in charging a capacitor of capacitance C by a sorce of EMF V, energy supplied by the source is QV and energy stored in the capacitor is 1/2QV. Justify the difference?
Energy supplied by the source is QV.
Energy stored in the capacitor is 1/2QV.
This is because the battery puts out energy QVb in the process of charging the capacitor to equilibrium at battery voltage Vb. But half of that energy is dissipated in heat in the resistance(I2R) of the charging pathway, and only QVb/2 is finally stored on the capacitor at equilibrium.
Part of the intuitive part that goes into setting up the integral is that getting the first element of charge dq onto the capacitor plates takes much less work because most of the battery voltage is dropping across the resistance R and only a tiny energy dU = dqV is stored on the capacitor. Proceeding with the integral, which takes a quadratic form in q, gives a summed energy on the capacitor Q2/2C = CVb2/2 = QVb/2 where the Vb here is the battery voltage.