How does a capacitor store energy ? And obtain the formula for the energy stored in the capacitor ?

Explanation

Take uncharged conductors 1 and 2 as shown in figure.

Imagine a process of transferring charge from conductor 2 to conductor 1 bit by bit so that at the ends, conductor 1 gets charge.
In transferring positive charge from conductor 2 to conductor 1 work will be done externally, because at any stage conductor 1 ts at a higher potential than conductor 2.
To calculate the total work by in a small step involving ttansfer of an infinitesimally amount of charge.
Consider the situation when the conductors 1 and 2 have charges Q. and -Q. respectively.
The potential difference V. between conductors 1 to 2 is `V=(Q)/(C)` where C is the capacitance of the system.
A small charge `deltaQ`. is transferred from conductor 2 to 1, then work done,
`deltaW=V.deltaQ`
`:. deltaW=(QdeltaQ)/(C)`
Total work done to bringing charge Q fron conductor 2 to 1 is obtain by integration,
`W= intdW`
`W= intdW`
`:.W=int_(0)^(Q)(Q)/(C)deltaQ :.W=(1)/(C)int_(0)^(Q).deltaQ=(1)/(C) [((Q.)^(2))/(2)]_(0)^(Q)`
`=(1)/(C)[(Q^(2))/(2)]:.W=(Q^(2))/(2C)`
Since, electrostatic force is conservative thb work is stored in the form of the potentia energy of the system of two conductors likE capacitors.
Hence, potential energy of capacitor can be seen as the energy stored in electric field between plates.
`:.` The potential energy ,
`U =(Q^(2))/(2C)`
other equivalent forms of this equation
If Q=CV then
`U=(1)/(2)CV^(2)`
and if `C=(Q)/(V)`
`U=(1)/(2)CV`