Ch24 ssm

Chapter 24
Capacitance
Conceptual Problems
5 • A parallel-plate capacitor is connected to a battery. The space between
the two plates is empty. If the separation between the capacitor plates is tripled
while the capacitor remains connected to the battery, what is the ratio of the final
stored energy to the initial stored energy?
Determine the Concept The energy stored in a capacitor is given by
QVU 2
1
= and the capacitance of a parallel-plate capacitor by .0 dAC ∈= We can
combine these relationships, using the definition of capacitance and the condition
that the potential difference across the capacitor is constant, to express U as a
function of d.
Express the energy stored in the
capacitor:
QVU 2
1
= (1)
Use the definition of capacitance to
express the charge of the capacitor:
CVQ =
Express the capacitance of a
parallel-plate capacitor in terms of
the separation d of its plates:
d
A
C 0∈
=
where A is the area of one plate.
Substituting for Q and C in equation
(1) yields: d
AV
U
2
2
0∈
=
Because
d
U
1
∝ , tripling the separation of the plates will reduce the energy stored
in the capacitor to one-third its previous value. Hence the ratio of the final stored
energy to the initial stored energy is 3/1 .
9 • A dielectric is inserted between the plates of a parallel-plate capacitor,
completely filling the region between the plates. Air initially filled the region
between the two plates. The capacitor was connected to a battery during the entire
process. True or false:
(a) The capacitance value of the capacitor increases as the dielectric is inserted
between the plates.
(b) The charge on the capacitor plates decreases as the dielectric is inserted
between the plates.
71

Chapter 2472
(c) The electric field between the plates does not change as the dielectric is
inserted between the plates.
(d) The energy storage of the capacitor decreases as the dielectric is inserted
between the plates.
Determine the Concept The capacitance of the capacitor is given by
d
A
C 0∈κ
= , the charge on the capacitor is given by CVQ = , and the energy
stored in the capacitor is given by 2
2
1
CVU = .
(a) True. As the dielectric material is inserted, κ increases from 1 (air) to its value
for the given dielectric material.
(b) False. Q does not change. C increases and V decreases, but their product
( ) is constant.CVQ =
(c) False. Because E = V/d and V decreases, E must decrease.
(d) False. The energy stored in the capacitor is given by 2
2
1
CVU = . With the
battery connected as the dielectric is inserted, V remains constant while Q
increases. Therefore U increases.
11 •• (a) Two identical capacitors are connected in parallel. This
combination is then connected across the terminals of a battery. How does the
total energy stored in the parallel combination of the two capacitors compare to
the total energy stored if just one of the capacitors were connected across the
terminals of the same battery? (b) Two identical capacitors that have been
discharged are connected in series. This combination is then connected across the
terminals of a battery. How does the total energy stored in the series combination
of the two capacitors compare to the total energy stored if just one of the
capacitors were connected across the terminals of the same battery?
Picture the Problem The energy stored in a capacitor whose capacitance is C and
across which there is a potential difference V is given by 2
2
1
CVU = . Let C0
represent the capacitance of the each of the two identical capacitors.
(a) The energy stored in the parallel
system is given by:
2
eq2
1
parallel VCU =
When the capacitors are connected in
parallel, their equivalent capacitance
is:
000parallel 2CCCC =+=

Capacitance 73
Substituting for and simplifying
yields:
eqC ( ) 2
0
2
02
1
parallel 2 VCVCU == (1)
If just one capacitor is connected to
the same battery the stored energy is:
2
02
1
capacitor1 VCU = (2)
Dividing equation (1) by equation
(2) and simplifying yields:
22
02
1
2
0
capacitor1
parallel
==
VC
VC
U
U
or
capacitor1parallel 2UU =
(b) The energy stored in the series
system is given by:
2
eq2
1
series VCU =
When the capacitors are connected in
series, their equivalent capacitance
is:
02
1
series CC =
Substituting for and simplifying
yields:
eqC ( ) 2
04
12
02
1
2
1
series VCVCU == (3)
Dividing equation (3) by equation
(2) and simplifying yields: 2
1
2
02
1
2
04
1
capacitor1
series
==
VC
VC
U
U
or
capacitor12
1
series UU =
Estimation and Approximation
13 •• Disconnect the coaxial cable from a television or other device and
estimate the diameter of the inner conductor and the diameter of the shield.
Assume a plausible value (see Table 24–1) for the dielectric constant of the
dielectric separating the two conductors and estimate the capacitance per unit
length of the cable.
Picture the Problem The outer diameter of a "typical" coaxial cable is about
5 mm, while the inner diameter is about 1 mm. From Table 24-1 we see that a
reasonable range of values for κ is 3-5. We can use the expression for the
capacitance of a cylindrical capacitor to estimate the capacitance per unit length
of a coaxial cable.

