# Magnetic field and respecetive force

1. Aug 10, 2013

### 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

Last edited: Aug 10, 2013
2. Aug 10, 2013

### Staff: Mentor

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

3. Aug 10, 2013

### joao_pimentel

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

Can you give me any formula or reference?

Thank you

4. Aug 10, 2013

### Staff: Mentor

See the link in my post.

5. Aug 10, 2013

### OmCheeto

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
6. Aug 10, 2013

### 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.

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

7. Aug 10, 2013

### 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?

8. Aug 10, 2013

### Staff: Mentor

@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.

9. Aug 11, 2013

### joao_pimentel

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

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 $B$

10. Aug 11, 2013

### Staff: Mentor

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.

11. Aug 11, 2013

### joao_pimentel

12. Aug 11, 2013

### Staff: Mentor

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
13. Aug 11, 2013

### 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