The divergence of a vector field tells us how many field lines goes into a volume element in relation to how many goes out. So if div B = 0 there should be the same amount of magnetic field lines going into a volume element as are going out of it right? But the Biot-Savart equations tells us that the magnetic field decreases by r^2 the further away you go. Could somebody help me out in pointing where I got the concept wrong?
Correct. Another way to think of it is that magnetic field lines only form closed loops - they can never begin or end. The fact that the B field lines never begin or end is not mutually exclusive with a change in the magnetic field intensity. In regions of higher magnetic field intensity, the field lines are "packed closer together", and in regions of low intensity the lines are further apart. Look at the attached image of the B-field of a current loop. The B-field lines are closed loops that are closer together near the current loop and further apart far from the current loop.
What does Div B mean? If you draw a box (any shape) anywhere in that magnet picture above, (imagine there are more lines than have been drawn, to make it more convincing) then count the lines going in and the lines going out, you would get the same total numbers. This works even if you include the magnet in the box. Loads of the lines are within the box but they don't count as they don't go in or out but the totals in and out will still be equal.
When you say div B = 0 mean that magnetic field lines only forms closed loops, is that in the moment a magnetic field begins and end (if that were to happen) there would be a change in what goes in in relation to what goes out and thus div B wouldn't be 0? That at least makes sense to me. The B described in Biot Savarts law is the same that is described in div B = 0 right? So from a pure mathematical standpoint div B shouldn't be 0 if it is decreased by r^2 the further in the coordinate system you go. If you get what I'm saying?
The picture that you have in your head of this is not quite right. If the lines are diverging fast then the lines 'going in' will all be concentrated over a small area and the lines 'going out' will be spread out over a greater remaining area. Draw it on paper rather than seeing it in your head and it may make more sense..
I get what you are saying, but your intuition is wrong. Take the case of a simple dipole field, where: [tex]\vec B = K(\frac{3\vec r (\vec m \cdot \vec r)}{r^5}-\frac{\vec m}{r^3})[/tex] where K is a constant, m is a constant vector in the direction of the dipole, and r is the radius vector. It satisfies divB = 0, but falls off as 1/r^3 at large r. Try it! If you plot it out, what you will see is closed loops which are closer together near the center and further apart as you move away from the center, as I said earlier.
hmm thanks. I might have gotten it (tried surfing around the web) I do get this though: [tex] \oint\limits_S \vec{B} \cdot d \vec{A} = 0 [/tex] So that helped a lot But from my understanding of what divergence is, I don't quite get it when it is in the form: div B = 0 (with my understanding of divergence being: How large the vectors are going into a volume element in relation to how large they are going out.)
I have two examples that will help you visually understand the divergence of a magnetic field and one example that helps understand divergence in general. 1) Imagine a charge going through a wire and the resulting magnetic field created according to the right hand rule. http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcur.html 2) Also imagine the pictures created when iron fillings are placed around the magnet and its poles. http://en.wikipedia.org/wiki/File:Magnet0873.png Magnetic fields "curl" around a point or line therefore they do not diverge to one specific place at all, especially due to the circular nature of their curl. When you flush a toilet you create a vector field that curls in the same fashion. Can you say that the field lines created by the water diverge to one point? The divergence of field lines created by a fluid vortex are actually a little higher than that of a magnetic field because of the spiral nature of a vortex. Vector field lines created by a fluid vortex (to an approximation) and magnetic fields do no diverge at a certain point. Therefore they have no divergence. Therefore, the divergence of these vector fields can be said to be "zero" (although a toilet flushing diverges more so than a magnetic field). Basically, vector fields cannot be said to have any level of divergence if they curl too much. As a vector field begins to curl infinitely into a circle (or some similar circular path), you can say that there is no level of divergence, hence the divergence of the field = 0. I hope these visual examples helped you out.
It sounds like you're imagining a box with a finite extent and getting confused because the magnetic field should be weaker on one side of the box or the other. What you should be imagining is that in the limit that the box takes on an infinitesimal extent, the magnetic field strength is the same on all sides.