It's early on Sunday morning, and I'm up and about due to the chorus of Sulphur-Crested Cockatoos that often make my suburb sound like the depths of Jurassic Park. While lying there in the din, I was thinking about yesterday's post, and found myself pondering the following:
"What would happen if I stopped using a cube, but instead had two parallel sheets? Now, in this case, I don't have a closed surface anymore, but if I make my sheets infinitely large, I am guessing that the total integral over the two sheets would converge to the result expected by Gauss's law."
Can you see why?
Anyway, this is not going to be a long post, as I haven't had any coffee yet, and have a paper to deal with, but I am very quickly going to scribble down the solution. Basically, I am going to replace my square piece with a circle, and then make the circle infinitely large in radius, and this will be the same as making a square sheet infinitely large.
Note to mathmos out there - I'm a physicist and this is how we do maths.
OK - the geometry.
OK - simplify and actually doing the integral (and by doing, I mean Mathematica again)
But as we noted, the sphere and the box are closed surfaces, but two sheets are not. Why does this work?
Anyway, I'm getting off the Gauss hobbyhorse, as there are astronomy results coming. Watch this space.