Questions regarding magnetic fields and magnetic field lines

Question

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.

Answer 1

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.

Mathematica source code for generating the .gif

Answer 2

Strictly, you're not quite correct when you talk of a "test North Pole away from the North Pole of the magnet". The reason the magnetic field form closed curves is because a magnetic field always comes in a N-pole-S-pole pairs. If you had magnetic monopoles then the lines of the magnetic field wouldn't form closed loops. Mathematically, it is a consequence of $\nabla . B = 0$.

Answer 3

Instead of picturing an imaginary "test North Pole" (which apparently does not exist in Nature) as your magnetic field detector, I think it would be better to imagine a tiny little magnetic compass or even better yet, a little electric current loop. The compass and current loop are magnetic dipoles. They orient themselves in the direction of the field, and you can even sense the strength of the field they are in by watching how quickly they align -- a compass in a strong field will "snap" into position faster than in a weak field.

If one uses either of these test devices to map out a magnetic vector field around some other magnetic dipole and then you build magnetic field lines by stepping through the field in the direction each of the field vectors at those points tell you to go, the field lines will automatically end up closer together where the field is stronger and farther from each other where the field is weaker. A correctly drawn magnetic field (and yes, the only way to do it correctly is in 3D) will show field lines with spacing indicating the relative strength of the field.

Right now I don't see a simple way to argue why magnetic field lines always come around to make a loop that closes on the point where you started. It just comes about because there are no "sources" or starting points of field lines in magnetism the way electric charges are sources of electric field lines.

That's why it's not a good idea to think of a "Test North Pole". Think instead of a "test North South Dipole" instead. I think that will help.

Answer 4

here are some intuitions (for Q 1 and 2):

magnetic field, like gravity field, or electric field, are potentials, and the lines are iso-potential (like iso-height curves on a terrain). By construction of the idea of height and iso-height on a continuous terrain, they form loops as slicing a terrain always does.

This means that being at a given location has some "potential energy" value, that can release usable energy when going at a lower place, or request some to get at a higher place. The gradient of potential, i.e. the vertical distance, can be measured by the number of iso-lines you cross. Thus, exactly like differences of altitude, or differences of voltage, the larger the gradient the strengther the energy you get. A region of high density of magnetic lines is the equivalent of a cliff for gravity field (i.e. potential energy of height).

Answer 5

I'm only going to address the first question. I think it would deserve its own post, by the way.

First doubt: Why do magnetic field lines form closed curves?

The reason why can be expressed in two equivalent ways.

The first way is to say that there are no magnetic monopoles, i.e. the magnetic equivalent of point charges does not exist (or at least what we can say is that we still haven't observed them). So the magnetic field will always be a dipolar field, which is characterized by closed lines (see figure below). A popular way to express this concept is to say that no matter how many times you break a magnet, you will always obtain two smaller magnets with identical properties (a North pole and a South pole).

The second way is more formal. The second of Maxwell's equations tells us that

$$\nabla \cdot \mathbf B = 0$$

By applying the divergence theorem we can rewrite this equation as

$$\int_S \mathbf B \cdot d \mathbf S = 0$$

where $S$ is some closed surface and $d\mathbf S$ is its normal vector. The interpretation of the following equation is that all the magnetic field lines that enter any given region must also leave that region and vice versa. This is not true for the electric field. Indeed, the corresponding equation for the electric field is

$$\nabla \cdot \mathbf E = \frac{\rho}{\epsilon} \neq 0$$

If you choose a closed surface with a point charge inside, there will be field lines going outwards but no line going inwards. This is impossible with magnetic field lines, and you can see that the reason is that there are no magnetic monopoles, which is the first way we used to express this concept. So, we have also seen why the two explanations are equivalent.