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The magnetization field or M-field can be defined according to the following equation: = Where d m {\displaystyle \mathrm {d} \mathbf {m} } is the elementary magnetic moment and d V {\displaystyle \mathrm {d} V} is the volume element ; in other words, the M -field is the distribution of magnetic moments in the region or manifold concerned.
The formula needed in this case to calculate m in (units of A⋅m 2) is: =, where: B r is the residual flux density, expressed in teslas. V is the volume of the magnet (in m 3). μ 0 is the permeability of vacuum (4π × 10 −7 H/m). [6]
The energy per unit volume in a region of free space with vacuum permeability containing magnetic field is: = More generally, if we assume that the medium is paramagnetic or diamagnetic so that a linear constitutive equation exists that relates and the magnetization (for example = / where is the magnetic permeability of the material), then it ...
Calculating the attractive or repulsive force between two magnets is, in the general case, a very complex operation, as it depends on the shape, magnetization, orientation and separation of the magnets. The magnetic pole model does depend on some knowledge of how the ‘magnetic charge’ is distributed over the magnetic poles.
Continuous charge distribution. The volume charge density ρ is the amount of charge per unit volume (cube), surface charge density σ is amount per unit surface area (circle) with outward unit normal n̂, d is the dipole moment between two point charges, the volume density of these is the polarization density P.
The magnetocrystalline anisotropy energy is generally represented as an expansion in powers of the direction cosines of the magnetization. The magnetization vector can be written M = M s (α,β,γ), where M s is the saturation magnetization. Because of time reversal symmetry, only even powers of the cosines are allowed. [2]
On top of the applied field, the magnetization of the material adds its own magnetic field, causing the field lines to concentrate in paramagnetism, or be excluded in diamagnetism. [1] Quantitative measures of the magnetic susceptibility also provide insights into the structure of materials, providing insight into bonding and energy levels.
The magnetization is the negative derivative of the free energy with respect to the applied field, and so the magnetization per unit volume is = , where n is the number density of magnetic moments. [1]: 117 The formula above is known as the Langevin paramagnetic equation.