# Why does paramagnetic liquid rise in a magnetic field?

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## Main Question or Discussion Point

When a magnetic field is applied across the left part of the tube , it's level rises as shown .

I would be grateful if someone could help me understand why this happens .

Thanks .

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A very interesting problem. The paramagnetic fluid consists of magnetic moments (normally in random directions) that can align themselves with the magnetic field which is a lower energy state. They will also get pulled into regions of stronger magnetic field. If my interpretation is correct, this result is likely to be maximized by putting the center of the magnet at about the level of the fluid in the left tube or even slightly above. The magnetic moments experiencing a force to be drawn into the strongest region of the magnetic field which is at the center of the magnet (between the poles). The magnetic moment vectors $\mu$ are constant in amplitude for each molecule, but can point in random directions. The energy in the magnetic field is $U= -\vec{\mu } \cdot \vec{B}$ and there is also a force that is $F=-\nabla U$. The magnetic field $\vec{B}$ is not uniform, so that $F=- \nabla U$ will be in the direction of increasing $\vec{B}$. (Had the magnetic field been uniform, the force $F$ would be zero). $\\$ Note: The reason for the energy being $U=-\vec{\mu} \cdot \vec{B}$, is that a magnetic moment $\vec{\mu}$ in a magnetic field $\vec{B}$ experiences a torque $\vec{\tau}$ given by $\vec{\tau}=\vec{\mu} \times \vec{B}$. Integrating the torque over angle $\theta$, [ $U= \int |\vec{\tau} | d \theta$, with $sin(\theta)$ function in the cross product, $(| \vec{\tau}|= |\vec{\mu} \times \vec{B}|=| \vec{\mu} | |\vec{B}| sin(\theta)$), getting integrated], gives the result $U=-\vec{\mu } \cdot \vec{B}$ , where the dot product is the $cos(\theta)$ part of $-cos(\theta)$ when integrating $sin(\theta)$. $\\$ Additional note: Each magnetic moment $\vec{\mu}$ is essentially a sub-microscopic current loop of current $I$ and area $A$. $\vec{\mu}=I \vec{A}$ where the direction of $\vec{A}$ is perpendicular to the loop. The basic E&M equations apply for the torque that a loop of current experiences in a magnetic field.

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Hi Charles ,

The magnetic moments experiencing a force to be drawn into the strongest region of the magnetic field which is at the center of the magnet (between the poles).

They will also get pulled into regions of stronger magnetic field.
Why would a tiny bar magnet in a non uniform magnetic field move towards a region of stronger magnetic field ?

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Hi Charles ,

Why would a tiny bar magnet in a non uniform magnetic field move towards a region of stronger magnetic field ?
That's what the equation $F=-\nabla U$ represents. There is a force whenever the potential energy $U$ changes as a function of position. The magnetic moment being aligned with the magnetic field $U=-\mu \cdot B$) is a low energy state (when $\theta =0$ it minimizes $U=-\mu B cos(\theta)$), but the energy is even lower (more negative) when the magnetic moment moves to a region of stronger magnetic field $B$, thereby, it gets pulled into a region of stronger $B$. $\\$ Additional item: This is why I anticipate the best results for putting the center of the magnet just above the fluid level. If you put the magnet (considerably) below the fluid level, there would be as many magnetic moments being puled downward into the magnetic field as there are magnetic moments getting pulled upward into the field. The forces would then balance out with no change in the fluid level.

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Very nice explanation .

Why do the two rods align differently ?

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This experimental result is new to me, but I think I may be able to explain it. In the paramagnetic material, the magnetic moments occur in the same direction as the magnetic field, and this is the lowest energy state, i.e. the lowest energy state occurs when the maximum amount of material is in the magnetic field, and internally the magnetic moments will rotate in a direction to line up in the same direction as the magnetic field.( In the paramagnetic state, there is no alignment of the magnetic moments until the external magnetic field is introduced). $\\$ Diamagnetism, as I know it, is a result that occurs because of free electrons in metals (such as copper and aluminum) that can move all over the material. The diamagnetism is the result of the motion of the charged particle (the electron) in the magnetic field, unlike paramagnetism where the atom or molecule has a magnetic moment associated with it. For diamagnetism, it sometimes follows a LeChatlier type principle where the system shifts to reduce the effects of the perturbation. When a magnetic field is applied to a diamagnetic material, the magnetic field tends to get screened very slightly. (For the paramagnetic and ferromagnetic materials, the internal magnetic field becomes enhanced upon application of the magnetic field). $\\$ And the one case they left out is the ferromagnetic case: A permanent magnet will align itself very quickly with the magnetic field, and for the case of iron that is not a permanent magnet, it will also get magnetized and line up just like the paramagnetic case, but with forces perhaps 10,000 x or stronger.

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Diamagnetism, as I know it, is a result that occurs because of free electrons in metals (such as copper and aluminum) that can move all over the material. The diamagnetism is the result of the motion of the charged particle (the electron) in the magnetic field,
Could you please elaborate on the diamagnetic case as to why the rod becomes perpendicular to the field ?

