Derive the equations of stored energy for series or parallel connection of many capacitors.

Explanation

For series connection :
Charge remains same Q on each capacitor
`:.` Total energy stored,
`U =(Q^(2))/(2).(1)/(C)`
`=(Q^(2))/(2)[(1)/(C_(1))+(1)/(C_(2))+cdots+(1)/(C_(n))]`
`=(Q^(2))/(2C_(1))+(Q^(2))/(2C_(2))+cdots+(Q^(2))/(2C_(n))`
`:. U = U_(1)+U_(2)+cdots+U_(n)`
For parallel connection :
Potential difference V remains same for each capacitor
`:.` Total energy stored ,
`U=(1)/(2) CV^(2)`
`=(1)/(2)[C_(1)+C_(2)+cdots+C_(n)]V^(2)`
`=(1)/(2) C_(1)V^(2)+(1)/(2)C_(2)V^(2)+cdots+(1)/(2)C_(n)V^(2)`
`:. U_(1)+U_(2)+cdots+U_(n)`
So the total energy stored in both the connection is the sum of energy stored in each capacitor.