### EM II Notes 2014_11_24: Leinard-Wiechert Potentials

There's sooo much going on today.  I'm back in the lab again, but I'm also studying for the last little bit of my EM II class.   Here are the EM notes for today.  Hopefully, I'll get a lab book up again in the morning.

Looking at the Leinard-Wiechert Potentials.

We'll have a particel mofin along hte path $\vec{r} = \vec{r_o}\left(t\right)$.  There is a quite lengthy explanation of IRFs, but I'll skip that for now and keep careful track of whether or not this comes back to bite me in the butt.  We define $\vec{R}\left(t^\prime\right) = \vec{r} - \vec{r_0}\left(t\right)$ which is the vector from the point charge at time $t^\prime$ to the observatin poitn $\left(\vec{r}, t\right)$.  This gives us a retarded time, $t^\prime$ determined by $t - t^\prime = R\left(t^\prime\right)$, where $R\left(t^\prime\right) = |\vec{R}\left(t^\prime\right)|$.  This makes far more sense if you translate one of the ever present ever invisible $1$s to a c to get $c\left(t - t^\prime\right) = R\left(t^\prime\right)$

The potentials in the IRf can be written as

$\phi = \dfrac{e}{R\left(t^\prime\right)}$, $\vec{A} = 0$.

A charge at rest will have 4-veclocity $U^\mu = \left(1, 0, 0, 0\right)$.

We can noew define the 4 potential to be $A^\mu = f U^\mu$.  We can also form a four vector version of $R^\mu$ as $R^\mu = \left(t - t^\prime, r - r_0\left(t^\prime\right)\right) = \left(t - t^\prime, \vec{R}\left(t^\prime\right)\right)$.  Looking at this, you should see a four space distance without the time axis turned negative.  In a sense, this fits because it isnt' squared yet.  In a sense it doesnt' because if it isn't squared, then the time componet shoudl have an $i$ out in front.  This is somewhat Wick rotated, to coin a somewhat fancy phrase.

In the special case where the charge truly isn't moving, then $f$ above shoudl be $e/R\left(t^\prime\right)$.  For the more general case where the charge is moving with four velocity $U^\mu$, we get

$f = \dfrac{e}{\left(-U^\nu R_\nu\right)}$, so $A^\mu = -\dfrac{eU^\mu}{U^\nu R^\nu}$

Here, we have rather mysteriously gotten our negative sign back in front of the time coordinate.  Ask the professor about this tomorrow.  The pertinent point is near equation 7.29.

Now, on to problem number 2
For 2.a., and b, see the Wake Forest notes:
\url{http://users.wfu.edu/natalie/s13phy712/lecturenote/lecture27/lecture27latexslides.pdf}

expression 17 and up give the appropriate curl.  If time allows take a look at Dr. Nevels article on graphically protraying E and B fields.  There's no reason everything shouldn't apply here since the L\&W potentials were derived classically.

The strategy is just to bludgeon through the curl of $\vec{A}$ equation and get the final result.  Then, bludgeon through the cross product of $\vec{E}$ and show that the results are equivalient.

### 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…