# Force on current in magnetic field

1. Aug 30, 2011

### KOSS

This question comes under the familiar topic "force on a current carrying wire in a magnetic field".

You all know the Lorentz force and $F=LI\times B$. So I'm teaching a high school class on this and a senior teacher tells me the students will need to explain the force on the wire for their exam in terms of the modified B field lines (due to sum of B fields of the current in the wire plus the magnet). I could not teach this because I did not know how to explain it.

Here's the sketch in words: B field N to S horizontal (left to right say). Current in wire coming at us out of page. RH Grip Rule field around the wire anticlockwise. This adds to the magnet field. Result: Field lines above wire partially cancel. Field lines below wire reinforce. High field below the wire, weak field above it.

From what I could make out the textbook seemed to be trying to explain the upwards motion of the wire being due to movement of the wire from "high magnetic field region to low magnetic field region". Almost as if it were a magnetic field pressure effect.

But I've always thought this force on a wire carrying a current in a B field was just simply the Lorentz force on the charges. How can magnetic pressure or "movement from high to low field intensity" be the cause? Is it additional to the Lorentz force? Is it different and does it dominate the Lorentz force?

I can see that the field around the wire is not uniform because of the superposition of the magnet's field wit the current's field, but is the lifting of the wire still not just the plain Lorentz force?

Any enlightenment would be welcome.

PS. I'll post the textbook text and figure in a mo' once I scan it.)

Last edited: Aug 30, 2011
2. Aug 30, 2011

### KOSS

Here's the textbook page (see attachment):

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• ###### force_on_current_carrying_wire.jpg
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3. Sep 19, 2011

### KOSS

I have found a few places where the force on the current carrying wire is discussed in terms of "tension" in the B field lines. However, my present understanding is that to make any sense of such language I would still need to talk to my students about the full Faraday tensor.

Does anyone reading this forum know of a better and clearer way to introduce the behaviour and influence of the magnetic field on the current carrying wire in terms of magnetic field tension that does not require a digression into the EM field tensor? Remember, this is high school physics, not university level.

4. Sep 19, 2011

### Phrak

This teacher is misled. So was the creator of the attached thumbnail drawing. Magnetic fields act in superpostion--that is, a direct sum. The magnetic field locally induced by the wire has no force back on the wire. So you need only consider the external field.

5. Sep 19, 2011

### entphy

Koss's question is a valid question, and the teacher is not misled. The fact is : both the explanations in terms of the Lorentz force on the current carrying conductor, and the force due to the interaction of the external and current carrying magnetic fields are equivalent. I hereby provide the proof that I just worked out as followed.

Starting with the Lorentz force on the current carrying conductor, here I assume the current is I$\hat{z}$ pointing in the +z direction into the paper (following the convention of the attached diagram by Koss, and B is thus a uniform field pointing in the -$\hat{y}$ direction),

F = LI$\hat{z}$$\times$B
By Ampere's Law, the current I = $\oint$H$\cdot$dl, H is the magnetic field intensity due to the current carrying wire and integration is along any closed loop around the wire. Substitute the expression into the Lorentz force and one could obtain

F = L$\oint$H$\cdot$dl$\hat{z}$$\times$B
$\Rightarrow$F = L$\int$$\nabla$$\times$H$\cdot$ds$\hat{z}$$\times$B
$\Rightarrow$F = L$\int$($\nabla$$\times$H)$\times$B($\hat{z}$$\cdot$ds)
$\Rightarrow$F = L$\int$-$\nabla$(H$\cdot$B)($\hat{z}$$\cdot$ds)+L$\int$(B$\cdot$$\nabla$)H($\hat{z}$$\cdot$ds)

The last equation evolves from the preceding one by taking the vector identity, and by virtue that B is a constant magnetic field thus all the curl and grad operations result in 0.

Hence, it is clear that the resultant force on current carrying wire is proportional to the area integral in space of -$\nabla$(H$\cdot$B)+(B$\cdot$$\nabla$)H, which is the force due the gradient of the interactions of the resultant fields by H and B in space. And both the explanations are equivalent.

