate, (aluminum like copper is not magnetic), oppose the original magnetic field created by the coil and cause it to levitate.  The coil gets very, very hot in the process because of all the energy from the wall current being turned into heat by the electrical resistance of the coil, (video 1).

http://youtu.be/5HnihTg1rso

What the post and video are demonstrating is a combination of Faraday's Law of Inductance, (a changing magnetic field will induce a changing electrical current in a conductor), and Lenz's Law, (the current produced in Faraday's Law will always run in a direction to create a new magnetic field that will oppose the change in the first magnetic field.)  Here's the second post I made with a video showing the levitating coil from the side...

Here's the same eddy current levitation demo seen from the side.  Notice that the coil won't stand still it drifts off to the edge.  This is an example of Earnshaw's theorem that states you can't get stable levitation if all you're using is a set of charges or magnetic fields that repel each other.
http://en.wikipedia.org/wiki/Earnshaw%27s_theorem
So, to review, Faraday's law says that a changing magnetic field will induce an electric current in a conductor, (like the aluminum plate shown here).  Lenz's law says the current will move in a direction that will create a second magnetic field that will oppose the change in the first.  That means the magnetic field created by the current in the aluminum repels the magnetic field created by the coil.  Earnshaw's theorem says the levitation won't be stable and off the coil goes.

http://youtu.be/9yMjqivs8Fk

Bruce pointed out that in the following video of the levitation of a superconductor, there are none of the stability issues I just mentioned above and asked why.  By the way, this is soooo not UFO levitation technology!  Everyone knows they use phase conjugated microwave beams :)

http://youtu.be/_zpsEO5t-TM

So, why's the levitating superconductor stable while the levitating coil wasn't?  The levitation mechanism, is exactly the same.  As the superconductor nears the magnetic field a current is setup inside it that exactly repels the magnetic field outside and voi la, levitation!  This is called the Meissner effect in superconductors and  it's also why you'll hear the phrase 'superconductors expel all magnetic fields from their interior."

So, again, why is the levitation stable?  As it turns out, the stability is also caused by eddy currents, they're just acting in a different direction, (sideways) than the eddy currents that cause levitation   Things become a little more clear if you repeat the expulsion phrase above more completely: "Type I superconductors expel all magnetic fields from their interior."  Type II superconductors actually allow some magnetic field to penetrate.  Picture 1 shows what's going on.

The type I superconductor on the left expels all magnetic field.  If you try to levitate a magnet above a type I superconductor  it will just slide off the edge like the coil over the aluminum plate[1].  Type II superconductors allow some magnetic field to penetrate at what are called pinning sites.  It's this pinned field that provides stable levitation.  It all goes back to Lenz's law again. Looking at picture 1, imagine one of the red magnetic field lines tries to move left or right across the superconductor.  The moving magnetic field line is a changing magnetic field.  That field induces a current inside the superconductor   According to Lenz's law the new current will setup a magnetic field that will oppose the change in the original magnetic field.  In other words, the newly induced field will push back on the magnet to keep it from sliding across the surface of the superconductor   You can see this stabilizing force at action in the video of a magnet suspended above a superconductor that NASA loaned us below.

http://youtu.be/D5O_vPpurkI

Paradoxically, (physicists love to say paradoxically), the same property that allows superconductors to expel magnetic fields can also  make them behave as magnets.  Watch the video below of a Type II superconductor that was cooled into its superconducting state with a magnet sitting directly on top of it.  Since it wasn't superconducting before it cooled, the field from the magnet could easily penetrate   When the superconductor entered its superconducting state  the field was stuck where it penetrated and the superconductor and magnet were stuck together.

http://youtu.be/-zhdomE6FXw

NASA is looking into using this propensity of magnets and superconductors to maintain the magnetic field lines between themselves once established as a way to assemble clusters of satellites in outer space[2], (picture 2).

So, to wrap it all up, eddy currents can be used both to levitate things using magnetic fields, and, in the case of superconductors, make that levitation stable.  Their are quantum mechanical reasons that superconductors don't resist electrical currents, but as for the levitation, it's just plain old eddy currents.

A Few Research Notes
Interestingly, a theory paper a few years ago purports to have shown that while flux pinning certainly is a way to get stability, it's not necessary[3].

References:
1.  On the instability of Type I superconductor levitation
http://dx.doi.org/10.1063%2F1.1721125
Šimon I. (1953). Forces Acting on Superconductors in Magnetic Fields, Journal of Applied Physics, 24 (1) 19. DOI:

2.  NASA on flux pinning and satellite control
http://www.spacecraftresearch.com/files/ShoerPeck_JAS2009.pdf

3.  Stability and pinning
http://dx.doi.org/10.1063%2F1.2794408
Perez-Diaz J.L. & Garcia-Prada J.C. (2007). Finite-size-induced stability of a permanent magnet levitating over a superconductor in the Meissner state, Applied Physics Letters, 91 (14) 142503. DOI:

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…

### The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1.  While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero.  This is typically explained in the following way.  While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare.  That doesn't stop us from looking for them though!

Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2].  Cabrera was utilizing a loop type of magnetic monopole detector.  Its operation is in concept very simpl…

### Unschooling Math Jams: Squaring Numbers in their own Base

Some of the most fun I have working on math with seven year-old No. 1 is discovering new things about math myself.  Last week, we discovered that square of any number in its own base is 100!  Pretty cool!  As usual we figured it out by talking rather than by writing things down, and as usual it was sheer happenstance that we figured it out at all.  Here’s how it went.

I've really been looking forward to working through multiplication ala binary numbers with seven year-old No. 1.  She kind of beat me to the punch though: in the last few weeks she's been learning her multiplication tables in base 10 on her own.  This became apparent when five year-old No. 2 decided he wanted to do some 'schoolwork' a few days back.

"I can sing that song... about the letters? all by myself now!"  2 meant the alphabet song.  His attitude towards academics is the ultimate in not retaining unnecessary facts, not even the name of the song :)

After 2 had worked his way through the so…