The two plates of a parallel-plate
capacitor attract each other, since they are oppositely charged.

When the plates are
allowed to move together so that they touch, the work done by the force of attraction is equal
to the original energy of the capacitor. If the plates were originally a distance d apart, the
work done is Fd where F is the force between them.

Therefore:

The charge on the insulated plate contributes half the field and that induced on the earthed plate the other half; the force on the charge on one plate is due to the field set up by the charge on the other. This explains the factor of ½ in the above formulae.