Equation you can get from given link.
https://en.wikipedia.org/wiki/Force_between_magnets
Dear Darshan,
The best way to get the differential equation like the one between two magnets is Gilbert model .
The attached publication covers the required answer.
Mansuripur’s paradox
David J. Griffiths∗
A recent article claims that the Lorentz force law is incompatible with special relativity. We discuss the “paradox” on which this claim is based. The resolution depends on whether one assumes a “Gilbert” model for the magnetic dipole (separated monopoles) or the standard “Amp`ere” model (a current loop). The former case was treated in these pages many years ago; the latter, as several
authors have noted, constitutes an interesting manifestation of “hidden momentum.”
II. GILBERT DIPOLES: NAMIAS’S RESOLUTION
First the paradox: In S0
(the “proper” frame) there is an ideal magnetic dipole m = m0 ˆx at (0, 0, d), and a
point charge q at the origin, both at rest. The torque on m is (obviously) zero. Now examine the same configuration in S (the “lab” frame), with respect to which
S 0 moves at constant speed v in the z direction (Fig. 1).In S the (moving) point charge generates electric and magnetic fields.
For more details, please see attached file.
Hoping this will be helpful,
Rafik
Regarding magnetic interaction between magnets, you may wish to relate theoretical hypothesis with confirmed experiments.
The ad hoc experimentally derived equation that accounts for magnetic interaction between 2 bar magnets is given in Resnick & Halliday. Physics. John Wyley & Sons, New York, 1967. (page 93):
$
F = \frac{3 \mu_0 \mu^2}{2 \pi d^4}
$
and the ad hoc experimentally derived equation for flat circular loudspeaker magnets is:
$
F = \frac{3 \mu_0 \mu^2}{4 \pi d^4}
$
As reported in 2013 in this paper:
Michaud A (2013)"On The Magnetostatic Inverse Cube Law and Magnetic Monopoles". International Journal of Engineering Research and Development e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com. Volume 7, Issue 5 (June 2013), PP.50-66.
http://www.ijerd.com/paper/vol7-issue5/H0705050066.pdf
Both reveal an inverse cube magnetic interaction between magnets.
The following experiment reported in 2014 in Nature magazine regarding two interacting electrons confirms the inverse cube relation between the magnetic aspects of electrons:
Shlomi Kotler et al. (2014) Measurement of the magnetic interaction between two bound electrons of two separate ions. Nature magazine. doi:10.1038/nature13403. Macmillan Publishers Ltd. 19 June 2014, Vol. 510, pages 376 to 380.
http://www.nature.com/articles/nature13403.epdf?referrer_access_token=yoC6RXrPyxwvQviChYrG0tRgN0jAjWel9jnR3ZoTv0PdPJ4geER1fKVR1YXH8GThqECstdb6e48mZm0qQo2OMX_XYURkzBSUZCrxM8VipvnG8FofxB39P4lc-1UIKEO1
Note that the circular magnet equation is a far fields equation that directly applies to pointlike behaving electromagnetic particles such as electrons.
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