Force between two bar magnets
The Gilbert model assumes that the magnetic forces between magnets are due to magnetic charges near the poles. This model produces good approximations that work even close to the magnet when the magnetic field becomes more complicated, and more dependent on the detailed shape and magnetization of the magnet than just the magnetic dipole contribution. Formally, the field can be expressed as a multipole expansion: A dipole field, plus a quadrupole field, plus an octopole field, etc. in the Ampère model, but this can be very cumbersome mathematically.
Calculating the attractive or repulsive force between two magnets is, in the general case, an extremely complex operation, as it depends on the shape, magnetization, orientation and separation of the magnets. The Gilbert model does depend on some knowledge of how the 'magnetic charge’ is distributed over the magnetic poles. It is only truly useful for simple configurations even then. Fortunately, this restriction covers many useful cases.
The force between two identical cylindrical bar magnets placed end to end at great distance x>>R is approximately calculated by the shown formulaRelated formulas
|F||Force between two bar magnets (N)|
|B0||flux density very close to each pole (T)|
|A||area of each pole (m2)|
|L||length of each magnet (m)|
|R||radius of each magnet (m)|
|μ0||Vacuum permeability (permeability of free space, permeability of vacuum or magnetic constant)|
|x||the separation between the two magnets (m)|