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Could you please elaborate on the diamagnetic case as to why the rod becomes perpendicular to the field ?
That one comes as a surprise to me, but I can understand why it might occur. There is nothing energetically favorable in the diamagnetic case to have a magnetic field running through the material. In this case, apparently the system decides, (by LeChatlier's principle), that it doesn't want the introduction of the magnetic field and in this case minimizes its introduction by rotating to the smaller distance of traversal by the magnetic field. (The physics of diamagnetic materials can get very complicated, but if you think of it qualitatively as e.g. copper or aluminum where you have all kinds of free electrons, it might be reasonable to assume that the electrons will have a weak response away from the magnetic field.) These diamagnetic effects, (except for superconductors), are quite weak, and I believe any rotation that occurs will occur very slowly. These experiments with the diamagnetic and paramagnetic materials require well balanced samples and need to be free of things like air currents and breezes, etc. The string that they are suspended from needs to be quite fine, etc. $\\$ Perhaps someone else can also offer some additional insight... Diamagnetic materials is a subject to which I only have a brief introduction.

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Diamagnetic case looks tricky .

TSny
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Not much that I can add to Charles' very nice discussion. I'm not very knowledgeable on the topic.

One minor point: All materials have some diamagnetic characteristics. However, in paramagnetic and ferromagnetic substances, the paramagnetism or ferromagnetism overwhelms the diamagnetism. Diamagnetic effects can be due to behavior of bound electrons in atoms as well as due to free electrons. More on this here: https://en.wikipedia.org/wiki/Diamagnetism

As to why needle shaped diamagnetic substances orient differently than paramagnetic substances in magnetic fields, I found this description:
http://www.pa.msu.edu/people/stump/EM/chap9/Needles.pdf

As to why needle shaped diamagnetic substances orient differently than paramagnetic substances in magnetic fields, I found this description:
http://www.pa.msu.edu/people/stump/EM/chap9/Needles.pdf
Yes I have seen the paper before . But my knowledge is quite limited . I don't understand what is $z$ of the dipole such that for values of $z$ paramagnetic materials are attracted and diamagnetic are repelled so as to orient perpendicular to the magnetic field .

In Diamagnetic materials , magnetic moments are induced (tiny electron current loops ) such that the magnetic fields produced by them opposes the external magnetic field .In this way , diamagnetic material repels the external magnetic field .

Reason for paramagnetic material tendency to move towards stronger magnetic field is explained in post #4.

But why does diamagnetic materials have a tendency to move towards lower magnetic fields unlike paramagnetic materials ?

That's what the equation $F=-\nabla U$ represents. There is a force whenever the potential energy $U$ changes as a function of position. The magnetic moment being aligned with the magnetic field $U=-\mu \cdot B$) is a low energy state (when $\theta =0$ it minimizes $U=-\mu B cos(\theta)$), but the energy is even lower (more negative) when the magnetic moment moves to a region of stronger magnetic field $B$
Could you please explain why does diamagnetic materials behave in opposite manner ( tendency to move towards weaker magnetic field ) ?

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Could you please explain why does diamagnetic materials behave in opposite manner ( tendency to move towards weaker magnetic field ) ?
I don't have a good explanation for it, but the effect is a very weak one. Even the screening of magnetic fields by diamagnetic materials, with the exception of superconductors is usually very weak. The best explanation I have follows from LeChatlier's principle, but it isn't conclusive. In any case, the effect you observe of the bar that turns at right angles to the field is likely to be a very weak one, so that it would take some very detailed calculations (calculations which I have not done), which take into account some second order and higher terms to explain the observed results. $\\$ The response of paramagnetic materials, and even ferromagnetic materials is more readily explained and the many of the calculations are rather straightforward. The calculations to quantitaively explain diamagnetic properties are in general much more detailed, and with the exception of superconducting materials, diamagnetic responses are normally very weak. $\\$ While we are on the subject of magnetism, one thing you might find of interest is this Insights article on magnetic surface currents that I authored. Below is a "link" to it. These calculations are of moderate or lesser difficulty, and give some interesting results. They are quite a lot simpler than most calculations I have seen to try to explain diamagnetic properties. https://www.physicsforums.com/insights/permanent-magnets-ferromagnetism-magnetic-surface-currents/

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TSny
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Could you please explain why does diamagnetic materials behave in opposite manner ( tendency to move towards weaker magnetic field ) ?
I don't know if this will help. In diamagnetic materials, the induced atomic magnetic moments are in the opposite direction of B. In paramagnetic materials, the magnetic moments tend to align in the same direction as B. A rough model of a magnetic moment is a current loop. So, for a nonuniform B field we get a picture as shown below.

