Math Notes: Cylindrical r Coordinate Value to Trace a Spherical r Dependent on z

OK, so that was quite the title.  I haven't done one of these in a while, but classes are about to start again and i figured I may as well get started deriving things again.  Plus, I had to do it for the can crusher magnet simulation code[1] for the experiment[3].  Here's what's really going on.  I have a Sage function that will give me the magnetic field in the z direction produced by a coil of wire that sits at z = 0 and a has a radius of 'rcoil'.  I'd like to know the magnetic field produced by the loop of wire along a circular path that is perpendicular to the plane of the current carrying coil.  A circular path perpendicular to the plane of a coil kind of begs for spherical coordinates, but the routine I have takes a z coordinate and a radius coordinate in the cylindrical coordinate system.  In the picture above, the circular path is shown, and the coil of wire is at the diameter of the circle and perpendicular to the page.  Note:  For those reading on G+, the equations aren't going to come out correctly, what with the lack of Latex support and all... apologies, but you can get the full version at, and garner the vast majority of it from the figure below.

At each height z above the coil the magnetic field will be the same all the way around the coil for a given spherical radius.  We can exploit this radial symmetry and only calculate the value of the magnetic field once instead of at each point around the circle.  Then, we can simply make a plot of the magnetic field vs z with the understanding that the radius of the sphere is constant and specified.  Here's the transform to get the value of the magnetic field dependent on our position on the z axis, given the radisu of the sphere we're interested in and the z coordinate, (of course... don't you just love saying that?).  The only trick is to figure out what cylindrical value of radius to plug in for each value of z to keep all our points on the spherical circle.
For any point on the sphere, notice that the cylindrical r value can be written as

$r = r_{coil}*cos\theta$,

where $r_{coil}$ is the constant radius of the coil which is also the constant radius of the sphere that we're interested in.  Next, we note that at each point along the spherical path

$\theta = asin\left(\dfrac{z}{r_coil}\right)$

Plugging this into the first expression, we get

$r = r_coil cos\left(asin\left(\dfrac{z}{r_{coil}}\right)\right)$

There's one last little modification.  Arcsin's of other functions always creep me out just a little, so fortunately, there's a trig identity we can use to get rid of this one.  The identity[2] is

$cos\left(arcsin\left(x\right)\right) = \sqrt{1-x^2}$

By plugging this in, we get the final expression

$r = r_{coil} \sqrt{1-\left(\dfrac{z}{r_{coil}}\right)^2}$

Now, for each value of z, I just call the magnetic field routine, passing it the radius of the coil, the value of z, and the expression for r above that depends only on the radius of the coil and z!  The sage code looks like so:

 Bzradius(rcoil, z, curren) = dbcoilbz(rcoil, z, rcoil*(1-(z/rcoil)^2)^0.5, curren)

This gives me nice little graphs like the following.

References:
1.  github Can Crusher simulation project
https://github.com/hcarter333/cancrusher

2.  Trig identities
https://en.wikipedia.org/wiki/Trigonometry_idenities#Compositions_of_trig_and_inverse_trig_functions

3.  The hray experiment
http://copaseticflow.blogspot.com/2014/07/simulations-everywhere-simulations-lab.html

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…