Fundamentals of Physics

David Halliday, Robert Resnick , Jearl Walker

Chapter 25

Capacitance - all with Video Answers

Educators

+ 4 more educators

Chapter Questions

Problem 1

The sewage outlet of a house constructed on a slope is 6.59 $\mathrm{mbe}$ low street level. If the sewer is 2.16 $\mathrm{m}$ below street level, find the minimum pressure difference that must be created by the sewage pump to
transfer waste of average density 1000.00 $\mathrm{kg} / \mathrm{m}^{3}$ from outlet to sewer.
$+70 \mathrm{pC}$ and $-70 \mathrm{pC}$ , which result in a 20 $\mathrm{V}$ potential difference
between them. (a) What is the capacitance of the system? (b) If the
charges are changed to $+200 \mathrm{pC}$ and $-200 \mathrm{pC}$ what does the capac-
itance become? (c) What does the potential difference become?

Salamat Ali

Salamat Ali

Numerade Educator

Problem 2

The capacitor in Fig, $25-25$ has a
capacitance of 25$\mu \mathrm{F}$ and is initially
uncharged. The battery provides a
potential difference of 120 $\mathrm{V}$ . After
switch $\mathrm{S}$ is closed, how much charge
will pass through it?

Ze-Han Lee

Numerade Educator

Problem 3

A parallel-plate capacitor has circular plates of 8.20 $\mathrm{cm}$
radius and 1.30 $\mathrm{mm}$ separation. (a) Calculate the capacitance.
(b) Find the charge for a potential difference of 120 $\mathrm{V}$ .

Salamat Ali

Numerade Educator

Problem 4

The plates of a spherical capacitor have radii 38.0 $\mathrm{mm}$ and
40.0 $\mathrm{mm}$ . (a) Calculate the capacitance. (b) What must be the plate
area of a parallel-plate capacitor with the same plate separation
and capacitance?

Ze-Han Lee

Numerade Educator

Problem 5

What is the capacitance of a drop that results when two
mercury spheres, each of radius $R=2.00 \mathrm{mm},$ merge?

Salamat Ali

Numerade Educator

Problem 6

You have two flat metal plates, each of area 1.00 $\mathrm{m}^{2}$ , with
which to construct a parallel-plate capacitor. (a) If the capacitance of the device is to be 1.00 $\mathrm{F}$ , what must be the separation
between the plates? (b) Could this capacitor actually be
constructed?

Ze-Han Lee

Numerade Educator

Problem 7

If an uncharged parallel-plate capacitor (capacitance $C )$ is
connected to a battery, one plate becomes negatively charged as electrons move to the plate face (area A). In Fig. $25-26$ , the depth $d$ from which the electrons come in the plate in a particular capacitor is plotted against a range of values for the potential difference $V$ of the battery. The density of conduction electrons in the copper plates is
$8.49 \times 10^{28}$ electrons/m $^{3} .$ The vertical
scale is set by $d_{s}=1.00 \mathrm{pm},$ and the horizontal scale is set by $V_{s}=20.0 \mathrm{V} .$ What is the ratio $C / A ?$

YL

Yeeren Low

Numerade Educator

Problem 8

How many 1.00$\mu \mathrm{F}$ capacitors must be connected in parallel to
store a charge of 1.00 $\mathrm{C}$ with a potential of 110 $\mathrm{V}$ across the
capacitors?

Ze-Han Lee

Numerade Educator

Problem 9

Each of the uncharged capacitors in Fig. $25-27$ has a capacitance of 25.0$\mu \mathrm{F}$ . A potential difference of $V=4200 \mathrm{V}$ is established when the switch is closed. How many
coulombs of charge then pass
through meter $\mathrm{A}$ ?

David Morabito

Numerade Educator

Problem 10

In Fig. $25-28$ , find the equivalent capacitance of the combination. Assume that $C_{1}$ is $10.0 \mu \mathrm{F}, C_{2}$
is $5.00 \mu \mathrm{F},$ and $C_{3}$ is 4.00$\mu \mathrm{F}$ .

Ze-Han Lee

Numerade Educator

Problem 11

In Fig. $25-29$ , find the equivalent capacitance of the
combination. Assume that $C_{1}=10.0 \mu \mathrm{F}, C_{2}=5.00 \mu \mathrm{F},$ and $C_{3}=$
4.00$\mu \mathrm{F}$

Salamat Ali

Numerade Educator

Problem 12

Two parallel-plate capacitors, 6.0$\mu \mathrm{F}$ each, are connected in
parallel to a 10 $\mathrm{V}$ battery. One of the capacitors is then squeezed so
that its plate separation is 50.0$\%$ of its initial value. Because of the squeczing, (a) how much additional charge is transferred to the capacitors by the battery and (b) what is the increase in the total
charge stored on the capacitors?

