Analytic geometry polar equations

Obviously, the geometry-related or the molecular shape parameters are difficult to quantify one-dimensionally. Single numbers reflecting molecular shape differences are adequate only in cases of rigid and planar solutes. They become significant in QSRR equations if a series of analytes considered comprises of compounds of similar size and polarity [42,43],... 

The value of the induced potential depends on not only the externally applied electric field but also the electric properties of both the solid surface and the liquid electrolyte and geometry of the solid surface. To date, more attention has been given to the induced potential on ideally polarizable surfaces. To obtain the induced potential of an ideally polarizable sphere, two assumptions are applied after the polarization, the electric field lines near the conducting surface are distorted and go around the conducting sphere (expressed as ,. = 0 as shown in Fig. 3) and the potential at the ideally polarized surface is identical (expressed as = 0 at r < a). Utilizing the assumptions above, Dykhin and Bazant solved the Laplace s equation [2, 3] and got the analytical solution of the induced potential on a polarized cylinder and sphere. The analytical solutions to the Laplace s equation are only limited to cylinder and sphere geometries, because the boundary conditions for the Laplace s equation are much easier in these cases. The obtained induced potential on an ideally polarizable sphere is... 

In a recent study we presented a calculation for the case of a linearly polarized laser incident obliquely upon a nematic film (i.e., where the laser propagation wave vector makes a finite angle with the director axis). We showed that, under physically reasonable assumptions (e.g., all the angles involved are small), the torque balance equations lend themselves to analytical solutions. Some transverse dependences of the reorientation as a function of cell geometry and optical director-axis configurations were discussed. 

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