It has been a busy week, with a talk to the Macarthur Astronomical Society on Monday, and to the Astronomical Society of New South Wales last night. And if you were in Blighty on Wednesday at 4am, you would have heard me join dr karl on Radio 5 Live's graveyard shift to talk about why high speed protons don;t become black holes, but that is the topic for another post.
As I mentioned, I've recently been teaching electromagnetism, and I like to take a little bit of a side-ways glance at derivations and equations. Why? Because sometimes those given in text books can appear a little too idealized or simplified.
One of the things that you have to talk about in electromagnetism is Gauss's law. Mathematically, Gauss's law can seem quite intimidating to a first year student, even those in our advanced class, but let's take a look at what it means in a simplified sense, and then something a little more complicated.
Right, the maths (and to our American cousins, it is maths, not math. Mathematics is the plural of the word mathematic).
permittivity of free space. What's the flippy thing on the left. Let's look at this in picture terms.
Michael Faraday, we talk about idea of the electric field, which we think of each charge in the universe being the source (if positive, sink if negative) of electric field lines (the blue lines in the above picture). The number of lines is dependent upon the amount of charge, so doubling the charge makes twice as many blue lines.
So, what is the right side of the equation telling us? Imagine we consider a surface around the charge (the red, badly drawn thing in the picture above), then the what the right hand size effectively does is count the number of blue lines going through the surface; in this case 4.
It should not take a huge amount of mental effort to understand that if you change the shape and size of the red surface, total number of blue lines crossing the surface is 4. There is a little thought needed for this as you can imagine deforming the surface so much that a blue line going out comes back in again, but then goes back out. So really the right hand side is the number of outward going blue lines minus the number of inward heading blue lines.
When students are introduced to this, you usually have the following picture.
Coulomb's law (that's the first bit up there, and yes, I know that I left the unit vector of on the right hand side). The integral simply becomes the size of the electric field over the surface, multiplied by the area of the surface (I've not worried about the vector dot product which we will return to in a minute). The simplicity of choosing a sphere makes the problem, well, simple.
But let's consider something a little more complex. Let's put the charge at the centre of a cube, with a side length of 2a.
For those that don't like maths, look away now. First, the geometry of the problem
Mathematica to do them for us. So, what do we find?
Gauss's law works for a cube (which, of course, we knew it would :). Right, back to the grindstone.