Magnetic field and respecetive force

[joao_pimentel]

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

 

[mfb]

It is complicated
"Metal" is not interesting - I guess you mean "ferromagnetic" (only some materials are ferromagnetic, iron is the most relevant example)

 

[joao_pimentel]

sorry. Yes I meant a ferromagnetic material like iron for example.

Can you give me any formula or reference?

Thank you

 

[mfb]

See the link in my post.

 

[OmCheeto]

It is complicated
"Metal" is not interesting - I guess you mean "ferromagnetic" (only some materials are ferromagnetic, iron is the most relevant example)

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" )

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.

 

[joao_pimentel]

So I suppose this is my answer

$$\mathbf{F}=\nabla \left(\mathbf{m}\cdot\mathbf{B}\right)$$

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

I suppose as well that $\mathbf{B}$ 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 $\mathbf{m}$ 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

 

[joao_pimentel]

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

 

[mfb]

@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 µ_r missing somewhere.

 

[joao_pimentel]

This leads to $F=\nabla (cB^2) = 2c B \nabla |B|$ (check this!)

$F=\nabla (cB \mathbb{.}B)=\nabla (c|B|^2)=c\nabla (|B|^2)=c\sum_{k=1}^3\frac{\partial (|B|^2)}{x_k}\mathbb{\vec{e_k}}=c\sum_{k=1}^3 2 |B|\frac{\partial (|B|)}{x_k}\mathbb{\vec{e_k}}=2c|B|\sum_{k=1}^3 \frac{\partial (|B|)}{x_k}\mathbb{\vec{e_k}}=2c|B|\nabla (|B|)$

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 $B$

 

[mfb]

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.

 

[joao_pimentel]

Can you kindly reference me a good sketch, cause honestly I can't distinguish good form bad sketch, when it concerns magnetic fields.

 

[mfb]

http://magician.ucsd.edu/Essentials_2/WebBook2ch1.html#x3-50001.3 looks reasonable for a bar magnet. As you can see, the magnetic field is very strong close to its poles, and weaker elsewhere.

 

[joao_pimentel]

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