The B field is stronger below each loop. If you pick a point of the current loop, such as P in the diagram, you can see that B is not vertical. B has a horizontal component. If you use the right hand rule to determine the direction of the magnetic force on the current due to the horizontal component of B, then you find that the diamagnetic loop is pushed upward away from the strong field region. The paramagnetic loop is pulled downward toward the stronger field.

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I don't know if this will help. In diamagnetic materials, the induced atomic magnetic moments are in the opposite direction of B. In paramagnetic materials, the magnetic moments tend to align in the same direction as B. A rough model of a magnetic moment is a current loop. So, for a nonuniform B field we get a picture as shown below.

View attachment 209380

The B field is stronger below each loop. If you pick a point of the current loop, such as P in the diagram, you can see that B is not vertical. B has a horizontal component. If you use the right hand rule to determine the direction of the magnetic force on the current due to the horizontal component of B, then you find that the diamagnetic loop is pushed upward away from the strong field region. The paramagnetic loop is pulled downward toward the stronger field.
@TSny I think this is quite helpful in explaining why the diamagnetic bar rotates to get as much material as possible away from the magnetic field. Thank you. A very satisfactory explanation. :) :) $\\$ And to quantify, the energy is $U=-\mu \cdot B$, but $\mu$ is in the opposite direction of $B$ in the diamagnetic material. (essentially the dot product becomes a minus sign because $cos(180^o)=-1$). $F=-\nabla U$ which will be opposite what is was for paramagnetic materials, where $F$ was in the direction of increasing $B$. The paramagnetic material wants to have as much material in the magnetic path as possible, while the diamagnetic is the opposite. $\\$ One thing that is missing from this explanation though is why the diamagnetic material should generate a magnetic moment state that is energetically unfavorable by being aligned opposite to the applied $B$ field, and thereby it becomes a state of higher energy... $\\$ And an additional note on diamagnetism, especially in metals: The De Haas-van Alphen effect, (which you can google), is an example of one type of diamagnetism, and a much detailed theory is required to quantify the effects. The diamagnetic susceptibility oscillates as a function of magnetic field strength, and thereby the explanation for it gets quite complex.

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I don't know if this will help. In diamagnetic materials, the induced atomic magnetic moments are in the opposite direction of B. In paramagnetic materials, the magnetic moments tend to align in the same direction as B. A rough model of a magnetic moment is a current loop. So, for a nonuniform B field we get a picture as shown below.

View attachment 209380

The B field is stronger below each loop. If you pick a point of the current loop, such as P in the diagram, you can see that B is not vertical. B has a horizontal component. If you use the right hand rule to determine the direction of the magnetic force on the current due to the horizontal component of B, then you find that the diamagnetic loop is pushed upward away from the strong field region. The paramagnetic loop is pulled downward toward the stronger field.
Wow ! Fabulous reasoning

Does a magnetic dipole feel any force near the middle of a bar magnet ?

Suppose the plane of the circle (current loop ) is parallel to the long axis of the bar magnet . The plane will also be parallel to the magnetic field of the magnet .

Now if I consider a tiny current element just like you have shown in post#15 , the force due to magnetic field will be in a direction perpendicular to the axis . A diametrically symmetric point will experience a force in opposite direction . Hence net force on a magnetic dipole near the middle of a bar magnet would be zero .

Does this look reasonable ?

I can't draw a sketch now .But if anything is unclear , shall draw a diagram later .

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If I understand the question, a magnetic moment=$\mu$=current loop, that is not aligned with the magnetic field, experiences a torque in the center of the magnet and, at least in a paramagnetic material, will try to align itself with the direction of the magnetic field to minimize the energy $U=-\mu \cdot B$. In the magnet with north (+) and south (-) poles, the magnetic field points from + to -. In the center of the magnet, the field is nearly uniform, and staying at the point of maximum field will also minimize its energy. If it were to drift (up or down) away from the point of maximum field, it would experience a force that wants to return it to the middle regardless of which direction it moves. And in the horizontal direction there is no force, because basically the field is very uniform in moving horizontally between the poles of the magnet. (at least to a reasonably good approximation.)

I think I agree with what you have said . I was actually thinking about the case where a bar magnet is brought close to the middle of a stationary bar magnet and whether the moving bar magnet feel any force .I tried to model the moving bar magnet as a magnetic dipole . I am not sure if you found my reasoning correct .

@TSny , what's your take on post#18 ?

A small magnet put between the magnetic poles will want to align itself so it has $(+).....- +..... (-)$, where the parentheses are for the stationary magnet. if you got it very near the center, it might tend to stay there, but it is an unstable equilibrium point and is likely to finish up as $(+)-+ ........ ...(-)$ , or $(+).......... -+(-)$, attaching itself to one pole or the other. $\\$ @Vibhor Your example with the small magnet is quite instructional in that it shows how the magnetic moment in the molecule of the paramagnetic material would respond. You should even be able to place it slightly below the center of the larger stationary magnet and see the upward pull that results. $\\$ You could perhaps even put a bunch of small magnets each inside a small plastic sphere and create a demo that would respond similarly to the liquid of your original post.