EM Notes Part I: The visual bit of relativistic EM fields pointing at the observer

This is kind of cool from yesterday's EM notes. Our professor pointed out that if you calculate the field from a relativistically moving electric charge, you'll always find that it's pointed straight at the point of observation. Anyone have any idea why? The argument could certainly be made that if you measure the field from a static charge that it will also be pointing straight at you. Then, there's also the realization that the Lorentz transformation only affects the E and B fields in a frame that are perpendicular to the frame's tangential velocity. I'm not sure that's either here or there since the point of observation can be anywhere. Here's the associated diagram for the curious.

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

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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 differe…

Division: Distributing the Work

Our unschooling math comes in bits and pieces. The oldest kid here, seven year-old No. 1 loves math problems, so math moves along pretty fast for her. Here’s how she arrived at the distributive property recently. Tldr; it came about only because she needed it.
“Give me a math problem!” No. 1 asked Mom-person.

“OK, what’s 18 divided by 2? But, you’re going to have to do it as you walk. You and Dad need to head out.”

And so, No. 1 and I found ourselves headed out on our mini-adventure with a new math problem to discuss.

One looked at the ceiling of the library lost in thought as we walked. She glanced down at her fingers for a moment. “Is it six?”

“I don’t know, let’s see,” I hedged. “What’s two times six? Is it eighteen?”

One looked at me hopefully heading back into her mental math.

I needed to visit the restroom before we left, so I hurried her calculation along. “What’s two times five?”

I got a grin, and another look indicating she was thinking about that one.

I flashed eac…

The Javascript Google URL Shortener Client API

I was working with the Google API Javascript Client this week to shorten the URLs of Google static maps generated by my ham radio QSL mapper. The client interface provided by Google is very useful. It took me a while to work through some of the less clear documentation, so I thought I'd add a few notes that would have helped me here. First, you only need to authenticate your application to the url shortener application if you want to track statistics on your shortened urls. If you just want the shortened URL, you don't need to worry about this. The worst part for me was that the smaple code only showed how to get a long url from an already shortened rul. If you follow the doucmentaiotn on the insert method, (the method for getting a shortened url from a long one), there is a reference to a rather nebulous Url resource required argument. It's not at all clear how to create one of these in Javascript. The following example code shows how:
var request = gapi.clie…

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