Elementary confusion about the magnetic dipole moment and field lines

[nonequilibrium]

simple magnetic dipole moment with impossible field lines? clueless.

Hello,

I'm taking a first year university course in electromagnetism. At a certain point we came upon the torque on a current loop in a uniform magnetic field. There we introduced the magnetic dipole moment $$\overline{\mu} = I\overline{A}$$ with I the current through the loop and A the surface in between the conductors that form the loop (so I suppose this definition is a special 2D-case). Then the torque on the loop turned out (surprisingly enough always) to be $$\overline{\tau} = \overline{\mu} \times \overline{B}$$.

Now in that whole discussion, the concept of magnetic dipole moment seemed to be something purely geometrical. Yet in the exercises there was vaguely mentioned that if "something" has a magnetic dipole moment, you can look at it as a little magnet, with the magnetic dipole moment vector pointing from N to S (the poles). Okay, I figured this comparison with a magnet was made because it behaved like a magnet (i.e. it alligns itself in a magnetic field). But after some online searching, I found "something" with a magnetic dipole moment actually "has" magnetic field lines, like an actual magnet. How does this suddenly arise out of the previous geometrical-like definition and role of this quantity? Was it silly of me to assume there could be something that behaved like a magnet but didn't have magnetic field lines? It does sound silly in sé.

If the confusion stopped there, it would be alright. But in the next chapter we saw the other elementary result that an electric circuit is a source of magnetic field lines. Now I figured, they must be the same field lines (although that would also be odd, having seen they exist before seeing they exist in the chapter devoted to it). But it turns out --after drawing those latter field lines for a random circular loop, i.e. the magnetic field originating from the current itself -- that they're exactly opposite to the field lines associated with the magnetic dipole moment of the object.

Are they connected? Is one negligible compared to the other? How do you even calculate the field lines due to the geometric-like magnetic dipole moment? Is my confusion a fair one, because I must say I'm getting the feeling I'm lacking a whole lot of understanding.

A genuine thanks to all helpers,
mr. vodka

 

[nonequilibrium]

I'm beginning to think my confusion arises out of a possible mistake from my professor. He claimed that the magnetic dipole moment vector tends to line up with the magnetic field vector and that the magnetic field of the dipole moment enforces the exterior magnetic field. Is this last remark true? If the opposite is true: that when it is aligned, the magnetic field arising from the magnetic dipole works AGAINST the exterior magnetic field, then that solves my confusion...

One more remark though: if the magnetic dipole moment vector of "something" always tends to align with the exterior magnetic field, then why does it not in the following example:

A magnetic bottle.

You can associate a magnetic dipole moment with the proton, which is a vector pointing to the left. This is clearly not pointing to the right, as does the magnetic field...

Thank you.

 

[Born2bwire]

From the quick read of your second post, your professor's statement is correct. It is energetically more advantageous for the dipoles to line up with the applied field. This reinforces the magnetic field inside the material and decreases it outside the material. This is expected because when we observe a B field penetrating into a magnetic material, we find that the B field is concentrated into the magnetic material. This corresponds with the previous statement that the external field is diminished while the internal field is reinforced.

As for the direction of your field lines, you are probably making a mistake with your right hand rules. Don't forget, when we talk about the microscopic dipoles of a material, we sometimes talk about the electrons that are orbiting while in circuits we generally discuss them using currents. The effective current of the microscopic dipoles flows in the opposite direction of the electrons.

Magnets are made up of dipoles. The basic classical model is that magnetic materials are made up of microscopic dipole currents. How these currents arise are immaterial and the best ideas we can come up with are generally at odds with known quantum theory. Suffice to say, we just assume that the molecules have a dipole moment. In classical theory this is from say an electron in the orbital. Normally these dipole moments are randomly oriented and thus we have no net field in the bulk. However, if we can get these dipole moments to point in the same general direction in a given volume, we have what is called a domain. These domains have a net magnetic field. In a ferromagnetic material, there are many domains but again normally randoly oriented. If we can line up the domains, then we have a magnet (like with a ferromagnetic material we can do this by applying a strong magnetic field, causing the dipole moments to align and persist). In materials where these domains do not persist, we still get what is called magnetization if the material is magnetic. This was what I described in the beginning.

 

[nonequilibrium]

Aha, I see what you're saying. But I still believe my professor made a mistake. As you say, the magnetic dipole aligns with the external B field and enforces the magnetic field inside the dipole and diminishes B outside the dipole. He claimed the opposite (unless I noted it wrongly...) His claim was that electric and magnetic dipoles are fundamentally different in the way that the former diminishes the external field and the latter enforces the external field. But that's wrong, then, isn't it? They BOTH work against the external field, with the only exception that the magnetic dipole enforces the magnetic field INSIDE (but I don't think he was talking about that, because if you were to replace the current-magnetic dipole with a small magnet, you wouldn't be able to see what happens inside the magnet and it would be enough the say the magnet aligns and just does nothing but decrease the field [except for a bit at both ends...]).

Thanks for your little explenation about the origin of it all. The alignment of dipoles then makes a lot of sense, cause otherwise there pretty much couldn't exist any big material magnets.

One further question though, as I noted in my 2nd post: you can view the proton as having a magnetic dipole (just like a current or the electrons in the classical model), but the weird thing is: in the magnetic bottle, the magnetic dipole vector is 180° opposite to the external field, while it should have a much lower potential energy if it was aligned like you'd expect a magnetic dipole to do. (I understand it's weird to imagine it actually flipping, but technically speaking, if you handle the proton as nothing but a magnetic dipole, you seem to have a very odd situation that's contradictory to "the magnetic dipole tends to align with the external field" -- is it maybe just in an unstable equilibrium?)

 

[sardar]

Hello mr. vodka,

Thank you for reaching out and sharing your confusion about the magnetic dipole moment and field lines. It is completely understandable to feel confused about these concepts, as they can be quite abstract and difficult to grasp at first.

Let's start with the definition of magnetic dipole moment. As you mentioned, it is a quantity that describes the strength and direction of a magnetic dipole, similar to how electric dipole moment describes the strength and direction of an electric dipole. In the case of a current loop, the magnetic dipole moment is defined as the product of the current and the area enclosed by the loop, as you correctly stated.

Now, when we say that something with a magnetic dipole moment behaves like a magnet, it means that it interacts with magnetic fields in a similar way to how a magnet would. This does not necessarily mean that it has physical poles or field lines like a magnet does. It is just a way of understanding and visualizing how this quantity interacts with magnetic fields.

To answer your question about the connection between the magnetic dipole moment and the magnetic field lines generated by a current-carrying loop, they are indeed related. The direction of the magnetic dipole moment is opposite to the direction of the magnetic field lines generated by the loop. This is because the magnetic dipole moment is a vector quantity, and the direction of this vector is determined by the right-hand rule.

Regarding your confusion about the field lines being opposite in direction, this is because they are representing different things. The field lines generated by the current-carrying loop represent the magnetic field produced by the current itself. On the other hand, the field lines associated with the magnetic dipole moment represent the magnetic field produced by the dipole moment. So, they are not the same field lines, but they are both present and contribute to the overall magnetic field in the region.

In terms of calculating the field lines due to the magnetic dipole moment, it can be done using the same principles as calculating the magnetic field due to a current-carrying loop. However, it is a more advanced topic and requires a deeper understanding of vector calculus and electromagnetism.

In conclusion, your confusion is completely valid and it is important to ask questions and seek clarification when learning new concepts. I would recommend discussing your confusion with your professor or a tutor who can provide a more in-depth explanation and help you better understand these concepts.

Best of luck in your studies. Keep asking questions