The magnetic field of larger magnets can be obtained by modeling them as a collection of a large number of small magnets called dipoles each having their own m. The magnetic interaction is described in terms of a vector fieldwhere each point in space and time is associated with a vector that determines what force a moving charge would experience at that point see Lorentz force.
The "magnetic flux density". If a ferromagnetic object such as a piece of iron is brought into a magnetic field, the "magnetic force" exerted on that object is directly proportional to the gradient of the magnetic field strength where the object is located.
Since it is based on the fictitious idea of a magnetic charge density, the pole model has limitations. Alternatively, the magnetic field can be defined in terms of the torque it produces on a magnetic dipole see magnetic torque on permanent magnets below.
First, measure the strength and direction of the magnetic field at a large number of locations or at every point in space. If a magnetized object is divided in half, a new pole appears on the surface of each piece, so each has a pair of complementary poles.
For simple magnets, m points in the direction of a line drawn from the south to the north pole of the magnet.