modeling the magnetic field of multiple magnet bars

laura

Hi all,

I have several magnets (bars) of well known size.

These magnets are placed on a table and they can rotate around their
center until they reach a stable state.

I want to know if there are some mathematical formulas which can
describe the interaction in this system.

I've found a formula which describes the force between two identical
cylindrical bar magnets placed end-to-end.

But, I need a more general formula where multiple magnets are involved
and are not always placed end-to-end.

Is there such formula?

thanks,
laura

HFarmer

laura wrote:

> Hi all,
>
> I have several magnets (bars) of well known size.
>
> These magnets are placed on a table and they can rotate around their
> center until they reach a stable state.
>
> I want to know if there are some mathematical formulas which can
> describe the interaction in this system.
>
> I've found a formula which describes the force between two identical
> cylindrical bar magnets placed end-to-end.
>
> But, I need a more general formula where multiple magnets are involved
> and are not always placed end-to-end.
>
> Is there such formula?
>

For a very large number of arbitrarily placed bar magnets? All I can think
of that is readily made would be the same analysis applied to the dipoles
in a ferromagnetic matterial. Where each magnet would act like an
individual dipole in a much larger composite system.

For a smaller number of magnets I would try the following. Assuming you
know the Hamiltonian formulation of mechanics... First we should write the
kinetic energy of the magnets in terms of their rotation.
T=1/2 I(/Omega)_i^2 Where each /Omega_i is the angular frequency of each
magnet. Then we need a Magnetic vector potential term which will contain
information on the changing magnetic field as each magnet rotates. Their
will also have to be a term that would relate how each magnets potential
energy would vary with respect to the other magnets due to its position on
the table.... This can get messy.

Ultimately the only way I know to handle an assembly of dipoles is with stat
mech. I survived stat mech with a B last year and I never want to face it
again unless forced to.:) Consult a stat. mech. book

Perhaps if you provide more specific information I can provide a more
specific answer.

> thanks,
> laura

--
"....a good profession as long as you don't earn a living at it." A.E
www.geocities.com/hontasfx

Igor Khavkine

laura wrote:
> Hi all,
>
> I have several magnets (bars) of well known size.
>
> These magnets are placed on a table and they can rotate around their
> center until they reach a stable state.
>
> I want to know if there are some mathematical formulas which can
> describe the interaction in this system.

It depends on what kind of magnet placement interests you. If your
magnets are small compared to the separation between them, then you
might be satisfied with the magnetic dipole approximation. Presume each
magnet is represented by a point-like dipole. Then you can calculate
the magnetic field produced by each magnet and the potential energy of
each magnet will be proportional to (m.B) (m - magnetic moment, B -
magnetic field at its location). The equations of motion for the
magnets can then be extracted from the combination of their potential
and kinetic energies.

If that approximation is not good enough for your purposes and you care
about the dimensions of the magnets. Then you may have to do something
much more sophisticated, like solving the magnetostatic field equations
(most likely numerically) for a given magnet arrangement. There are
software packages specialized for this purpose.

Hope this helps.

Igor

Ken S. Tucker

laura wrote:
> Hi all,
>
> I have several magnets (bars) of well known size.
>
> These magnets are placed on a table and they can rotate around their
> center until they reach a stable state.

> I want to know if there are some mathematical formulas which can
> describe the interaction in this system.
>
> I've found a formula which describes the force between two identical
> cylindrical bar magnets placed end-to-end.
>
> But, I need a more general formula where multiple magnets are involved
> and are not always placed end-to-end.
>
> Is there such formula?
>
> thanks,
> laura

I'm thinking you might treat each pole of the
bar magnet as a "monopole", much like you
could assign them "+" or "-" charge.

>From that, you set a fixed constant Radius "R"
between the monopoles at each end of the bar,
and, as you specify, they rotate about a center,
which defines all their possible positions.
The variables to be solved are the differences
in the radius "r" separating different bars
monopoles since "R" is fixed.

If I understand the physics correctly, the bar
configuration will assume a least potential energy
geometry.
Following electrostatics, that energy is proportional
to the product of the monopole strengths and
inverse to radius, (force is inverse to radius squared).

I think that provides enough information to produce
a specific energy equation (Igor or Timo might correct
me) that can then be minimized using standard
"maximum-minimum" differential calculus to compute
the "r"s.

Is that a viable procedure?
Regards
Ken S. Tucker

HFarmer

Ken S. Tucker wrote:

>
> laura wrote:
>> Hi all,
>>
>> I have several magnets (bars) of well known size.
>>
>> These magnets are placed on a table and they can rotate around their
>> center until they reach a stable state.
>
>> I want to know if there are some mathematical formulas which can
>> describe the interaction in this system.
>>
>> I've found a formula which describes the force between two identical
>> cylindrical bar magnets placed end-to-end.
>>
>> But, I need a more general formula where multiple magnets are involved
>> and are not always placed end-to-end.
>>
>> Is there such formula?
>>
>> thanks,
>> laura
>
> I'm thinking you might treat each pole of the
> bar magnet as a "monopole", much like you
> could assign them "+" or "-" charge.
>
>>From that, you set a fixed constant Radius "R"
> between the monopoles at each end of the bar,
> and, as you specify, they rotate about a center,
> which defines all their possible positions.
> The variables to be solved are the differences
> in the radius "r" separating different bars
> monopoles since "R" is fixed.

You left out the rotational variable of each and every bar magnet.
Every single one of them would need a variable to describe their angular
displacement. There would also be a term for the rotational energy of the
bars. If we neglect any frictions that are in the system then this becomes
even more important. As one magnet swigs it will perturb others and
perhaps cause them to swing. So on and so forth. Also your "r" would vary
with respect to the angular varyable for each pole of the magnet. There
would be cross terms in the Hamiltonian. Cross terms for each and every
magnets relationship to each and every other magnet... So on and so forth.

I would really like to know approximately how many magnets are we talking
about? If it is say ten. Then this problem can be solved. I cringe at
how hard it would be. If it's more than that I really think you would be
better served with statistics. Is it really mission critical to your
application to have full knowledge of where each and every pole is at all
times?

Here is how you would do this statistically. First you set up a hamiltonian
for the simplest part of the system. Say three magnets interacting on your
table. Write the Hamiltonian operator and solve for the eigenstates. Use
the spectrum of the Hamiltonians eigenstates to construct the partition
function of the system ( Z ). From this one can model all the macroscopic
properties of the system as a whole. Magnetizeation, Energy (temperature)
etc. A good reference just for a start would be
http://en.wikipedia.org/wiki/Statistical_mechanics.

A better reference would be a text book.

Is this anywhere near what you are looking for?


>
> If I understand the physics correctly, the bar
> configuration will assume a least potential energy
> geometry.
> Following electrostatics, that energy is proportional
> to the product of the monopole strengths and
> inverse to radius, (force is inverse to radius squared).
>
> I think that provides enough information to produce
> a specific energy equation (Igor or Timo might correct
> me) that can then be minimized using standard
> "maximum-minimum" differential calculus to compute
> the "r"s.
>
> Is that a viable procedure?
> Regards
> Ken S. Tucker

--
"....a good profession as long as you don't earn a living at it." A.E
www.geocities.com/hontasfx