Imagine the given charge Q divided into very small parts q, q1, q2 ..., and these little parts successively carried up to the conductor. The first very small quantity q is brought up without any appreciable work and produces by its presence a small potential V'. To bring up the second quantity, accordingly, we must do the work q'V', and similarly for the quantities which follow the work q''V'', q'''V''', and so forth. Now, as the potential rises proportionately to the quantities added until the value V is reached, we have, agreeably to the graphical representation of Fig. 38, for the total work performed,
W = 1/2QV,
which corresponds to the total energy of the charged conductor. Using the equation Q = CV, where C stands for capacity, we also have,
W = 1/2CV2, or W = Q2/2C.
It will be helpful, perhaps, to elucidate this idea by an analogy from the province of mechanics. If we pump a quantity of liquid, Q, gradually into a cylindrical vessel (Fig. 39), the level of the liquid in the vessel will gradually rise. The more we have pumped in, the greater the pressure we must overcome, or the higher the level to which we must lift the liquid. The stored-up work is rendered again available when the heavy liquid Q, which reaches up to the level h, flows out. This work W corresponds to the fall of the whole liquid weight Q, through the distance h/2 or through the altitude of its centre of gravity. We have
W = 1/2Qh.
Further, since Q = Kh, or since the weight of the liquid and the height h are proportional, we get also
W = 1/2Kh2 and W = Q2/2K.
Fig. 38.