What is the force between two magnets and how is it calculated?

[Hassan2]

When two cylindrical magnets are brought close together so that the North pole of one is close to the south pole of the other, there is an attraction force between them.

But what is the nature of this force?

I used to think that this force is nothing but Lorentz force that the magnetic field of one magnet exerts on the the equivalent current densities of the other. If it was so, there must be no force on the circular face of the magnet because the net current density is zero there.

So what is the nature of the fore?

The force is usually understood and calculated from the energy points of the view and the principle of virtual work. Can it be explained by Maxwell's equations as well?

Thanks.

 

[Drakkith]

I'm not quite sure what you are asking. Maxwells equations describe the electromagnetic force, which includes your standard everyday magnet. What do you mean by "the nature of the force"? Could you elaborate on what you are asking?

 

[Hassan2]

Thanks Darkkith,

How can we calculate the local magnetic force (magnetic force on each point of the magnet) using Maxwell's equations?

In case the the example's magnets, the forces are outward normal to the close surfaces.

Added: The question is general for magnetic force and the magnets example is just to simplify the problem.

 

[Drakkith]

Thanks Darkkith,

How can we calculate the local magnetic force (magnetic force on each point of the magnet) using Maxwell's equations?

In case the the example's magnets, the forces are outward normal to the closes surfaces.

I don't think I can answer that, as I don't know how. All I could suggest is the wikipedia article here:
http://en.wikipedia.org/wiki/Maxwells_equations
http://en.wikipedia.org/wiki/Gauss's_law_for_magnetism

Do you know the math required to do the calculations?

 

[Hassan2]

Thanks.
But the links are not about magnetic force but about magnetic ( electromagnetic) field.
I have been working on this topic for about one year and have enough experience in the math.

One of the link says the Maxwell's equations together with the Lorentz force law, form the foundation of classical electrodynamics. That means we should be able to describe the force using Lorentz force law. But I can't find out how.

 

[Simon Bridge]

To understand how to use Maxwels equations to get the force of one permanent magnet on another you need to understand how the permanent magnets get to be magnetic in the first place. Note - the magnetic force is a special case of the electromagnetic field.

For general magnetic fields it is a matter of solving the equations with the parameters of the system as boundary conditions.

 

[Hassan2]

I have an average level of knowledge of magnetism in microscopic scale ( micromagnetics) and I think is enough to understand magnetic the force but I can't figure it out yet.

Magnetic flux lines can be visualized by distribution of iron filings on a flat sheet pierced by a wire carrying a current. In this case, the small pieces of iron are first magnetized and virtually become magnets, then the torque applied on these small magnets by the magnetic field aligns them in the direction of the field, but it doesn't apply a net force on them. For example if the paper had zero frictions, the iron filings wouldn't move along the flux lines.

If I know the distribution of the magnetic field in a piece of iron, I can use the Maxwell stress tensor and calculate the total force on it. But for example in deformation analysis, I need to have local forces. I am also aware of some methods to calculate the local force too but I am looking for a physical meaning of these methods.

If a cylinder made of a soft magnetic material put close of a permanent magnet with with its circular face close to one of the magnet's pole, the methods predict that the face would protrude ( bulge). I can't find out why there are forces on the ( center ) of the face.

Thanks

 

[Simon Bridge]

Presumably you realize there have to be forces on the center of the face: it is in a magnetic field! So I figure you don't understand how the distribution of forces across the surface lead to a bulge?

the methods predict

What methods? How would you go about determining that the "face would protrude" ... that usually give you a hint as to how this happens. "Why" is a different kind of question.

How are you used to modelling macroscopic magnets in micromagnetics?
Meantime Maxwle's Equation for permanent magnets. (Multipole model.)

 

[marcusl]

The phenomena that give rise to magnetism in a solid are electron spin and long-range interactions, for which quantum mechanics is required to adequately describe them. The electromagnetic force is mediated (in quantum electrodynamics) by photons.
In the classical model, the magnetic force is just the electric Coulomb force described by special relativity as seen in a moving frame. It is more convenient, however, to talk of elemental dipoles dispersed in the medium. And for calculational purposes, we envision an effective (but fictitious) surface current flowing on the outside of the cylindrical magnet, or equivalently, an effective (but fictitious) surface magnetic charge on each pole face. The attraction between two magnets is then calculated as the attraction between the effective currents or pole charges.

 

[Hassan2]

Well my confusion is about the relation between the magnetic field and the force which it exerts on a piece of iron.

The net force excreted on a cube of iron in a uniform field is zero. But there are forces on the opposite faces of the cube which cancel each other out.

We have Lorenz law which relates the force and B and J ( current density), but this law seems to be irrelevant here.

Do we have another low for magnetic force, for example relating M ( magnetization ) and B field?

One of methods I referred to is based on the principle of virtual work but they may only be applicable to numerical methods. For example in the finite element method, choose a node on the face, and give it a virtual displacement δu. Only the neighboring neighboring elements would change. This causes a change δW in the magnetic energy. then F=-δW/δu.

 

[Hassan2]

Thanks marcusl,

You answered my question. In elementary textbooks, this topic is not touched properly.

Thanks again.

 

[Dickfore]

It's a tough problem. Permanent magnets are to be treated as rigid bodies with, in general 6 degrees of freedom each. They not only exert force, but a torque on each other (try putting them at an angle with the same poles near each other and see what happens).

Two permanent magnets are characterized by permanent magnetization (magnetic moment per unit volume) M.

You can start with the formula for the potential energy between two point dipoles $\mathbf{m}_1$, and $\mathbf{m}_2$, separated by a radius-vector $\mathbf{x}$:
$$U_{12} = \frac{\mu_0}{4 \pi} \, \frac{\mathbf{m}_1 \cdot \mathbf{m}_2 - 3 (\mathbf{m}_1 \cdot \hat{\mathbf{x}})(\mathbf{m}_2 \cdot \hat{\mathbf{x}})}{\vert \mathbf{x} \vert^3}$$
Then, use the principle of superposition to integrate over the volume of each magnet:
$$U = \frac{\mu_0}{4 \pi} \, \int_{V_1}{\int_{V_2}{\frac{\mathbf{M}_1 \cdot \mathbf{M}_2 - 3 (\mathbf{M}_1 \cdot \hat{\mathbf{x}})(\mathbf{M}_2 \cdot \hat{\mathbf{x}})}{\vert \mathbf{x} \vert^3} \, d^3x_2} \, d^3x_1}, \ \mathbf{x} = \mathbf{x}_0 + \mathbf{x}_2 - \mathbf{x}_1$$

This potential energy is a function of the mutual separation $\mathbf{x}_0$ of two fixed points on each magnet, say, their centers of mass, as well as the angles a fixed coordinate system attached to each of the magnets builds with a laboratory coordinate system. Differentiating w.r.t. these variables yields the force, as well as the respective torques.