Paradox regarding energy of dipole orientation

[Meir Achuz]

Justin:
All you have shown is that A.j gives the right answer.
If you dot B_0 with B from a dipole, you will get the same wrong answer as in your first post.

"Please stop claiming these methods are different without math to back up your statements." shows you have not taken my advice.
You are now on your own.

 

[Sam Park]

However, even taking those two situations as solved, as I mentioned before, "It seems though that you guys are suggesting that a magnetized object would anti-align to an external field, where as a loop with frozen in charges sent spinning would align with an external field." This is because there is no internal energy being used up to maintain the magnetic dipole of a magnetized object (is there?). So for this case where the dipole remains constant without expenditure of energy as with the two loop cases given above, U=+m.B seems to require it to anti-align. So there is still something I am missing here.

There is an important difference between the energy calculations for a magnetic dipole produced by “free current” and one produced by a magnetized subtance. The distinction is captured by the difference between the B and the H fields. Outside any magnetic material, B and H are strictly proportional to each other, but inside magnetic material they are quite different. The potential energy density of a magnetic field is really B.H/2, and reduces to B^2/(2 mu) only outside of any magnetic material.

Take for example a spherical shell of uniformly magnitized material. Outside the shell the B and H fields are essentially identical, being that of a perfect dipole, for which the field energy is indifferent to orientation. Inside the sphere, there is a uniform field, but the direction of the H field is opposite the direction of the B field in this interior region. This gets back to my first post in this thread, asking if you were really sure the components of the cross-term B_f and B_d really had the same sign. I should have said that, in the region that counts (inside the magnetized sphere), the field that counts is the H_d field, which points in the opposite direction of the B_d field.

The reason we need to use H is that a magnetic field B is caused by two different types of currents, which are sometimes called “free current” and “bound current”. The free currents are the ones you would usually identify as electrical currents in a system, produced by hooking up a battery to a wire, or some similar method. In contrast, the “bound currents” are the ones we infer must be present in atoms comprising a permanently magnetized substance. Unlike free currents, bound currents don’t do any work, so when we perform calculations involving the energy of the magnetic field we need to distinguish between the fields produced by these two kinds of current. This is why we need to use the field energy expression based on the H field when considering magnetized objects.

 

[JustinLevy]

I don't know if this is related, but you may like to take a look the Feynman lectures discussion on this. Whether it's +/-m.B depends on whether you count the energy to set up the external B field or something like that (I'm garbling it).

Edit: Volume 2, Chapter 15

I think I have that around here somewhere. Thanks, I'll check it out.

Justin, You are continuing to misinterpret and misrepresent my statements. I'm not talking about the A.j versus B^2 method.

I'm sorry if there is a miscommunication. Your post seemed to be claiming that using the A field gave the correct answer and the B field did not. Even now, going back and rereading your post, I get the same impression ... that you are saying that using the A field gives the correct answer and the B field does not. Can you please reread your previous post and clarify?

If you would had simply read the link clem gave you 3 weeks ago then you would have spared yourself a lot of time. The mathematical contents of the paper is perfectly mainstream and perfectly relevant for this issue despite the author's interpretation of DE+BH as a probability density only.

As stated before, I READ THAT PAPER. Please everyone stop bringing up that paper. It is not useful for this situation. Here's what they say about the dipole problem:
"In fact, [ $F =\nabla (\mu \cdot B)$ ] should be considered a separate force law for permanent magnetic dipoles on a par with the Lorentz force on charge or the j×B force on currents, since it cannot be derived from those force laws."

Their answer is a non-answer. Their "solution" to the dipole problem is to postulate a new force. If you want to take that as an answer, fine, but you don't need to reference that paper, for stating it outhand yourself is giving as much justification as they gave it.

Secondly, while I agree much of their math is straight-forward and correct, their paper is very non-mainstream. For some reason they have some serious philosophical problems with EM radiation carry energy and go through all kinds of manipulations to try to justify it, but fall very short. In doing so they take a classical theory and turn it into a theory with manifestly non-local interactions, come to the "conclusion" that EM radiation doesn't carry energy or momentum, and completely disregard the stress energy tensor as physical despite its use in GR (instead opting for a "energy in the charges" approach which would make GR tortuous).

