Typically, the magnetic force between two magnets would be calculated by the following formalism, and with this knowledge.. Firstly,  magnets have magnetic fields leaving their poles and extending into space. When only a single magnetic, say A, exists, all of its magnetic field exits one pole and the field fills space, predominantly near the magnet. It then is drawn to the other pole where it is absorbed. These fields, in accord with the maxwell stress tensor can be used to calculate forces. For the single magnet, it results in a pull (tension) on the pole faces, which is balanced by the magnets (iron) physical strength to handle this tension.

When a second magnet, say B, is added to the system, now the field lines can either go to the same magnet, or to the opposite magnetic A. Now a number of field lines go between A and B. Magnetic field lines have stresses in accord with the Maxwwell stress tensor: a tension (pull) along the field lines, and an outward pressure in the perpendicular direction (which is why the fields extend widely into space - the fields push outward in this direction.

To calculate the force between the two magnetics: one must integrate over the field structures, so as to examine the connecting field lines. The force is the sum of the tension forces (B^2/8 pi) on the field lines between the two. If the magnets are just straight bars, and anti-alligned in space, the two magnets attract each other ,because of the pull along the fields yield a tension between the two. If the bars are placed in the same direction, then the fields do not connect, and the perpendicular maxwell stress tensor in this case, yields a repulsive force, of comparable magnitude. This can also be calculated by the perpendicular tension between the two sets of field lines.

If you want to be precise, then it's probably not such an easy calculation to do!

Notice that a magnetic moment does not experience a net force in a homogeneous magnetic field. Rather, the force (which is undoubtedly present :-)  is given by the product of moment and field gradient. So you would have to be able to compute the field distribution (in the presence of the magnet, the exact form of which may be subject to the value of the magnetic field in case the (differential)susceptibility of the magnet is nonzero, and then integrate the resulting thing over the volume of the magnet.

For a simple estimation, assume that the field distribution of the electromagnet is unperturbed by the presence of the magnet and that the magnet can be represented by its total magnetic moment (magnetization times volume), at the position of the center of mass of the magnet. In this simplified form, the product of field gradient and total moment would be what you are looking for [to be quantitative, you have to get your units right and consistent].

As to what the gradient of the magnetic field amounts to, it will depend on the geometry and making of the electromagnet. You would either have to model or to measure it. Probably, there do exist approximative formulae for some typical geometries. Again, the very much simplest version would be to represent the magnet by a point dipole and to associate its field with the dipole field of this point dipole positioned at its center. But this, clearly is only valid in the far field and most probably not at the distances you're interested in.

I think both our answers, mine and Kai's have some good aspects and bad aspects.

As far as mine goes, it relies upon an integration over the field line structure, which is a more complicated , but from my point of view, needed. aspect to obtain an "exact" solution.

Kai's answer is simpler, but i am afraid might not work for the configuration chosen.

Namely, a point dipole magnet experiences no net force when it sits in a uniform field. It only experiences a torque.in such a field (this would be the twist which twists a compass magnet to orient itself to align towards the magnetic poles when on Earth.

Ken, we seem to have been writing at the same time... 

Since a moment experiences no force, the field gradient is needed (I described that).

And yes, with the rather close proximity between the two objects, my approximative approach may result in a lousy approximation.

What is suitable will also depend on what (level of accurateness) the OP is after.

Yet another approach could be to determine/calculate B(r) and H(r)  [by whatever method and level of precision] for the two objects separately, superimpose (add) the fields for a given relative position and calculate the magnetostatic energy from integrating over B(r)H(r) [mind any prefactors to get the scales and units right]. Then change the relative position a bit and repeat the calculation. Divide the energy difference by the distance to get the component of the force along that path.

I have no idea whether that's more or less work to do than what you proposed, Ken. It should, however be equivalent in the end....

I myself use the integral method by summing the contribution of each equivalent bonded magnetic current (Ampere equivalence) or magnetic pole (Coulomb equivalence) to the total magnetic fields H and B. The force can then be obtained thanks to a second surface integration using the generalized Maxwell stress tensor. You can find these methods in the litterature but feel free to contact me on this subject if you want.

Dear M Harshad Kamble

A few years ago, I have done the calculations of forces induced by magnetic fields (The nature of the sources of fields has no importance for the calculation). The best way is the use of the energy principle. This method uses no simplification.

But, if you want to use this method, you have to use finite element method.

If you have not such a software. A free software exists and it is limited to 2D modeling. That will not be an issue if your study can be assimilated to a 2D axisymmetric problem.

Let me know if you want additional information.

Best regards

Harshad Kamble: Have a look at FEMM free software package.

Please have a look at the recent paper:

Parametric resonance induced chaos in magnetic damped driven pendulum by Giorgi Khomeriki, http://arxiv.org/abs/1511.04593

The calculations presented there are simplified but nevertheless quite useful for the situation described, similar to yours.

Thank you all for your reply and suggestions!!!

Basically, the magnetic field carries tension (and pressure) forces

with the maxwell stress tensor showing the values....

if u just have a surface (half way between the magnets, where the field

is all aligned between the two), then just use surface tension = B^2/4 pi,

and add up the B values *area and that is the result..

the particular source of the magnetic field is a red herring. (unimportant)...

and distance apart is small so not important..-- fields are assumed aligned between

the two.(perpendicular to face of magnets)..

                     Dear Harshad

To calculate the force is not  enough to know the magnitude of the magnetic field of the permanent magnet at one point. It is necessary to calculate the magnetic field of the permanent magnet in the whole space. This magnetic field will depend on the shape of the magnet, its size ...... .In any textbook you will find the formula for the force, which acts on the current in the magnetic field. There will be an integral ......

I do not write in more detail, as the calculation of the field of permanent magnet system is well-known problem. On this subject there exist thousands of published works. I and my staffs conducted dozens of such calculations for the purpose of magnetic tomography. Follow the advice of Johan K. Frémery use FEMM. But FEMM solve two-dimensional problems. Use ANSYS.

"FEMM solve two-dimensional problems ..."

... and rotationally symmetric 3D problems as well, and unlike ANSYS is for free!

Yes Sir Fremerey. I agree with you. 

Magnetic force  F=grad(mB) where B is field strength from electromagnet and m is dipol moment of permanent magnet. For approximation dipole-dipole interaction approach can be used

Hello. I am looking for the same thing. Did you find the response? If you have a publication already, please let me know in order to search it. Thanks in advance.

Hi 

i utilised COMSOL platform to obtain the force. Please find the

Article Modelling of an uniaxial single-sided magnetically actuated ...

This problem is not essentially different from the more fundamental problem of calculating the force between magnets in general. In that case, we have summarized the essential formulary in this short presentation:

Presentation Why do magnets attract or repel?

hi

i would like to know if you find out the solution.

i have a project and i am going to generate vibration through magnet and an electromagnet. could you please explain for me how can i calculate force between magnet and electromagnet.(i want to generate 700hz and 0.5mm amplitude vibration)

Any suggestions will be highly appreciated.

In order to calculate the force between a permanent magnet and an electromagnet, try to characterize them by their respective magnetic dipole moments and use Eq. (15) of our review:

Presentation Why do magnets attract or repel?