I may not have been rigorous enough in the mathematical derivation. But I hope it suffice to unite the explanations hence not necesary to involve the electromagnetic field tensor.

Last edited: Sep 19, 2011
6. Sep 19, 2011

### Phrak

I have one question. In which direction is the force on a current carrying wire directed due to its own induced magnetic field?

7. Sep 19, 2011

### entphy

I guess the question is which direction is the force on a current carrying wire directed due to its own induced magnetic field interacting with the external applied magnetic field, as there would be no force without the 2 magnetic fields interacting. The direction of the force should then be the Lorentz force, and consistent either by the applying Fleming's left hand rule or simply work out the resultant field intensity in the space around the wire. The force should direct from high field intensity region to the low field intensity region.

P.S. I have made an amendment in the original post above to include a missing term that was left out in the derivation.

8. Sep 20, 2011

### Phrak

No. Magnetic fields don't interact. B_total = B_1 + B_2, a direct sum. There is no product term.

9. Sep 20, 2011

### entphy

Magnetic fields certainly interact with each other. Otherwise how do 2 magnets attract or repel each other? If one places a small bar magnet in a larger magnetic field, it's clear that the north and south pole of the small bar magnet would be attracted to the opposite poles of the external field respectively. This is how a simple compass works in the first place.

If there is no interaction between the magnetic fields, as you insist, then lets just ignore the magnetic field of the small bar magnet. Then this small bar magnet would be of no difference from a piece of unmagnetized metal and behaves as such in the magnetic field. It will get magnetized accordingly regardless how the original magnetic poles are oriented and gets attracted to the magnetic field. Just like picking a paper clip or nail using a magnet, it does not depend on how the paper clip or nail were oriented. This is clearly not the case for the small bar magnet. So how could there be no interaction between magnetic fields?

Yes, magnetic and electric fields are linear and can be superimposed linearly, i.e. the direct vectorial sum with another magnetic and electric fields result in the net resultant field. But this does not mean there is no interactions between magnetic fields.

10. Sep 20, 2011

### Phrak

This is getting a little out of hand. You obviously haven't studied physics; yet insistant on your preconceptions. I personally have no problem what-so-ever in well thought opinions that deviate from accepted physics, but your's is not one of them. On the other hand, you are misleading KOSS, if he's still around watching this thread, with your fanciful ideas when an answer to his question should be seen as directed toward an answer in accepted ideas.

Last edited: Sep 20, 2011
11. Sep 20, 2011

### entphy

I am truly aghast by how pathetic you are in this well respect forum that suppose to serve nothing but scientific truth. Obviously your understanding of physics is so poor or none but yet you are so ashame of being exposed, you had to resort to retoric to get your way out. Do you seriously think by insisting on your misconception you could turn the wrong to right? If you are so right about your non-interacting magnetic fields, why can't you argue with logic and with established scientific principle and point out the inconsistent part? Instead of offering such useless comment to "disprove" something?

I feel oblige to reply eventhough I found it extremely stupid to worth my time to respond at all, because I seriously do not want KOSS to be misled by you, and his/her students at stake!

My points as followed, and I bet you don't have the guts to confront them point by point logically except your hysterical retoric.

1. The force on the current carrying conductor is a very well established phenomenon that can be explained by the Lorentz force on the moving charge in the conductor, or equivalently the interaction of the magnetic fields. Obviously you don't even have the high school level of physics, read Nelkon & Parker Advanced Level Physics Chapter 10, section "Interaction of Magnetic Fields". The explanation is right on the spot by the interaction of the external magnetic field and the magnetic field due to the current carrying wire. I suppose you would probably like to write to Nelkon & Parker to tell them they are wrong all the time. But I advise you to publish your theory in any respected journal first why magnetic fields are not interacting and there is no equivalent explanation for such an elemental physical observation!

2. I have just worked out a more rigorous mathematical derivation to show both models are exactly equivalent. It is ready to be published anytime but I won't until you show yours why they are not equivalent, and point out why the step by step logical mathemetical manipulation from Lorentz force to interacting magnetic fields are incorrect.

