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Magnetic field and respecetive force

  1. Aug 10, 2013 #1
    Magnetic field of a permanent magnet and respecetive force

    Hello

    I just want you to explain me a bit of physics, cause I am a lay.

    How to relate (which formula) the magnetic field of a permanent magnet, the vector B at each point (x,y,z), with the force applied to a certain particle of metal, with no speed, within that field?

    Thank you
     
    Last edited: Aug 10, 2013
  2. jcsd
  3. Aug 10, 2013 #2

    mfb

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    It is complicated
    "Metal" is not interesting - I guess you mean "ferromagnetic" (only some materials are ferromagnetic, iron is the most relevant example)
     
  4. Aug 10, 2013 #3
    sorry. Yes I meant a ferromagnetic material like iron for example.

    Can you give me any formula or reference?

    Thank you
     
  5. Aug 10, 2013 #4

    mfb

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    See the link in my post.
     
  6. Aug 10, 2013 #5

    OmCheeto

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    I think it's even more complicated than that. That link is about forces between magnets.

    I just did an experiment with my 1 Tesla rare earth magnets and a pair of nails.

    The nails weigh about 1/2 gram, are 1.5 cm long, 2 mm in diameter, and were originally not magnetized.

    The magnet is a smooth edged cube, the edges measuring roughly 4.1 mm.

    The magnet is able to lift a nail off the table from a distance of 1 cm.

    The magnet nail combination was not able to life the 2nd nail from the table until the distance was ≈1 millimeter.

    When the magnet was removed from the first nail, the nails stayed attached. I had created magnets!

    Trying to determine the strength of the residual nail flux density, I was only able to determine that a separation of 0.1 mm resulted in nail #1 not being able to support nail #2.

    The last measurement I did, was to remove the magnet, flip the poles, and slowly bring it towards the nails. When the magnet was 2.5 cm from nail #1, nail #2 was released. I'm guessing that the field strength of the two nails can be deduced from this measurement. (Perhaps I should turn this problem over to micromass, for another "Math Challenge" :tongue2: )

    The nails were still both magnetized after this portion of the experiment, as each could support the others weight.

    But introducing unmagnetized nail #3, neither was able to budge it.

    Anyways, the problem with this problem, as I see it, is that the magnetization of the ferromagnetic material is influenced, and changed by the permanent magnets, making this a really dynamic problem. If I flip the poles of the permanent magnet, and bring it to the two nails, their magnetic fields reverse.

    Problems with this experiment:
    Like many nails, these had flat heads and pointy tails. Geometry is probably critical.
    When I find my dremel tool, I'll redo the experiment.
     
    Last edited: Aug 10, 2013
  7. Aug 10, 2013 #6
    So I suppose this is my answer

    [tex]\mathbf{F}=\nabla \left(\mathbf{m}\cdot\mathbf{B}\right)[/tex]

    where [itex]\mathbf{m}[/itex] is the vector of the magnetic dipole moment, which has the direction from south pole to north magnetic pole.

    I suppose as well that [itex]\mathbf{B}[/itex] at each euclidean point is the tangent of all those lines we see going around the magnet.

    http://upload.wikimedia.org/wikipedia/commons/b/bb/Magnetic_field_due_to_dipole_moment.svg

    Can you provide me any image with the forces at each euclidean point, considering that [itex]\mathbf{m}[/itex] doesn't change neither direction nor magnitude, as it is not intuitive to calculate the dot product and the respective gradient?

    Thank you

    PS: Please correct me if anything is wrong
     
  8. Aug 10, 2013 #7
    PS: I can see that close to poles, F is higher as m is aligned with B (cross product is maximum) and there is a great change in B, which provokes the gradient to be high in magnitude, but it would be nice to see one picture of the vector F at each point :)
    Can you give any reference?
    Thanks in advance
     
  9. Aug 10, 2013 #8

    mfb

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    @joao_pimentel: Careful, m is from one object and B is from the other object.

    If your piece of iron/steel is not magnetized initially, its m will depend on the position. As an approximation, it will be proportional to B coming from the magnet (as long as B is not too strong, ~2T for iron/steel).

    This leads to ##F=\nabla (cB^2) = 2c B \nabla |B|## (check this!)
    Looking at the dimensions, I expect that c is a multiple of V/µ0 where V is the volume of the magnet and µ0 is the vacuum permeability. There might be a factor of µr missing somewhere.
     
  10. Aug 11, 2013 #9
    ##F=\nabla (cB \mathbb{.}B)=\nabla (c|B|^2)=c\nabla (|B|^2)=c\sum_{k=1}^3\frac{\partial (|B|^2)}{x_k}\mathbb{\vec{e_k}}=c\sum_{k=1}^3 2 |B|\frac{\partial (|B|)}{x_k}\mathbb{\vec{e_k}}=2c|B|\sum_{k=1}^3 \frac{\partial (|B|)}{x_k}\mathbb{\vec{e_k}}=2c|B|\nabla (|B|)##

    Considering ##2c|B|## a real positive number, the only term which will change the direction of ##F## is ##\nabla|B|##. Though, I cannot see how ##|B|## changes over space, because those lines in the pictures don't give notion of magnitude of ##B##
     
  11. Aug 11, 2013 #10

    mfb

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    With a good sketch and as a rough estimate, a high line density corresponds to a large |B|.
    For a real magnet, you need some map of the field strength.
     
  12. Aug 11, 2013 #11
  13. Aug 11, 2013 #12

    mfb

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    http://magician.ucsd.edu/Essentials_2/WebBook2ch1.html#x3-50001.3 [Broken] looks reasonable for a bar magnet. As you can see, the magnetic field is very strong close to its poles, and weaker elsewhere.
     
    Last edited by a moderator: May 6, 2017
  14. Aug 11, 2013 #13
    Hi, thank you very much for reference, nevertheless I suppose I won't be able to trace the directions of F at each point. I'll continue searching if I find anything
     
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