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Energy Stored in a Capacitor
The energy stored in a capacitor is said to reside in the dielectric permeated by the electric field. Figure 217 shows a simple capacitor in which a charge of q coulombs is stored in the electric field. The top plate carries a net charge of +q coulombs and the bottom plate a net charge of  q coulombs. The voltage between the top plate and bottom plate of the capacitor is v volts.
If a differential charge +dq is carried from the bottom plate to the top plate, there will be a differential increase of energy dw.
Let Then dF = force on the differential charge dq expressed in newtons
The differential energy required to carry the differential charge from the bottom plate to the top plate is given by the line integral along any path between the plates
When Eq. 247 is substituted in Eq. 248 the result is
However, the line integral of the electric field intensity along any path from one plate to the other gives the voltage between plates; hence
Substitution of Eq. 250 in Eq. 249 yields
In order to determine the energy required to store a charge q in the field of a capacitor when the capacitor is initially uncharged, Eq. 251 is integrated as follows
but from Eq. 225 we get
and from Eqs. 252 and 253 there results
Making use of Eq. 253 in 254, we obtain the more commonly used expression for energy stored in a capacitor, i.e.


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