Research On Electric Polarization And Mutual Interaction Of Two Conducting Spheres In The Potential Of A Point Electric Dipole

The electric polarization of a body or dielectric medium refers to the chargeredistribution caused by an external electric field, and the degree of polarization isrepresented by the dipole moment, which relies on the geometry shape and dielectricproperties of the body. When a single conducting sphere is immersed in a uniformelectric field, only a dipole moment will be induced. However, when there exist twospheres, higher order multipoles will be induced on each sphere due to the mutualinduction of them. This will make the mutual interaction of the spheres becomecomplex possible. Analyses for the problem, in principle, can be made by solvingLaplace’s equation of electric potential in bi-spherical coordinate or the method ofmultipole expansion. However, sophisticated mathematical tricks are required and theresults often have to be expressed in definite integrals. This reduces the availability ofthe results. To simplify the analytical process, we here adopt the image dipole method.The dependence of the dipole moment on the center-to-center separation of the spheresis determined by solving the difference equations that have been built up for the imagedipoles, and the interaction force of the spheres is calculated.Firstly, the imaging rule of the dipole moment is founded for the polarization oftwo conducting spheres in a homogeneous electric field. By means of the mutualmapping images of the dipole moment between the two spheres, the dependence of thetotal dipole moment of each conducting sphere on the distance between each other isdetermined. Based on such a circumstance, the interaction force between the twospheres is calculated. Our analysis shows that two conducting spheres attract each otherwhen the external electric field parallels to the orientation between the centres of thespheres, while they repel each other when the exotic electric field perpendiculars to thedirection between the centers of the spheres.Subsequently, the polarization interaction between the two conducting spheres inthe point dipole filed is analyzed. The difficulty of analysis will be improved greatly asa result of the inhomogeneous field distributions of point dipole. As an illustration, wehere focus on analyzing the typical case that the point dipole is placed between the twoconducting spheres. Meanwhile, a rigorous analysis formula is given correspondingly.The calculated results indicate that the mutual effect between the two conductingspheres will be enhanced significantly in the case of the above proposed model, whichleads to the total dipole moment and interaction force increase rapidly as the separationdecrease. Finally, the asymptotic behavior of the total dipole moment is analyzed as theseparation between the two spheres is remote or close to each other. On the one hand,the mutual effect becomes weak between the two spheres when the center-to-centerseparation of spheres become far enough. The results reveal that the dipole momentdecays with-3power of the center-to-center separation of spheres despite ofhomogeneous electric field or the point dipole field. On the other hand, the mutualeffect becomes large when the two spheres become close enough, which results in thedipole moment increases with the distance between the two spheres decrease rapidly.Also, the corresponding dipole moment as a function of distance between the twospheres is more complicated. Furthermore, the concise asymptotic expressions are givenfor these two cases.