I have reproduced a proof from a textbook in this thread already showing the relation between the "energy in the fields" and the "energy in the charges" methods. No further discussion about the relation between these methods should be needed. No further discussion of that paper should be needed (or wanted) either.

Vq+A.j is DE+BH only under certain conditions since (see eq.49) of the link above.

Vq+A.j = DE+BH + surface term + time dependent term

The paper calculates the surface term in two ways see eq.39 and eq.42.

You complain that I keep interpretting your posts as saying only one of the two energy calculations methods is allowed here. Yet you keep bringing it up in every post.

I showed the relation between the two. I've even explicitly done the calculation for both cases and got the same answer. These are the same here. If we are in agreement, why do you keep bringing it up? It is clear that the "solution" to this problem does not lie in our choice of calculating with the "energy in the fields" or the "energy in the charges" methods. So let's please move on.

The real issue is the constant current assumed in the calculations and the energy that has to be put in the loop to keep the current constant.

Okay, let's consider an example that doesn't have a constant current so we don't need to worry about the energy necessary to keep that going.

Consider a spinning ring of charge, initially anti-aligned to a magnetic field of magnitude B supplied by an external source. Initially the magnetic moment of the ring has a magnitude m1. Thus the energy of orientation is U = -m1 B. Let it align with the field, and the magnitude of the dipole moment is now m2, with the energy U=+m2 B. This energy must have come from somewhere, and with no current source to blaim it must have come from the Kinetic energy or from the self energy in the dipole field. The kinetic energy is proportional to M (the mass of the ring) and the self energy of the dipole field is proportional to m1^2. Each of these can be adjusted separately as well as the external field. So what prevents us from making a situation where m1.B > the kinetic energy + self energy of the dipole field ? In that case, it is clear that the total energy would be minimized with the dipole anti-aligned. If it goes to torque here, there isn't even enough energy to bring this to rest, let alone align the dipole with the field ... so what happens?

It might help you to read the derivation of Eq. (4.102) in Jackson (2nd Ed.) (or another text) where he derives the positive gradient for the electric case at constant V. The same procedure applies in the magnetic case at constant current.

With some source of energy providing the necessary work to keep the current the same, I think I understand that case now. It is the magnetized material and non-current source cases that still seem strange to me.

You could also read the section in that preprint on a permanent moment which does have -mu.B.

As noted several times already, I have read that paper. It is unpublished, non-mainstream, and when it gets to the situations relavent here, just states answers (and even postulates new forces) instead of working anything out. Let us please drop that paper.

Justin:
All you have shown is that A.j gives the right answer.
If you dot B_0 with B from a dipole, you will get the same wrong answer as in your first post.

No, I have shown that BOTH give the same answer. Not only have I worked this out explicitly, I have even reproduced the textbook proof that shows the relation between the "energy in the fields" and the "energy in the charges" methods. The methods give the same results here.

If you have some philosophical issue with the surface terms like those preprint authors, then just don't consider infinite sources. The answer to this problem has nothing to do with A.j versus B^2 methods. This has been shown many times, so please please let us move on from that.

"Please stop claiming these methods are different without math to back up your statements." shows you have not taken my advice.
You are now on your own.

Those methods are equivalent for this problem. This has been shown explicitly by working it out both ways for this problem, as well as a general proof from a textbook. If you still wish to claim otherwise, then you are wrong. Saying so does not make me unwilling to learn. It just makes me unwilling to ignore textbooks and explicit mathematical examples, and instead believe otherwise because someone says so without any math to back it up.