Chapter 2474
The capacitance of a cylindrical
dielectric-filled capacitor is given
by: ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
1
2
0
ln
2
R
R
L
C
∈πκ
where L is the length of the capacitor,
R1 is the radius of the inner conductor,
and R2 is the radius of the second
(outer) conductor.
Divide both sides by L to obtain an
expression for the capacitance per
unit length of the cable: ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
1
2
1
2
0
ln2ln
2
R
R
k
R
RL
C κ∈πκ
If κ = 3:
( )
nF/m1.0
mm5.0
mm5.2
lnC/mN10988.82
3
229
≈
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅×
=
L
C
If κ = 5:
( )
nF/m2.0
mm5.0
mm5.2
lnC/mN10988.82
5
229
≈
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅×
=
L
C
A reasonable range of values for C/L,
corresponding to 3 ≤ κ ≤ 5, is: nF/m0.2nF/m1.0 ≤≤
L
C
15 •• Estimate the capacitance of the Leyden jar shown in the Figure 24-34.
The figure of a man is one-tenth the height of an average man.
Picture the Problem Modeling the Leyden jar as a parallel-plate capacitor, we
can use the equation for the capacitance of a dielectric-filled parallel-plate
capacitor that relates its capacitance to the area A of its plates and their separation
(the thickness of the glass) d to estimate the capacitance of the jar. See Table 24-1
for the dielectric constants of various materials.
The capacitance of a dielectric-filled
parallel-plate capacitor is given by: d
A
C 0∈κ
=
where κ is the dielectric constant.

Capacitance 75
Let the plate area be the sum of the
area of the lateral surface of the jar
and its base:
2
base
area
lateral 2 RRhAAA ππ +=+=
where h is the height of the jar and R is
its inside radius.
Substitute for A and simplify to
obtain:
( )
( )
d
RhR
d
RRh
C
+
=
+
=
2
2
0
2
0
∈πκ
ππ∈κ
If the glass of the Leyden jar is Bakelite of thickness 2.0 mm and the radius and
height of the jar are 4.0 cm and 40 cm, respectively, then:
( )( ) ( )[ ]
nF2.3
mm0.2
cm0.4cm402
mN
C
10854.8cm0.49.4 2
2
12
=
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅
×
=
−
π
C
The Storage of Electrical Energy
19 • (a) The potential difference between the plates of a 3.00-μF capacitor
is 100 V. How much energy is stored in the capacitor? (b) How much additional
energy is required to increase the potential difference between the plates from
100 V to 200 V?
Picture the Problem Of the three equivalent expressions for the energy stored in
a charged capacitor, the one that relates U to C and V is 2
2
1
CVU = .
(a) Express the energy stored in the
capacitor as a function of C and V:
2
2
1
CVU =
Substitute numerical values and
evaluate U:
( )( ) mJ0.15V100F00.3
2
2
1
== μU
(b) Express the additional energy
required as the difference between
the energy stored in the capacitor at
200 V and the energy stored at
100 V:
( ) ( )
( )( )
mJ0.45
mJ0.15V200F00.3
V100V200Δ
2
2
1
=
−=
−=
μ
UUU

