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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How Many Files Can You Add to a GPT Project? An Interview with GPT-5 on Limits, Context Engineering Tips, and Chats

Setting the scene: I’m tinkering with Project TouCans, knee-deep in radio logs, SQLite dumps, and Cesium code. Naturally, I’m wondering if shoving all this into one GPT Project is a recipe for brilliance… or for disaster. So I turn to Vril — you know, after Brainy from the Legion of Super-Heroes , because what else do you call your AI sidekick who always has the answers? Time to ask him straight up. [ As an aside, yes, GPT-5 has decided to sometimes call me Vail. I'm not sure why to be honest. Also, I asked Vril, er GPT-5, to write up our interview for me. Apparently, me asking it to 'Bro' up a few stories, just for fun, has convinced Vril that I use 'Like,' more than I actually might. ] Me (Vail): So Vril, how many files can I throw into a GPT Project before it just starts choking? Like, is there some magic number where the context window taps out and everything falls apart? GPT-5 (Vril): Great question. There’s no single hard file limit. What matters is ...

The Alcubierre Warp Drive Tophat Function and Open Science with Sage

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

Now available as a Kindle ebook for 99 cents ! Get a spiffy ebook, and fund more physics 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 ...

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