Energy Stored in a Capacitor

The energy stored in a capacitor is said to reside in the dielectric permeated by the electric field. Figure 2-17 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.

Figure 2-17. Capacitor with a charge

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
E = electric field intensity at some point between the plates in volts per meter

Then dF = force on the differential charge dq expressed in newtons

[2-47]

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

[2-48]

When Eq. 2-47 is substituted in Eq. 2-48 the result is

[2-49]

However, the line integral of the electric field intensity along any path from one plate to the other gives the voltage between plates; hence

[2-50]

Substitution of Eq. 2-50 in Eq. 2-49 yields

[2-51]

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. 2-51 is integrated as follows

[2-52]

but from Eq. 2-25 we get

[2-53]

and from Eqs. 2-52 and 2-53 there results

[2-54]

Making use of Eq. 2-53 in 2-54, we obtain the more commonly used expression for energy stored in a capacitor, i.e.

[2-55]