Chapter 2476
23 •• An air-gap parallel-plate capacitor that has a plate area of 2.00 m2
and
a separation of 1.00 mm is charged to 100 V. (a) What is the electric field
between the plates? (b) What is the electric energy density between the plates?
(c) Find the total energy by multiplying your answer from Part (b) by the volume
between the plates. (d) Determine the capacitance of this arrangement.
(e) Calculate the total energy from 21
2U CV= , and compare your answer with
your result from Part (c).
Picture the Problem Knowing the potential difference between the plates, we
can use E = V/d to find the electric field between them. The energy per unit
volume is given by 2
02
1
Eu ∈= and we can find the capacitance of the parallel-
plate capacitor using .0 dAC ∈=
(a) Express the electric field
between the plates in terms of their
separation and the potential
difference between them:
kV/m100
mm1.00
V100
===
d
V
E
(b) Express the energy per unit
volume in an electric field:
2
02
1
Eu ∈=
evaluate u:
( )
33
2
2
2
12
2
1
mJ/m3.44mJ/m27.44
kV/m001
mN
C
10854.8
==
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅
×= −
u
(c) The total energy is given by:
( )( )( )
J
uAduVU
μ5.88
mm1.00m2.00mJ/m27.44 23
=
=
==
(d) The capacitance of a parallel-
plate capacitor is given by: d
A
C 0∈
=
evaluate C:
( )
nF7.17nF71.17
mm1.00
m2.00
mN
C
108.854 2
2
2
12
==
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅
×
=
−
C
(e) The total energy is given by: 2
2
1
CVU =

Capacitance 77
evaluate U:
( )( )
).(withagreementinJ,5.88
V100nF17.71
2
2
1
c
U
μ=
=
Spherical Capacitors
49 •• A spherical capacitor consists of a thin spherical shell that has a
radius R1 and a thin, concentric spherical shell that has a radius R2, where R2 > R1.
(a) Show that the capacitance is given by C = 4π∈0R1R2/(R2 – R1). (b) Show that
when the radii of the shells are nearly equal, the capacitance approximately is
given by the expression for the capacitance of a parallel-plate capacitor,
C = ∈0A/d, where A is the area of the sphere and d = R2 – R1.
Picture the Problem We can use the definition of capacitance and the expression
for the potential difference between charged concentric spherical shells to show
that ( ).4 12210 RRRRC −= ∈π
(a) Using its definition, relate the
capacitance of the concentric
spherical shells to their charge Q and
the potential difference V between
their surfaces:
V
Q
C =
Express the potential difference
between the conductors:
21
12
21
11
RR
RR
kQ
RR
kQV
−
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=
Substitute for V and simplify to
obtain: ( )
12
210
12
21
21
12
4
RR
RR
RRk
RR
RR
RR
kQ
Q
C
−
=
−
=
−
=
∈π
(b) Because R2 = R1 + d: ( )
22
1
1
2
11121
RR
dRRdRRRR
=≈
+=+=
because d is small.
Substitute for R1R2 and R2 − R1 to
obtain: d
A
d
R
C 0
2
04 ∈∈π
=≈

Chapter 2478
Disconnected and Reconnected Capacitors
53 •• A 100-pF capacitor and a 400-pF capacitor are both charged to
2.00 kV. They are then disconnected from the voltage source and are connected
together, positive plate to negative plate and negative plate to positive plate.
(a) Find the resulting potential difference across each capacitor. (b) Find the
energy dissipated when the connections are made.
Picture the Problem (a) Just after the two capacitors are disconnected from the
voltage source, the 100-pF capacitor carries a charge of 200 nC and the 400-pF
capacitor carries a charge of 800 nC. After switches S1 and S2 in the circuit are
closed, the capacitors are in parallel between points a and b, and the equivalent
capacitance of the system is . The plates to the right in the
diagram below form a single conductor with a charge of 600 nC, and the plates to
the left form a conductor with charge −Q = −600 nC. The potential difference
across each capacitor is
400100eq CCC +=
eqCQV = . In Part (b) we can find the energy dissipated
when the connections are made by subtracting the energy stored in the system
after they are connected from the energy stored in the system before they are
connected.
+200 nC 200 nC−
1 2
1
2
+800 nC 800 nC−
S S1 2
a b
200 nC−+200 nC
+800 nC800 nC−
C = 100 pF
C = 100 pF
C = 400 pF
C = 400 pF
(a) When the switches are closed and
the capacitors are connected together,
their initial charges redistribute and
the final charge on the two-capacitor
system is 600 nC and the equivalent
capacitance is 500 pF:
−
eq
+600 nC600 nC
C = 500 pF
The potential difference across each
capacitor is the potential difference
across the equivalent capacitor:
kV20.1
pF500
nC600
eq
===
C
Q
V

