Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.) Granted Application number EP06829239A Other languages German ( de) Google Patents Magnetic levitation system The approximation of a medium or an array of charges by only dipoles and their associated dipole moment density is sometimes called the point dipole approximation, the discrete dipole approximation, or simply the dipole approximation.- Google Patents EP1958323A1 - Magnetic levitation system In a common model of this type, the distant charges are treated as a homogeneous medium using a dielectric constant, and the nearby charges are treated only in a dipole approximation. The nearby charges then give rise to local field effects.
A related approach is to divide the charges into those nearby the point of observation, and those far enough away to allow a multipole expansion. A more general version of this model (which allows the polarization to vary with position) is the customary approach using electric susceptibility or electrical permittivity.Ī more complex model of the point charge array introduces an effective medium by averaging the microscopic charges for example, the averaging can arrange that only dipole fields play a role. In particular, as in the example above that uses a constant dipole moment density confined to a finite region, a surface charge and depolarization field results. The simplest approximation is to replace the charge array with a model of ideal (infinitesimally spaced) dipoles. To the accuracy of this dipole approximation, as shown in the previous section, the dipole moment density p( r) (which includes not only p but the location of p) serves as P( r).Īt locations inside the charge array, to connect an array of paired charges to an approximation involving only a dipole moment density p( r) requires additional considerations. In particular, truncating the expansion at the dipole term, the result is indistinguishable from the polarization density generated by a uniform dipole moment confined to the charge region. By truncating this expansion (for example, retaining only the dipole terms, or only the dipole and quadrupole terms, or etc.), the results of the previous section are regained.
If observation is confined to regions sufficiently remote from a system of charges, a multipole expansion of the exact polarization density can be made. This linear dielectric example shows that the dielectric constant treatment is equivalent to the uniform dipole moment model and leads to zero charge everywhere except for the surface charge at the boundary of the sphere.
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