### Electrical Displacement Field Media Boundary Conditions

Today's post is a little plain Jane, (as far as I know), in that it's just a review of our classes' derivation of the boundary conditions for the electrical displacement field at an interface between two materials.  Here's a question.  Does anyone know of anything, cool or clever to take away from the following derivation?  At the moment, it seems necessary, but not tantalizing.

First, we'll need the Divergence Theorem, (picture 1),

stating that the volume integral of the divergence of a vector field is equal to the surface integral of the same field with respect to the normal of the surface that bounds the volume.

Given the above tools, and starting with a boundary between two materials, and the typical pillbox, (picture 2),

where delta A is the area of the top and bottom of the box, n is the unit vector normal to the top of the pillbox and the pillbox straddles the boundary between the two materials.  In the end we'll let the sides of the pillbox shrink to zero, pulling the two faces of the pillbox onto the interface, so the volume is zero.

We're using Gauss's Law, (picture 3),

where rho is the surface charge density and D is the displacement field.  Taking the volume of both sides we wind up at, (picture 4),:

This is where the divergence theorem comes in to get rid of the volume integral on the left, (picture 5),

while at the same time letting the box walls shrink to nothing, (arbitrarily squat), the charge enclosed in the pillbox becomes just a surface charge density on the pillbox and the integral of the displacement field normal to the bounding surface is just the difference of the normal components of the displacement fields on the two sides of the boundary as seen in the above formula.   This finally gives a condition that must be met between the two displacement fields on opposite sides of the boundary, (picture 6):

References:
Stokes' Theorem:
http://en.wikipedia.org/wiki/Stokes%27_theorem

Picture of the Day:

 From 1/20/13

### 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…