;-------------------------------------------

Sam,
I thought you could work out problems with that involve M in two ways:

Method 1:
Given M and the free current:
- Calculate the bound current
- Calculate the magnetic field using these current densities
- Calculate whatever you need using the magnetic field now

method 2:
Given M and the free current:
- Calculate H
- Calculate whatever you need using B and H now

I thought these two methods (use the bound current, or use H) were equivalent. In particular, when I had to take an EM course, I remember them requiring us to do many problems using both methods to apparently drive home that either way worked. So shouldn't I get the same energy either way?

What you brought up seems to clearly answer no. It seems obvious now, but I will need to read more to jive these two thoughts in my head. This however seems to raise another issue since using the relations already derived, and given that H is defined as $H= \frac{1}{\mu_0} B - M$ we arrive at the relation between B_external and M inside the sphere as $H= \frac{1}{\mu_0} B_{ext} + (\frac{2}{3} M) - M = \frac{1}{\mu_0} B_{ext} - (\frac{1}{3} M)$. This seems to raise another "magnitude of the energy" issue.

Thanks for the input. I think you've sketched together enough that I can see all the answers are in your suggestions and examples there ... I just need to sort through this some more. At least now it is abundantly clear how the magnetized and non-magnetized situations arrive at different signs.

 

[Hans de Vries]

I'm sorry if there is a miscommunication. Your post seemed to be claiming that using the A field gave the correct answer and the B field did not.

Justin, I think everybody here agrees about the following numbers:

jA = 1 (mechanical energy transformed into electromagnetic field energy)
B^2 = 2/3 (electromagnetic field energy inside the volume)
S = 1/3 (surface term of the electromagnetic field energy)
T = 0 (time dependent term)

The miscommunication here is that you equate the "B^2 method" with B^2+S+T while
others simply mean B^2=2/3. Read for instance clem's last post and your response to it.


Regards, Hans

 

[Hans de Vries]

As stated before, I READ THAT PAPER. Here's what they say about the dipole problem:

"In fact, [ $F =\nabla (\mu \cdot B)$ ] should be considered a separate force law for permanent magnetic dipoles on a par with the Lorentz force on charge or the j×B force on currents, since it cannot be derived from those force laws."

Their answer is a non-answer. Their "solution" to the dipole problem is to postulate a new force.

This the whole point. $F =\nabla (\mu \cdot B)$ is the correct force (with the right sign)
on the dipole when the Lorentz force is applied on a circular current !

The author assumes a mysterious sign flip (stemming from elementary
particle physics). There are however no sign flips. The sign is always like this.
There are no mysterious sign flips in the force.

Consider a spinning ring of charge

An increase in flux (increase in B) through the ring induces an electric field
along the loop $\partial B/\partial t = -\nabla\times E$ which opposes the current in the loop and
slows down the spinning ring.

The mechanical energy from the angular momentum of the ring decreases
by a factor $\mu\cdot B$ while the total field energy increases by the same factor.

This will reduce the current in the loop, but the amount by which the current
decreases depends on the mass (-> angular momentum) of the spinning ring.
A very heavy spinning ring will provide the kinetic energy without hardly slowing
down and the current remains almost the same.

A static non-homogeneous magnetic field B can not increase the total
energy of the dipole (it is a conservative field), but it can transform the
kinetic energy from the angular momentum into kinetic energy from linear
motion.

Note that none of the energy calculations done or mentioned this tread
contains such a "kinetic energy from linear momentum" because a static
situation is always assumed. The calculation of the various energies of
an accelerating dipole is much more involved.

Without doing such calculations all we can tell is that apparently all the
kinetic energy that becomes available from the slowing down of the
spinning ring ($\mu\cdot B$) would be available for an increase of the linear
momentum via $F =\nabla (\mu \cdot B)$


Regards, Hans

 

[gabbagabbahey]

Sam,
I thought you could work out problems with that involve M in two ways:

Method 1:
Given M and the free current:
- Calculate the bound current
- Calculate the magnetic field using these current densities
- Calculate whatever you need using the magnetic field now

method 2:
Given M and the free current:
- Calculate H
- Calculate whatever you need using B and H now

I thought these two methods (use the bound current, or use H) were equivalent. In particular, when I had to take an EM course, I remember them requiring us to do many problems using both methods to apparently drive home that either way worked. So shouldn't I get the same energy either way?