The evidence is clear and conclusive open to everyone with intellectual honesty to examine!

12. Sep 20, 2011

### entphy

I am truly aghast by how pathetic you are in this well respect forum that suppose to serve nothing but scientific truth. Obviously your understanding of physics is so poor or none but yet you are so ashame of being exposed, you had to resort to retoric to get your way out. Do you seriously think by insisting on your misconception you could turn the wrong to right? If you are so right about your non-interacting magnetic fields, why can't you argue with logic and with established scientific principle and point out the inconsistent part? Instead of offering such useless comment to "disprove" something?

I feel oblige to reply eventhough I found it extremely stupid to worth my time to respond at all, because I seriously do not want KOSS to be misled by you, and his/her students at stake!

My points as followed, and I bet you don't have the guts to confront them point by point logically except your hysterical retoric.

1. The force on the current carrying conductor is a very well established phenomenon that can be explained by the Lorentz force on the moving charge in the conductor, or equivalently the interaction of the magnetic fields. Obviously you don't even have the high school level of physics, read Nelkon & Parker Advanced Level Physics Chapter 10, section "Interaction of Magnetic Fields". The explanation is right on the spot by the interaction of the external magnetic field and the magnetic field due to the current carrying wire. I suppose you would probably like to write to Nelkon & Parker to tell them they are wrong all the time. But I advise you to publish your theory in any respected journal first why magnetic fields are not interacting and there is no equivalent explanation for such an elemental physical observation!

2. I have just worked out a more rigorous mathematical derivation to show both models are exactly equivalent. It is ready to be published anytime but I won't until you show yours why they are not equivalent, and point out why the step by step logical mathemetical manipulation from Lorentz force to interacting magnetic fields are incorrect.

The evidence is clear and conclusive open to everyone with intellectual honesty to examine!

13. Sep 20, 2011

### entphy

I am truly aghast by how pathetic you are in this well respect forum that suppose to serve nothing but scientific truth. Obviously your understanding of physics is so poor or none but yet you are so ashame of being exposed, you had to resort to retoric to get your way out. Do you seriously think by insisting on your misconception you could turn the wrong to right? If you are so right about your non-interacting magnetic fields, why can't you argue with logic and with established scientific principle and point out the inconsistent part? Instead of offering such useless comment to "disprove" something?

I feel oblige to reply eventhough I found it extremely stupid to worth my time to respond at all, because I seriously do not want KOSS to be misled by you, and his/her students at stake!

My points as followed, and I bet you don't have the guts to confront them point by point logically except your hysterical retoric.

1. The force on the current carrying conductor is a very well established phenomenon that can be explained by the Lorentz force on the moving charge in the conductor, or equivalently the interaction of the magnetic fields. Obviously you don't even have the high school level of physics, read Nelkon & Parker Advanced Level Physics Chapter 10, section "Interaction of Magnetic Fields". The explanation is right on the spot by the interaction of the external magnetic field and the magnetic field due to the current carrying wire. I suppose you would probably like to write to Nelkon & Parker to tell them they are wrong all the time. But I advise you to publish your theory in any respected journal first why magnetic fields are not interacting and there is no equivalent explanation for such an elemental physical observation!

2. I have just worked out a more rigorous mathematical derivation to show both models are exactly equivalent. It is ready to be published anytime but I won't until you show yours why they are not equivalent, and point out why the step by step logical mathemetical manipulation from Lorentz force to interacting magnetic fields are incorrect.

The evidence is clear and conclusive open to everyone with intellectual honesty to examine!

14. Sep 20, 2011

### KOSS

Hi folks, yes I am still following this thread.

I thank both Phrak and entphys for their contributions. I'm not so sure there is a deep disagreement. Many physical problems have equally valid, yet seemingly different, explanations. Maybe there is just a langauge issue in the discussion above?