Capacitance 79
(b) The energy dissipated when the
capacitors are connected is the
difference between the energy stored
after they are connected and the
energy stored before they were
connected:
afterbeforedissipated UUU −= (1)
beforeU is given by:
( ) i212
1
222
1
112
1
21before
VQQ
VQVQ
UUU
+=
+=
+=
where Vi is the charging voltage.
afterU is given by: QVU 2
1
after =
where Q is the total charge stored after
the capacitors have been connected and
V is the voltage found in Part (a).
Substitute for and in
equation (1) and simplify to obtain:
beforeU afterU ( ) QVVQQU 2
1
i212
1
dissipated −+=
Substitute numerical values and evaluate :dissipatedU
( )( ) ( )( ) J640kV20.1nC600kV00.2nC800nC200 2
1
2
1
dissipated μ=−+=U
59 •• Capacitors 1, 2 and 3, have capacitances equal to 2.00 μF, 4.00 μF, and
6.00 μF, respectively. The capacitors are connected in parallel, and the parallel
combination is connected across the terminals of a 200-V source. The capacitors
are then disconnected from both the voltage source and each other, and are
connected to three switches as shown in Figure 24-42. (a) What is the potential
difference across each capacitor when switches S1 and S2 are closed but switch S3
remains open? (b) After switch S3 is closed, what is the final charge on the
leftmost plate of each capacitor? (c) Give the final potential difference across each
capacitor after switch S3 is closed.
Picture the Problem Let lower case qs refer to the charges before S3 is closed
and upper case Qs refer to the charges after this switch is closed. We can use
conservation of charge to relate the charges on the capacitors before S3 is closed
to their charges when this switch is closed. We also know that the sum of the
potential differences around the circuit when S3 is closed must be zero and can
use this to obtain a fourth equation relating the charges on the capacitors after the
switch is closed to their capacitances. Solving these equations simultaneously will
yield the charges q1, q2, and q3. Knowing these charges, we can use the definition

Chapter 2480
of capacitance to find the potential difference across each of the capacitors.
(a) With S1 and S2 closed, but S3
open, the charges on and the potential
differences across the capacitors do
not change. Hence:
V200321 === VVV
(b) When S3 is closed, the charges
can redistribute; express the
conditions on the charges that must
be satisfied as a result of this
redistribution:
1212 QQqq −=− ,
2323 QQqq −=− ,
and
3131 QQqq −=− .
Express the condition on the
potential differences that must be
satisfied when S3 is closed:
0321 =++ VVV
where the subscripts refer to the three
capacitors.
eliminate the potential differences:
0
3
3
2
2
1
1
=++
C
Q
C
Q
C
Q
(1)
find the initial charge on each
capacitor:
( )( ) C400V200F00.211 μμ === VCq ,
( )( ) C800V200F00.422 μμ === VCq ,
and
( )( ) C1200V200F00.633 μμ === VCq
Let q = q1. Then: q2 = 2q and q3 = 3q
Express q2 and q3 in terms of q1 and
q:
12 qqq += (2)
and
qqq 213 += (3)
Substitute in equation (1) to obtain: 0
2
3
1
2
1
1
1
=
+
+
+
+
C
qq
C
qq
C
q
or
0
F00.6
2
F00.4F00.2
111
=
+
+
+
+
μμμ
qqqqq
Solving for and evaluating q1 gives: ( ) C255C40011
7
11
7
1 μμ −=−=−= qq

Capacitance 81
Substitute in equation (2) to obtain: C145C255C4002 μμμ =−=q
Substitute in equation (3) to obtain: ( ) C545C4002C2553 μμμ =+−=q
(c) Use the definition of capacitance
to find the potential difference across
each capacitor with S3 closed:
V127
F00.2
C255
1
1
1 −=
−
==
μ
μ
C
q
V ,
V3.36
F00.4
C145
2
2
2 ===
μ
μ
C
q
V ,
and
V8.90
F00.6
C545
3
3
3 ===
μ
μ
C
q
V
General Problems
79 •• A parallel combination of two identical 2.00-μF parallel-plate
capacitors (no dielectric is in the space between the plates) is connected to a
100-V battery. The battery is then removed and the separation between the plates
of one of the capacitors is doubled. Find the charge on the positively charged plate
of each of the capacitors.
Picture the Problem When the battery is removed, after having initially charged
both capacitors, and the separation of one of the capacitors is doubled, the charge
is redistributed subject to the condition that the total charge remains constant; that
is, Q = Q1 + Q2 where Q is the initial charge on both capacitors and Q2 is the
charge on the capacitor whose plate separation has been doubled. We can use the
conservation of charge during the plate separation process and the fact that,
because the capacitors are in parallel, they share a common potential difference.
Find the equivalent capacitance of
the two 2.00-μF parallel-plate
capacitors connected in parallel:
F00.4F00.2F00.2eq μμμ =+=C
find the charge on the equivalent
capacitor:
( )( ) C400V100F00.4eq μμ === VCQ
Relate this total charge to charges
distributed on capacitors 1 and 2
when the battery is removed and the
separation of the plates of capacitor
2 is doubled:
21 QQQ += (1)