Ze-Han Lee

Numerade Educator

Problem 13

A 100 pF capacitor is charged to a potential difference of $50 \mathrm{V},$ and the charging battery is disconnected. The
capacitor is then connected in parallel with a second (initially
uncharged) capacitor. If the potential difference across the first capacitor drops to $35 \mathrm{V},$ what is the capacitance of this second
capacitor?

Salamat Ali

Numerade Educator

Problem 14

In Fig. $25-30$ , the battery has a potential difference of $V=10.0 \mathrm{V}$
and the five capacitors cach have a capacitance of 10.0$\mu \mathrm{F}$ . What is
the charge on (a) capacitor 1 and
(b) capacitor 2$?$

Ze-Han Lee

Numerade Educator

Problem 15

In Fig, $25-31,$ a 20.0 $\mathrm{V}$ battery is connected across capacitors
of capacitances $C_{1}=C_{6}=3.00 \mu \mathrm{F}$ and $C_{3}=C_{5}=2.00 C_{2}=2.00 C_{4}=4.00 \mu \mathrm{F}$ . What are (a) the equivalent capacitance $C_{\mathrm{eq}}$ of the capacitors and (b) the charge stored by $C_{\mathrm{eq}} ?$ What are $(\mathrm{c}) V_{1}$ and $(\mathrm{d}) q_{1}$ of capacitor $1,$ (e) $V_{2}$ and $(\mathrm{f}) q_{2}$ of capacitor $2,$ and $(\mathrm{g}) V_{3}$ and (h) $q_{3}$ of capacitor 3$?$

Salamat Ali

Numerade Educator

Problem 16

Plot 1 in Fig. $25-32 a$ gives the charge $q$ that can be stored on capacitor 1 versus the electric potential $V$ set up across it. The
vertical scale is set by $q_{s}=16.0 \mu \mathrm{C}$ , and the horizontal scale is set
by $V_{x}=2.0 \mathrm{V} .$ Plots 2 and 3 are similar plots for capacitors 2 and 3, respectively. Figure $25-32 b$ shows a circuit with those three
capacitors and a 6.0 $\mathrm{V}$ battery. What is the charge stored on
capacitor 2 in that circuit?

Ze-Han Lee

Numerade Educator

Problem 17

In Fig. $25-29$ , a potential difference of $V=100.0 \mathrm{V}$ is applied across a capacitor arrangement with capacitances $C_{1}=10.0 \mu \mathrm{F}$
$C_{2}=5.00 \mu \mathrm{F},$ and $C_{3}=4.00 \mu \mathrm{F}$ . If capacitor 3 undergoes electrical breakdown so that it becomes equivalent to conducting wire, what
is the increase in (a) the charge on capacitor 1 and (b) the potential
difference across capacitor 1?

Stanley Enemuo

Numerade Educator

Problem 18

Figure $25-33$ shows a circuit section of four air-filled capacitors that is connected to a larger circuit. The graph below the section shows
the electric potential $V(x)$ as a function of position $x$ along the lower
part of the section, through capacitor 4 . Similarly, the graph above the section shows the electric potential $V(x)$ as a function of position $x$
along the upper part of the section, through capacitors $1,2,$ and $3 .$ Capacitor 3 has a capacitance of 0.80$\mu \mathrm{F}$ . What are the capacitances of (a) capacitor 1 and (b) capacitor 2$?$

Ze-Han Lee

Numerade Educator

Problem 19

In Fig, $25-34,$ the battery has potential difference $V=9.0$ $\mathrm{V}, C_{2}=3.0 \mu \mathrm{F}, C_{4}=4.0 \mu \mathrm{F},$ and all the capacitors are initially uncharged. When switch $\mathrm{S}$ is closed, a total charge of 12$\mu \mathrm{C}$ passes through point $a$ and a total charge of 8.0$\mu \mathrm{C}$ passes through point
b. What are (a) $C_{1}$ and $(\mathrm{b}) C_{3}$ ?

Salamat Ali

Numerade Educator

Problem 20

Figure $25-35$ shows a variable "air gap" capacitor for manual tuning. Alternate plates are connected together; one group of plates is fixed in position, and the other group is
capable of rotation. Consider a capacitor of $n=8$ plates of alternating polarity, cach plate
having area $A=1.25 \mathrm{cm}^{2}$ and separated from adjacent plates by distance $d=3.40 \mathrm{mm} .$ What is the
maximum capacitance of the device?

