I went through certain statements regarding magnetic fields and M.G lines which I am not able to understand.
First doubt: Why do magnetic field lines form closed curves?
I went through some answers over the internet and on Phys. SE, but I could not understand them as they were a little difficult for me. The reason I thought of when I tried to think about this question is that as you move the test North Pole away from the North Pole of the magnet, it gradually is attracted by the South Pole of the magnet, and therefore, the magnetic field line moves closer and closer towards the South Pole, so we can say that it forms a loop. However, I am not still able to satisfy myself. So is this reasoning correct? Or if not, can someone explain this to me in a more easier way?
Second doubt: Why do we say that the strength of the magnetic field is
more where the lines are closer together?
Now when I try to think of this, I am just reminded that these lines are plotted using a test North Pole. So how can we say that where lines are closer, strength is more? Those lines just represent the direction, how is this justified? Isn't the strength of the field related to the closeness of the test North Pole to the magnet?
Third: Why do iron fillings acquire exactly the design of the magnetic
field?
I am pretty sure that you all know which experiment I am talking about. Now, I just cannot understand the reason behind this. Shouldn't the fillings just stick to the cardboard in a random design, because they are just getting attracted, what else is happening?
Last doubt: The diagram of the magnetic field lines that we see (the
2D diagram with many curves), is that diagram 3D in reality?
I hope these are not stupid questions, I really can't understand these.
Best Answer
The premise is false!
Take the following image I generated as an example. The black circles here are two current loops arranged haphazardly. The blue line is a single magnetic field line, plotted for a really long length. It's still going, and it isn't ending any time soon.
The only statement of importance is that $\nabla \cdot \vec{B}=0$. This can be interpreted differently: the divergence of a vector field at a point can be approximated by the flux into a very small sphere of volume $V$ at that point:$$\nabla \cdot \vec{B}=\lim_{V\to 0}\frac{\oint_S\vec{B}\cdot d\vec{a}}{V} $$ ($S$ denotes the surface of the sphere volume $V$ centered at the point in question, and $d\vec{a}$ denotes a vector area element). Therefore, if a magnetic field line penetrates the tiny sphere and ends, and has some magnitude, then $\nabla \cdot \vec{B}\neq 0$ and you've violated a Maxwell law!
But a magnetic field line can actually end. For example, imagine two single loop solenoids on top of each other, pointing in opposite directions. As derived on this page, we might have:
$$B_z=-\frac{\mu_0 R^2 I}{2((z-a)^2+R^2)^{3/2}}+\frac{\mu_0 R^2 I}{2((z+a)^2+R^2)^{3/2}}$$
At $z=0$, the field is zero. At $z<0$, the field is positive and along the z axis. At $z>0$, the field is negative and along the z axis. So clearly the field line heads towards zero, but never reaches it.
The following page defines $B=\sqrt{\vec{B}\cdot \vec{B}}$ and $\hat{b}=\vec{B}/B$, and proves that as you walk along a field line:
$$\frac{dB}{d\ell}=\hat{b}\cdot \nabla B=-B \nabla \cdot \hat{b}$$
If the field lines are converging then $\nabla \cdot \hat{b}<0$ and so $B$ is increasing in magnitude, and if the field lines are diverging then $\nabla \cdot \hat{b}>0$ and so $B$ is decreasing in magnitude. So there's your vector calculus proof.
J.D Callen, Fundamentals of Plasma Physics, chapter 3
This is more complicated. Each iron filing forms a little magnet that attracts its neighbors, so the iron filings can't fill up space and instead join end to end in directions induced by the magnetic field. So they form lines. Which field lines are chosen depends on the whole, ugly dynamics of the situation.
Yep, Maxwell's equations in their vector calculus form work only in 3D, so the lines you get, in general, are three dimensional lines.
Mathematica source code for generating the .gif