Chapter 2482
Because the capacitors are in
parallel:
21 VV = and
2
2
22
1
2
2
2
1
1 2
' C
Q
C
Q
C
Q
C
Q
===
Solve for Q1 to obtain:
2
2
1
1 2 Q
C
C
Q ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
= (2)
Substitute equation (2) in equation
(1) and solve for Q2 to obtain: ( ) 12 21
2
+
=
CC
Q
Q
evaluate Q2: ( )
C133
1F00.2F00.22
C400
2 μ
μμ
μ
=
+
=Q
Substitute numerical values in
equation (1) or equation (2) and
evaluate Q1:
C2671 μ=Q
85 ••• An electrically isolated capacitor that has charge Q on its positively
charged plate is partly filled with a dielectric substance as shown in Figure 24-51.
The capacitor consists of two rectangular plates that have edge lengths a and b
and are separated by distance d. The dielectric is inserted into the gap a distance
x. (a) What is the energy stored in the capacitor? Hint: the capacitor can be
modeled as two capacitors connected in parallel. (b) Because the energy of the
capacitor decreases as x increases, the electric field must be doing work on the
dielectric, meaning that there must be an electric force pulling it in. Calculate this
force by examining how the stored energy varies with x. (c) Express the force in
terms of the capacitance and potential difference V between the plates. (d) From
where does this force originate?
Picture the Problem We can model this capacitor as the equivalent of two
capacitors connected in parallel, one with an air gap and other filled with a
dielectric of constantκ. Let the numeral 1 denote the capacitor with the dielectric
material whose constant is κ and the numeral 2 the air-filled capacitor.
(a) Using the hint, express the energy
stored in the capacitor as a function
of the equivalent capacitance :eqC
eq
2
2
1
C
Q
U =
The capacitances of the two
capacitors are: d
bx
C 0
1
∈κ
= and
( )
d
xab
C
−
= 0
2
∈

Capacitance 83
Because the capacitors are in
parallel, the equivalent
capacitance, , is the sum of C( )xC 1
and C2:
( ) ( )
( )
( )[ ]ax
d
b
xax
d
b
d
xab
d
bx
CCxC
+−=
−+=
−
+=+=
10
0
00
21
κ
∈
κ
∈
∈∈κ
Substitute for in the expression
for U and simplify to obtain:
eqC
( )[ ]axb
dQ
U
+−
=
12 0
2
κ∈
(b) The force exerted by the electric
field is given by:
( )[ ]
( )[ ]{ }
( )
( )[ ]2
0
2
1
0
2
0
2
12
1
1
2
12
1
axb
dQ
ax
dx
d
b
dQ
axb
dQ
dx
d
dx
dU
F
+−
−
=
+−−=
⎥
⎦
⎤
⎢
⎣
⎡
+−
−=
−=
−
κ∈
κ
κ
∈
κ∈
(c) Rewrite the result in (b) to obtain:
( )
( )[ ]
( )
( )
d
Vb
C
d
b
Q
ax
d
b
d
b
Q
F
2
1
2
1
12
1
2
0
2
eq
02
2
2
0
02
∈κ
∈
κ
κ
∈
∈
κ
−
=
⎟
⎠
⎞
⎜
⎝
⎛
−
=
+−⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
−
=
Note that this expression is independent
of x.
(d) The force originates from the fringing fields around the edges of the capacitor.
The effect of the force is to pull the polarized dielectric into the space between the
capacitor plates.

Ch24 ssm

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