magnetic fields
Ok, so I was studying about magnetism. And
the book mentioned:
Magnetic field lines are directed from south pole towards the north pole and thus a closed and continuous curve is formed.
Here is an image from the book:
Then the diagrams indicated that magnetic field lines are directed from north pole towards the south pole.
Here are the diagrams:
Now which one do I believe? Both the images are from Selina Publishers' Physics…
Thanks in advance for support..
A magnetic field configuration corresponds to a knot when for two magnetic field lines given by the parametric curves: $\mathbf{x}_1(s)$ and $\mathbf{x}_2(s)$, the Gauss linking number
$$L\{x_1, x_2\} = \int ds_1 ds_2 \frac{d \mathbf{x_1}(s_1)}{ds_1} .\frac{\mathbf{x_2}(s_1) - \mathbf{x_2}(s_2)}{|\mathbf{x_1}(s_1) - \mathbf{x_2}(s_2|^3}\times \frac{d\mathbf{x_2}(s_2)}{ds_2}$$
is nonvanishing. This integral is an knot invariant and does not depend on a smooth deformation of the magnetic field lines (i.e., without cutting and reconnection of the field lines). Knotted configuration solutions of Maxwell equations are described by: Irvine and Bouwmeester in the following review article, based on a previous work by: Ra$\tilde{\rm{n}}$ada.
In order to answer your second question, let me describe a brief history of the application of knots to physics:
The possible connection of knots to elementary particles was originally suggested by Lord Kelvin in 1867 who speculated that atoms might be knotted vortex tubes in ether. This suggestion was the motivation of the mathematical theory of knot theory treating the analysis and classification of knots. Physicists returned to consider knots and knot invariants in the 1980s. Let me mention the two seminal works by Polyakov and Witten. Both works treat the relation of knots to the Chern-Simons theory. This subject has applications in string theory and condensed matter physics but not directly in particle physics.
However, the situation significantly changed due to the discovery of knotted stable and finite energy solutions in many models of classical field theories used in particle physics. This direction was initiated by Faddeev and Niemi, where they describe a knotted solution in the $O(N)$ sigma model in $3+1$ dimensions. Please, see the following review by Faddeev. Later, they argued that such solutions might also play an important role in low energy QCD. There are many works following Faddeev and Niemi's pioneering work.
Now, as very well known, stable, finite energy solutions of nonlinear classical field theories are called solitons. The most famous types of solitons in gauge field theories are monopoles and instantons.
The soliton solutions are not unique, for example a translation of a soliton in a translation invariant theory is also a soliton (remains a solution of the field equation). Also, there exist rotationally invariant solutions which when rotated in space or around certain directions, and there are also internal degrees of freedom (which correspond for example to isospin). The collection of these degrees of freedom is called the moduli space of the soliton. Thus the soliton can move and rotate and change its internal state, this is why it corresponds to a particle.
These degrees of freedom (moduli) can be quantized and solitons after quantization, can describe elementary and more generally non-elementary particles. A wide class of solitons are associated with topological invariants (topological quantum numbers) which are responsible for its stability. One of the sucessful soliton models is known as the Skyrme model and its solitons approximate the proton and the neutron and also heavier nuclei.
Thus, in summary, these knotted solutions correspond to particles because they are solitons.
First doubt: Why do magnetic field lines form closed curves?
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.
Second doubt: Why do we say that the strength of the magnetic field is more where the lines are closer together?
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
Third: Why do iron fillings acquire exactly the design of the magnetic field?
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.
Last doubt: The diagram of the magnetic field lines that we see (the 2D diagram with many curves), is that diagram 3D in reality?
Yep, Maxwell's equations in their vector calculus form work only in 3D, so the lines you get, in general, are three dimensional lines.
Best Answer
You believe both. They are both correct and they do not contradict each other at all.
The text says that the lines go from south to north inside the magnet. The images show the lines going from north to south outside the magnet. The fact that magnetic field lines form continuous closed loops means that south to north inside the magnet implies north to south outside and vice versa.