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The pith balls repel each other because of the electrical repulsive force between the two charged balls.  Let the distance between them be D units.  Let a mica sheet be introduced in the center of these.  let the charge be equal to q units.
                 Force = K q² / D²    This is balanced by the tension in the string and gravity.

The mica sheet also develops equal charge (opposite charge) on each side of the sheet on its surface.  Thus these equal charges are separated now only by D/2 distance.  The force So the pith ball is repelled so that the distance between the mica sheet and each pith ball will be D units.

Hence, the distance between them doubles.

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