Interaction Forces Between Two Far-spaced Drops

When solving problems involving the calculation of forces of electrostatic interaction between two conducting particles, approximations of dipole, dipole-coulomb, and coulomb interactions are frequently used. Within the framework of these approximations, particle interactions are considered as interactions of two electric dipoles, a dipole and a point charge, or two point charges. Here, the sizes of the particles define only the values of the dipole moments. Therefore, such approximations may only be used if the distance between the particles is much greater than the sum of their radii. However, in practice, these approximations are frequently used when the gap between the surfaces of the particles is comparable to or less than the size of the particles. Consequently, this raises doubts concerning the accuracy of cited approximations for various distances between particles. 

The forces acting on two distant conducting spherical particles are set out in ref. 

Compare now the exact values of the factors fi, given by Eqs. (12.46) and (12.47), with the approximate ones given by Eq. (12.49). For this, we introduce relative deviations  

To increase the accuracy of the definition of factors fi in cases when the particles are located relatively far apart, one should take advantage of the method of reflections, which allows one to obtain an approximate solution in the form of a power series, r. In this way, terms of higher order are taken into account in Eqs. (12.49). Omitting simple but bulky calculations, accurate to terms of r , we obtain  

Comparison of factors with the exact values shows that for A 5 for all k the difference between them does not exceed 2% and for A 3 does not exceed 3-5%. In Fig. 12.6, the dashed lines show the dependences of the errors aj upon A when are taken instead of 

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