Two capacitors with capacity C1 and C2 are charged to potential V1 and V2 respectively and then connected in parallel. Calculate the common potential across the combination, the charge on each capacitor, the electrostatic energy stored in the system and the change in the electrostatic energy from its initial value.

1.Suppose C1 , C2 are capacities of 2 condensers charged to potentials V1 and V2 respectively. Then total charges before sharing=  C1V1 + C2 V2 
If V is the common potential on sharing charges, then total charge after sharing=   C1V + C2 V = ( C1 + C2 ) V
As no charge is lost in the process of sharing, therefore:-
( C1 + C2 ) V = C1V1 + C2 V2 
So, common potential is given by :-
V = C1V1 + C2V2C1 + C2     .........(1)
2.   Charge on each capacitor after sharing of charges is given by :- 
So, charge on first capacitor is given by :- Q1 = C1
and charge on second  capacitor is given by :- Q2 = C2 V
where V is the common potential given by eq. 1

3.As energy stored in capacitor is given by E = 1/2Cv2
Total final  electrostatic energy stored in the system of two capacitors, when both are connected in parallel combination is 
E212(C1 + C2) V2
putting value of V from eq 1 in above equation, we get :
E2(C1V1 + C2V2)22 (C1 + C2)       ........(1)
4. Initial electrostatic energy before sharing of charges in both the conductors is :
E112C1 V12 + 12C2 V22   .......(2)
Change in electrostatic energy is given by :
ΔE = E1 - E2on solving, we get :ΔE  = C1C2 (V1 - V2)22 (C1 + C2)

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common potential = total initial charge / equivalent capacitance {V=q/c}

V =(q1+q2) / (C1+C2)

V = (C1V1 +C2V2)/(C1+C2)

charge on each capacitor after connecting is

q1=C1V =C1 (C1V1+C2V2)/(C1+C2)

Similarly , q2=C2V

electrostatical energy stored in the system

(1/2) Ceq V2

=(1/2) [(C1+C2)] [(C1V1+C2V2 ) / ( C1+C2)]2

=(1/2) [(C1V1)2 + (C2V2)2 + 2C1C2V1V2 ] / [C1+C2]

Change in electrostatical energy

Initial energy = 1/2C1V12 +1/2C2V22

final energy =as above

change in energy = final - initial

ONSOLVING we get Del (E)=(1/2) . {C1C2/(C1+C2)} . (V1-V2)2

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