W9_parallel_resonance.eps

138 Week 4: Capacitance

is the potential difference between the plates. Then thenextfork full of charge that he moves over,
he will have to do work:
∆W=V∆Q (227)

The work theblue devildoes charging up the plates isequalto the change in the potential energy
of the charged plates^44. We make the chunk of charge being moved differentially small, and write:

dU=V dQ=

Q

C

dQ (228)

and can easilyintegrate both sidesto find the total energy stored on the capacitor when we begin
withnocharge and charge it up to a total chargeQ 0 :

U=

∫

dU=

1

C

∫Q 0

0

Q dQ=

1

2

Q^20

C

(229)

We can thus easily write the total energy storedthree ways:

U=^1

2

Q^20

C

=^1

2

CV 02 =^1

2

V 0 Q 0 (230)

(where note, we useQ 0 =CV 0 to go from the first to the second, then use it again to go to the
third).

dU = VdQ Q Q

Slope 1/C

V(Q)

V 0

0

U = Area = 1/2 Q V 00

Figure 41: The energy as the area underneath the curveV(Q) =Q/C.

Of these, the third form is perhaps the most revealing and convenient. If we plotV(Q) =Q/C,
we get astraight line of slope 1 /C. The integral ofdU=V dQis just theareaunder this straight
line at the particular valuesQ 0 andV 0 =Q 0 /C. This, in turn, is just the area of a triangle – one
half the base times the height. Which is, as you can easily see in figure 41, 1/ 2 Q 0 V 0. It’s also a
good time to remind you that wedidan integral of this sort in the chapter on potential and energy,
except this time we didn’t distribute the chargeQin a ball, we left it in a thin layer on the surface
of the capacitor plate(s) so that it is even easier (and gives us the promised factor of 1/2 instead of
3 /5).

Energy Density

A very important question to ask is: just whereisall of this energy in the capacitor stored? We
did a lot of work charging up the capacitor, and all of the work we canget back comes from charge
we’ve stored in this way being drivenby the electric field of the charge itselfback into equilibrium
as the separated charges neutralize and the field collapses. It is thereforereasonableto guess that
the energy is storedin the electric field we createas we rearrange the charge in the first place.

Can we write the energy of the capacitor in terms of the field strength? Yes we can! For simplicity,
we’ll as usual in this chapter consider the parallel plate capacitor tosee how, and then note that

(^44) Think of the workyou dolifting a book over your head being equal to theincreasein its gravitational potential
energy – the work done by gravity, or the electric field in the case of the capacitor, is the opposite of the work done
by you or the devil.