To determine the energy stored in
capacitor, consider initially two uncharged conductors 1 and 2. Imagine next a
process of transferring charge from conductor 2 to conductor 1 bit by bit, so
that at the end, conductor 1 gets charge Q.
By charge conservation, conductor 2 has
charge –Q at
the end.
In transferring positive charge from
conductor 2 to conductor 1, work will be done externally, since at any stage
conductor 1 is at a higher potential than conductor 2.
To calculate the total work done, we first
calculate the work done in a small step involving transfer of an infinitesimal
(i.e., vanishingly small) amount of charge. Consider the intermediate situation
when the conductors 1 and 2 have charges Q' and –Q' respectively. At this stage, the potential difference V' between
conductors 1 to 2 are Q' /C,
where C is
the capacitance of the system.
Next imagine that a small charge δQ' is transferred from conductor 2 to 1. Work done in this step (δW' ),
resulting in charge Q' on conductor 1 increasing to Q' + δQ' , is
given by
The total work done (W) is the sum of the
small work (δ W) over the very
large number of steps involved in building the charge Q' from
zero to Q.
The same result can be obtained directly
from Eq. by integration
The surface charge density s is related to the electric field E between the plates,
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