Solution DASA S-1

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According to the method of the image the field inside the capacitor is equivalent of the field of an infinite Chain of point charges with alternating signs.

+Q                   -Q       p;               +Q                 - Q                         + Q     

-2d                      -d       p;                0                         d                           2d

During the displacement  x  of the sphere the image charge +Q Move into the same direction, the image -Q into the opposite one. It is not necessary to consider the image +Q, because their distance from the sphere are NOT changed by the displacement.

If we consider the pair of image -Q with the same distance from the sphere we get the electric force.

F_e  =  Q²/ 4πε_o ∑ [1/(kd-2x) - 1/(kd+2x)]                                                                                                             k= 1,3,5........

      =   Q²/ 4πε_o ∑ 8x(2n+1)d/[(2n+1)²d²- 4x²]²                                                                                                    n= 0,1,2,3.......

for small displacement  x  we have approximately

F_e  =  2Q²x/πε_od³ ∑ 1/[2n+1]³

       = 1.05*2Q²x/πε_od³

^{As long as this force is smaller than the respective component of gravitation, we have a stable equilibrium}

                  2Q²x/πε_od^{3  < mgx/L}

^=>        Q < 2.28 x 10 ^-10 coulomb

       ans   Q = 2.25 x 10 ^-10 coulomb  (approx)

ans is accepted by +-       0.05 x 10 ^-10