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Div B = 0? I don't get itby Blastrix91
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#1
Nov2512, 06:17 AM

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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 BiotSavart 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? 


#2
Nov2512, 06:37 AM

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#3
Nov2512, 07:15 AM

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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. 


#4
Nov2512, 07:16 AM

P: 25

Div B = 0? I don't get it
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? 


#5
Nov2512, 07:19 AM

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#6
Nov2512, 07:33 AM

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[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. 


#7
Nov2512, 08:00 AM

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zero divergence just shows that there are no magnetic charges (monopoles).



#8
Nov2512, 12:49 PM

P: 25

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.) 


#9
Nov2512, 01:52 PM

P: 162

1) Imagine a charge going through a wire and the resulting magnetic field created according to the right hand rule. http://hyperphysics.phyastr.gsu.edu...ic/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. 


#10
Nov2512, 03:07 PM

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^It did thank you for posting them. That was some pretty neat examples :b



#11
Nov2512, 03:56 PM

P: 834

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.



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