magnetic charge on magnetic poles (SI unit: ampere-meter) μ is the permeability of the intervening medium (SI unit: tesla meter per ampere, henry per meter or newton per ampere squared) r is the separation (SI unit: meter). The pole description is useful to practicing magneticians who design real-world magnets, but real magnets have a pole distribution more complex than a single north and south. Therefore, implementation of the pole idea is not simple. In some cases, one of the more complex formulas given below will be more useful. Force between two nearby magnetized surfaces of area A The mechanical force between two nearby magnetized surfaces can be calculated with the following equation. The equation is valid only for cases in which the effect of fringing is negligible and the volume of the air gap is much smaller than that of the magnetized material, the force for each magnetized surface is: where: A is the area of each surface, in m2 H is their magnetizing field, in A/m. μ0 is the permeability of space, which equals 4π×10−7 T·m/A B is the flux density, in T The derivation of this equation is analogous to the force between two nearby electrically charged surfaces, which assumes that the field in between the plates is uniform. Force between two bar magnets The force between two identical cylindrical bar magnets placed end to end at great distance is approximately: where B0 is the flux density very close to each pole, in T, A is the area of each pole, in m2, L is the length of each magnet, in m, R is the radius of each magnet, in m, and x is the separation between the two magnets, in m relates the flux density at the pole to the magnetization of the magnet. Note that these formulations assume point-like magnetic-charge distributions instead of a uniform distribution over the end facets, which is a good approximation only at relatively great distances. For intermediate distances, numerical methods must be used. Force between two cylindrical magnets For two cylindrical magnets with radius , and length , with their magnetic dipole aligned, the force can be computed analytically using elliptic integrals. In the limit , the force can be approximated by, Where is the magnetization of the magnets and is the distance between them. For small values of , the results are erroneous as the force becomes large for close-to-zero