### Gauss's Sheets

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.

So, the reason why we choose a circle is that the problem then, due to circular symmetry, boils down to be a one-dimensional integral in the radius, r.

OK - simplify and actually doing the integral (and by doing, I mean Mathematica again)

Adding up the integral over the two sheets, top and bottom, we get the same answer as we did when we integrated over the sphere and the box.

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.

"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.

So, the reason why we choose a circle is that the problem then, due to circular symmetry, boils down to be a one-dimensional integral in the radius, r.

OK - simplify and actually doing the integral (and by doing, I mean Mathematica again)

Adding up the integral over the two sheets, top and bottom, we get the same answer as we did when we integrated over the sphere and the box.

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.

Thanks for the Headache Geraint!..LOL. It's been a while since my brain had to carry out calculations, anywhere close to this!

ReplyDeleteI learn so much more,when you are there in person.

How was the release of your research (on Wednesday)received?

I'm a big fan of recreational mathematics, especially if it is stuff I haven't learnt before.

ReplyDeletePaper is with the referee, so still waiting. I will post about it once we have the decision.