I'll try to summarize what I've learned form this thread so far. Correct me please if I'm still misguided :-)

A. Lorentz force explanation.
---------------------------
My take on Phrak is that "yes" magnetic fields do not interact by themselves, they add in superposition, just as entphys remarks. Both of you I think, agree that magnetic fields, once superposed, do act on moving charges. So we all agree that the Lorentz force explanation is valid. My present understanding of this explanation is that since without the external magnetic field the current carrying wire does not self-interact, the effect of adding the external magnetic field cannot (by linearity of superposition, i.e., adding up total field) result in any more of an effect than the Lorentz force due to the external B field---as if the field of the current carrying wire did not self-interact.

So teaching my students this version of the force on the wire is OK. Brilliant.

B. Field tension explanation.
---------------------------
Without resorting to tensors, entphys gives a sound derivation of the force on the wire in terms of the superposed fields. This is nice because it gives a concrete form of the Lorentz force in terms of the gradients of the fields. But in the absence of moving charges there is, of course no mass, hence no force, so this does not imply any magnetic field "interaction", it is simply an equivalent way of describing the Lorentz force without using the current, and instead using the magnetic field H generated by the current. All I need to do is explain to my students that to have the field H there needs to be a current, hence charges, hence a force on them---which can either be calculated simply using the Lorentz force and the Right Hand Rule, or equivalently, but by more complicated math, by using the B and H field gradients.

The picture implied by the textbook is thus not totally misleading, it is indeed "as if" there is a tension in the field lines, but of course these are fictions if we take QED seriously, which my students already understand, so all should be good for my classroom discourse I think.

Thanks to entphys, I can now explain this to my students in simple terms (sadly they are not familiar with vector algebra). In terms of physical analysis, I can say: "you have two options, either use the fact there is a current which cannot self-interact to get the force, OR consider the superposition of the fields but then the current is implied as the source of the inner field H, so you can't go and reintroduce the current, it drops out and is replaced by H in the force law in this view."

I might add that iif one used both, i.e., first calculating H and then also adding I again, you'd be over-counting the force! Indeed you'd go in a non-physical endless loop that way if you had to keep recalculating H after each re-inclusion of I, which is madness.

I should add that the real problem with the textbook is that it is jumping about two semesters worth of high school physics, since (a) the students are not yet exposed to vector algebra, and (b) they know the Biot-Savart law (constant I form), but were not this year expected to know Maxwell's equations nor Ampere's law. So it's a problem with the school curriculum (a perennial issue I have here with our public high school system). (I almost routinely get told off by higher "authorities" if I go ahead and teach my students extra physics, ..."they are not supposed to know this yet!" the senior teachers will say to me.)

PS. Maybe entphys could delete a couple of those repeat posts? (so it doesn't look like u r flaming) ;-)

15. Sep 20, 2011

### entphy

Hi Koss,

Actually I didn't post it 3 times intentionally. The fact is the system was stuck when the 1st post was uploaded so I shut down the browser and opened it again. It turned out then 3 posts were uploaded. I tried to delete the 2 repeated posts but could not find the delete/edit button. Any advice?

Just to clarify myself : the Lorentz force in the current carrying conductor does not come from the "self-interaction" of the magnetic field induced, that can't generate a force. If one just place a current carrying wire in space without any external magnetic field, there will be no force on the wire at all even though the magnetic field of the wire is still permeating the space.

But if there is an external magnetic field imposed, the 2 magnetic fields would then interact and resulted in the force. The magnetic fields are vectorial fields that can be added linearly, resulted in cancellation and reinforcement (be it complete or partial) over certain regions throughout the space. Now the net magnetic field will have some regions high in magnetic field density due to reinforcement, and some regions low in magnetic field density due to cancellation. High/low magnetific field density region is also high/low in magnetic energy density stored in space. If such non-uniformity in magnetic energy density is not symmetrically distributed in space, the gradient will manifest itself as a net force.

16. Sep 20, 2011

### davenn

yeah... if you click on the edit button for each of those posts in turn. When your in edit mode you will see a delete button at the bottom right of the text window.. then follow the commands :)

cheers
Dave

17. Sep 20, 2011

### entphy

Thanks Davenn, but I could only see the edit button for my latest post, not the rest of posts above. Not sure if this is due to my browser setting, but it seems that I could only edit the last post. :(

18. Sep 20, 2011

### KOSS

I just thought of another way to explain this qualitatively to high school students, they just need a smattering of special relativity. (Yeah, I'll get told off for doing this, but what the heck... my students can learn a little more than they are expected to know for exams, it won't kill them.)

