W9_parallel_resonance.eps

Week 4: Capacitance 139

the result can be shown to hold in the more general case of varying fields using more calculus in a
later course. In this course, we will limit ourselves toverifyingthat the result isconsistentwith the
energy computed for e.g. spherical or cylindrical capacitors, or with just the energy stored creating
a ball of charge like the one above. This isn’t quite a proof that it is general, but it certainly seems
as though it makes it more likely.

Consider, then, the energy stored in a parallel plate capacitor andwrite it in terms of the electric
field strength:

U =^1

2

CV^2 =^1

2

ǫ 0 A

d

(Ed)^2

=

1

2

ǫ 0 E^2 (Ad) =

1

2

ǫ 0 E^2 (Vol) (231)

whereAdis the volume of the region in between the plates where the field is nonzero in our idealized
picture (neglecting fringing fields). If we divide both sides of this equation by the volume, we obtain:

ηe=dU

dV

=^1

2

ǫ 0 E^2 (232)

theenergy density of the electromagnetic field.

Now, as noted, we have no good reasonyetto think that this is general and holds for varying
electric fields, but it certainly might, so we try it to see if it does. Let’sapply it to the case we just
solved, the energy of a ball of uniform charge. We write:

dU = ηedV=

1

2

ǫ 0 E(r)^24 πr^2 dr

U=

∫

dU =

∫

ηedV=

1

2

ǫ 0

∫∞

0

E(r)^24 πr^2 dr

=

1

2 (4πǫ^0 )

{∫R

0

(

keQ

R^3 r

) 2

r^2 dr+

∫∞

R

(

keQ

r^2

) 2

r^2 dr

}

=^1

2

1

ke

k^2 eQ^2

{∫R

0

r^4

R^6

dr+

∫∞

R

1

r^2

dr

}

=^1

2

keQ^2

{

1

5 R

+^1

R

}

=^1

2

keQ^26

5 R

=

3

5

keQ^2

R (233)

exactlyas we obtained at the end of Week/Chapter 3! This is a rather complicated variation in
E~, and yet it gives us exactly the right answer. This is strong evidencethat our form is general
(although as noted this evidence is not proof and a proper derivation of this expression is beyond
the scope of this course). You will obtain still more evidence by verifying this expression for some
other arrangements of charge in your homework.

4.3: Adding Capacitors in Series and Parallel

At this point, we know how to compute the capacitance of our three“simple” geometries, and know
in principlehow to proceed for more complicated cases (although the integralsand so on may be
very difficult in the general case, as always). Once we’ve either computed or, even better,measured
the capacitance of a capacitor, we won’t really care much what the geometry is. We can start to treat
a capacitor as an “object” in its own right, and give it asymbolto use in designing e.g. electrical
circuits. Our “standard symbol” for a capacitor will be a pair of stylized “plates” viewed edgewise,
with a wire running into each plate.