What you brought up seems to clearly answer no. It seems obvious now, but I will need to read more to jive these two thoughts in my head. This however seems to raise another issue since using the relations already derived, and given that H is defined as $H= \frac{1}{\mu_0} B - M$ we arrive at the relation between B_external and M inside the sphere as $H= \frac{1}{\mu_0} B_{ext} + (\frac{2}{3} M) - M = \frac{1}{\mu_0} B_{ext} - (\frac{1}{3} M)$. This seems to raise another "magnitude of the energy" issue.

Thanks for the input. I think you've sketched together enough that I can see all the answers are in your suggestions and examples there ... I just need to sort through this some more. At least now it is abundantly clear how the magnetized and non-magnetized situations arrive at different signs.

My previous post was in error (I hadn't fully read all of the posts to see that you were in fact having a very different issue); Sam's resolution of the "paradox" is correct. For a detailed discussion of it, see R.H. Young, Am. J. Phys 66, 1043 (1998) --- In which Young answers Griffiths' question on why the Hamiltonian is $-\vec{\mu}\cdot\vec{B}$--- and references cited therein.

 

[Meir Achuz]

This the whole point. $F =\nabla (\mu \cdot B)$ is the correct force (with the right sign)
on the dipole when the Lorentz force is applied on a circular current !

The author assumes a mysterious sign flip (stemming from elementary
particle physics). There are however no sign flips. The sign is always like this.
There are no mysterious sign flips in the force.

Read it again.
The force on a magnetic dipole is always $F =\nabla (\mu \cdot B)$.
But the energy on a current loop where the current is kept fixed is given by
$U=\mu \cdot B$, with the force $$F=\nabla U$$. The energy of a permanent magnetic dipole in a magnetic field is given by $U=-\mu \cdot B$, with the force $$F=-\nabla U$$.
The force $F =\nabla (\mu \cdot B)$ for a permanent magnet cannot be derived from the Lorentz force because a permanent magnet is not due to current loops but to the electron's magnetic moment, which is unrelated to current loops.

 

[Hans de Vries]

Read it again.
The force on a magnetic dipole is always $F =\nabla (\mu \cdot B)$.
But the energy on a current loop where the current is kept fixed is given by
$U=\mu \cdot B$, with the force $$F=\nabla U$$.

The force is per definition $$F=-\nabla U$$ and there is no sign change which would
violate the conservation of energy. A static magnetic field is conservative and
can only transform one kind of kinetic energy into another, in this case rotational
kinetic energy into translational kinitic energy as described in post #75.

The energy of a permanent magnetic dipole in a magnetic field is given by $U=-\mu \cdot B$, with the force $$F=-\nabla U$$.
The force $F =\nabla (\mu \cdot B)$ for a permanent magnet cannot be derived from the Lorentz force because a permanent magnet is not due to current loops but to the electron's magnetic moment, which is unrelated to current loops.

According to the Dirac equation there IS a current loop associated with the electron's
magnetic moment. See for instance: Sakurai, Advanced Quantum Mechanics eq(3.205)
and further in the section on the Gordon Decomposition.

A paper which discusses this in the context see:
http://jayryablon.files.wordpress.com/2008/04/ohanian-what-is-spin.pdf"

According to the prevailing belief, the spin of the electron or some other
particle is a mysterious internal angular momentum for which no concrete
physical picture is available, and for which there is no classical analog.
However, on the basis of an old calculation by Belinfante [Physica 6 887
(1939)], it can be shown that the spin may be regarded as an angular
momentum generated by a circulating flow of energy in the wave field
of the electron. Likewise, the magnetic moment may be regarded as
generated by a circulating flow of charge in the wave field. This provides
an intuitivelyl appealing picture and establishes that neither the spin nor
the magnetic moment are “internal” — they are not associated with the
internal structure of the electron, but rather with the structure of the field.
Furthermore, a comparison between calculations of angular momentum in
the Dirac and electromagnetic fields shows that the spin of the electrons
is entirely analogous to the angular momentum carried by a classical
circularly polarized wave.

see equation (22)Regards, Hans

 

[Hans de Vries]

According to the Dirac equation there IS a current loop associated with the electron's
magnetic moment. See for instance: Sakurai, Advanced Quantum Mechanics eq(3.205)
and further in the section on the Gordon Decomposition.