Recall, this whole issue arose because of that annoying textbook attempt to "explain" the Lorentz force on a curent in a magnetic field as due somehow due to a superposed Bmagnet+Hwire field intensity gradient.

So anyway, suppose we consider the rest frame of the charges, all moving for simplicity in lock step along the wire. Then the charges only see an electric field from each other, which shoves them along the wire, not perpendicularly. Then we Lorentz transform the external B field, and we should get an electric field perpendicular to I and B, so this gives the upwards Lorentz force, it is a Coulomb force viewed from the charge rest frame. Any remaining B field component will have no effect in this frame since in this frame the charges have zero velocity. Done.

19. Sep 20, 2011

### KOSS

Thanks again entphy for the clarification.

Just so I get it all straight, my interpretation of the mathematics is as follows. The fact that there is an asymmetric field intensity initially does not necessarily imply any force right? There is a force only upon whatever might be a source of field, i.e., the wire and/or the magnet in this case. The force will act only on the sources, and will be such as to restore symmetry or equivalently lower overall energy in the field configuration.

It's probably a moot point, but if (for the sake of argument) there was an asymmetric magnetic energy density in empty space there would be nothing to act on to remove the asymmetry. What I mean is, any departure from minimum energy, or maximum entropy, can only be restored if there are sources to act upon. In this sense the fields themselves do not "interact", which is all I suspect Phrak was trying to say (perhaps?). How one would get an asymmetric energy distribution in the first place without sources is of course moot. (No flame bait intended ;-)

20. Sep 21, 2011

### entphy

I hope I have captured the question correctly. Yes, having a non-uniform field does not always result in net force. If one creates a non-uniform field using for example an odd asymmetric shape magnet, the force would be acting on itself and the action and reaction simply cancel each other out by Newton’s 3rd law of motion. Thus no net force or simply no force. To see this more clearly, consider a system of 2 parallel wire sections that carry currents (not necessary the same magnitude and direction), the forces on each of the wires are always the same and opposite in magnitude. The force could be obtained simply by considering the Lorentz force on one wire (or equivalently the interaction of 2 magnetic fields generated by both wires), and the force on the other wire be deduced from Newton’s 3rd law of motion directly. Of course, if one works out the force on the other wire applying the same Lorentz force concept, the force would just come out the same as concluded from Newton’s 3rd law of motion. They are consistent. And if one further expresses the magnetic field density due to one of the current carrying wire in terms of the current using Biot-Savart law, the expressions for the Lorentz forces on each wire would simply come out symmetrical. Now if both wires are actually connected in the same circuit, that the current flow continuously from one wire to another (simply configurable), one wouldn’t expect the whole circuit to move/rotate by itself simply because the 2 parallel sections in the circuit generate magnetic fields pushing/pulling each other because action and reaction cancel each other. But if they are not built on the same circuit but allows to move under the force, then they would move relative to each other depending on the relative directions of currents.
The bottom line is magnetic fields do interact and the field is simply the extension of the source, or how a source interact with its world. Whatever magnetic field it may be, it has to be traced to some sort of current source, be it spinning or orbiting electron or a collective motion of charge.
I would support your idea of exposing the high school student to a little of Special Relativity, anyway the math required is only algebra (Lorentz transformation) and it is more of a paradigm shift for the common sense in our low speed world. And if the introduction to Special Relativity intrigues the interest of student to explore further, that would be successful education.
But I am just not entirely sure if transferring to the rest frame of the moving electron would simplify the physics sufficiently. I would thought that in the rest frame of the electron, the apparent positive charge of the nucleus would be moving in the opposite direction, constituting the same current that also generates a magnetic field as far as the electron is concern in addition to the Lorentz transformation on the uniform external magnetic field. This is my 2 cents but you may have already thought through this more thoroughly to provide enlightenment.