### Lorentz Force Special Rel Paradox and Stored Energy

This week I've been reading about the apparent paradox [1]between the Lorentz force law, (the law that explains how an electric charge is affected by a magnetic field), and special relativity reported last year.  In the article describing the paradox [2], I came across the phrase 'hidden momentum'.  I hadn't heard this term in relation to electromagnetism before, (neither had some of my professors), so I embarked on a little research project to learn more about it.  As an aside, if anyone would like to comment with a high level overview of the 'hidden momentum' concept, it would be greatly appreciated, (Thanks!)

Stored Energy
The first thing we need to know is that electric and magnetic fields store energy.  Put very basically, electric charges would rather rearrange so that an object is electrically neutral, and magnetic fields would rather not change.  Changing either a distribution of charges, or a magnetic field requires energy and the energy used to make the change winds up being stored in the new charge distribution or magnetic field.  One of the most impressive examples of energy being stored in a electric charge distribution is the Van de Graaff generator.  Charge is moved via a moving belt onto a large metal sphere.  It's just a big version of the same process that takes place when you rub your feet on the carpet and then generate a spark when you touch a door knob.  The energy used to move the charge is stored in the electric field created by the charge.  When enough energy is stored, it can all be released quickly creating a spark:

The picture above is from the Van de Graaff generator at the Boston Science Museum.  If you ever get a change, go!  It's awesome, and my high school classmate works there, (yeah, I'm proud!)

As I said, energy can also be stored in a magnetic field.  An example of this can be found in the ignition coils used in cars[3].  Electric energy (relatively), slowly creates a magnetic field that stores the energy.  The original magnetic field is then forced to rapidly collapse.  This collapse releases the stored energy very quickly.  The resulting pulse of energy is used to create the spark that ignites the gas in the cars engine.

That's all for today!  More tomorrow.

References;
1.  Popular science press on the special relativity paradox and its resolution

http://prl.aps.org/abstract/PRL/v108/i19/e193901

3.  Induction Coils on Wikipedia
http://en.wikipedia.org/wiki/Induction_coil#How_it_works

Blogger said…

### 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…