A paper which discusses this in the context see: http://jayryablon.files.wordpress.com/2008/04/ohanian-what-is-spin.pdf"

Regards, Hans

For an intuitive picture see the introductory chapter of my book (section 1.9)
http://physics-quest.org/Book_Chapter_EM_basic.pdf

Or the chapter on the Gordon decomposition itself (section 18.2 and 18.3)
http://physics-quest.org/Book_Chapter_Gordon_Decomposition.pdfRegards, Hans
(P.S. Some work in progress here regarding the decomposition of the axial current I see now)

 

[Meir Achuz]

The force is per definition $$F=-\nabla U$$ and there is no sign change which would
violate the conservation of energy.

I am a bit surprised that you say that so late in this thread. The thread started with the fairly simple demonstration that $$U=+\mu\cdot{\bf B}$$ for a loop with a constant current in a B field. This required $${\bf F}=+\nabla U$$ to get the correct form that
$${\bf F}=+\nabla(\mu\cdot{\bf B})$$. F also equals $$\nabla U$$ for the force of one capacitor plate on the other, when the capacitor is kept at constant voltage. This is shown in most EM texts. In every case, $${\bf F}=+\nabla(\mu\cdot{\bf B})$$, but the energy can have either sign depending on whether or not a constant voltage or constant current source is providing energy to keep the voltage or current constant.

For an electron, $$U=-\mu\cdot{\bf B}$$ because there is no external source feeding energy to the electron. This is independent of any speculation about the source of the moment.

 

[Hans de Vries]

I am a bit surprised that you say that so late in this thread. The thread started with the fairly simple demonstration that $$U=+\mu\cdot{\bf B}$$ for a loop with a constant current in a B field. This required $${\bf F}=+\nabla U$$ to get the correct form that $${\bf F}=+\nabla(\mu\cdot{\bf B})$$. F also equals $$\nabla U$$ for the force of one capacitor plate on the other, when the capacitor is kept at constant voltage. This is shown in most EM texts. In every case, $${\bf F}=+\nabla(\mu\cdot{\bf B})$$, but the energy can have either sign depending on whether or not a constant voltage or constant current source is providing energy to keep the voltage or current constant.

For an electron, $$U=-\mu\cdot{\bf B}$$ because there is no external source feeding energy to the electron. This is independent of any speculation about the source of the moment.

This is all correct Meir,
My original response was to Justin Levi's attack on section 4 of this paper:

http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.3421v3.pdf

Which discusses the magnetic moment of elementary particles. Which have, as you
say, $$U=-\mu\cdot{\bf B}$$ because they are passive.

As stated before, I READ THAT PAPER. Please everyone stop bringing up that paper. It is not useful for this situation. Here's what they say about the dipole problem:

"In fact, [ $F =\nabla (\mu \cdot B)$ ] should be considered a separate force law for permanent magnetic dipoles on a par with the Lorentz force on charge or the j×B force on currents, since it cannot be derived from those force laws."

Their answer is a non-answer. Their "solution" to the dipole problem is to postulate a new force. If you want to take that as an answer, fine

From "Here's what they say about the dipole problem" I concluded that the
author says that Justin's "paradox": $$U=+\mu\cdot{\bf B}$$ was explained by telling that
"elementary particles obey different laws".

Re-reading section 4 shows that the author correctly states that $$U=-\mu\cdot{\bf B}$$
So the author of the paper is correct here, besides the other independent
discussion about the nature of the magnetic moment of elementary particles.Regards, Hans.