Ze-Han Lee

Numerade Educator

Problem 21

In Fig. 25-36, the capacitances are $C_{1}=1.0 \mu \mathrm{F}$ and $C_{2}=3.0 \mu \mathrm{F},$ and both capacitors are charged to a potential difference of $V=100 \mathrm{V}$ but with opposite polarity as shown. Switches $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$ are now closed. (a) What is now the potential difference between points $a$ and $b ?$ What now is the charge
on capacitor (b) 1 and (c) 2$?$

Salamat Ali

Numerade Educator

Problem 22

In Fig.$25-37 $V= $10 \mathrm{V}, C_{1}$$=10$ $\mu \mathrm{F},$ and $C_{2}=C_{3}=20 \mu \mathrm{F}$ . Switch $\mathrm{S}$ is first thrown to the left side until capacitor 1 reaches equilibrium. Then the switch is thrown to the right. When equilibrium is again reached, how much charge is on capacitor 1$?$

Ze-Han Lee

Numerade Educator

Problem 23

The capacitors in Fig, $25-38$ are initially uncharged. The capacitances are $C_{1}=4.0 \mu \mathrm{F}, C_{2}=8.0 \mu \mathrm{F},$ and $C_{3}=12 \mu \mathrm{F}$ ,
and the battery's potential difference is $V=12 \mathrm{V} .$ When switch $\mathrm{S}$ is closed, how many electrons travel through (a) point $a$
(b) point $b,(\mathrm{c})$ point $c,$ and (d) point $d ?$ In the figure, do the electrons travel up or down through (e) point $b$ and
(f) point c?

Sheh Lit Chang

University of Washington

Problem 24

Figure $25-39$ represents two air-filled cylindrical capacitors connected in series across a battery with potential $V=10 \mathrm{V}$ of $1.5 \mathrm{cm},$ and a length of 5.0 $\mathrm{cm} .$ Capacitor 2 has an inner plate radius
of $2.5 \mathrm{mm},$ an outer plate radius of $1.0 \mathrm{cm},$ and a length of 9.0 $\mathrm{cm}$ . The outer plate of capacitor 2 is a conducting organic membrane that can be stretched, and the capacitor can be inflated to increase the plate separation. If the outer plate radius is increased to 2.5 $\mathrm{cm}$ by inflation, (a) how many electrons move through point $P$ and (b) do they move toward or away from the battery?

Ze-Han Lee

Numerade Educator

Problem 25

In Fig. $25-40,$ two parallel-plate capacitors (with air between the plates)
are connected to a battery. Capacitor 1
has a plate area of 1.5 $\mathrm{cm}^{2}$ and an electric field (between its plates) of magnitude
2000 $\mathrm{V} / \mathrm{m} .$ Capacitor 2 has a plate area of 0.70 $\mathrm{cm}^{2}$ and an electric field of magnitude 1500 $\mathrm{V} / \mathrm{m} .$ What is the
total charge on the two capacitors?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 26

Capacitor 3 in Fig. $25-41 a$ is a variable capacitor (its capacitance $C_{3}$ can be varied). Figure $25-41 b$ gives the electric potential $V_{1}$ across capacitor 1 versus $C_{3}$ . The horizontal scale is set by $C_{3 s}=12.0 \mu \mathrm{F} .$ Electric potential $V_{1}$ approaches an asymptote of 10 $\mathrm{V}$ as $C_{3} \rightarrow \infty .$ What are (a) the electric potential $V$ across the
battery, $\left($ b) $C_{1},$ and $(\mathrm{c}) C_{2} ?\right.$

Ze-Han Lee

Numerade Educator

Problem 27

Figure $25-42$ shows a 12.0 $\mathrm{V}$ battery and four uncharged capacitors of capacitances $C_{1}=1.00 \mu \mathrm{F}$
$C_{2}=2.00 \mu \mathrm{F}, C_{3}=3.00 \mu \mathrm{F},$ and $C_{4}=$ 4.00$\mu \mathrm{F}$ . If only switch $\mathrm{S}_{1}$ is closed,
what is the charge on (a) capacitor 1
(b) capacitor $2,$ (c) capacitor $3,$ and
(d) capacitor 4$?$ If both switches are closed, what is the charge on (e) capacitor $1,$ (f) capacitor $2,(\mathrm{g})$ capacitor
$3,$ and $(\mathrm{h})$ capacitor 4 ?

Salamat Ali

Numerade Educator

Problem 28

Figure $25-43$ displays a 12.0 V battery and 3 uncharged capacitors of capacitances $C_{1}=4.00 \mu \mathrm{F}$ ,
$C_{2}=6.00 \mu \mathrm{F},$ and $C_{3}=3.00 \mu \mathrm{F}$ . The
switch is thrown to the left side until capacitor 1 is fully charged. Then the
switch is thrown to the right. What is
the final charge on (a) capacitor $1,$
(b) capacitor $2,$ and $(c)$ capacitor 3$?$

Ze-Han Lee

Numerade Educator

Problem 29

What capacitance is required to store an energy of 10 $\mathrm{kW} \cdot \mathrm{h}$
at a potential difference of 1000 $\mathrm{V} ?$

Salamat Ali

Numerade Educator

Problem 30

How much energy is stored in 1.00 $\mathrm{m}^{3}$ of air due to the "fair
weather" electric field of magnitude 150 $\mathrm{V} / \mathrm{m} ?$

Ze-Han Lee

Numerade Educator

Problem 31

$\mathrm{A} 2.0 \mu \mathrm{F}$ capacitor and a 4.0$\mu \mathrm{F}$ capacitor are connected
in parallel across a 300 $\mathrm{V}$ potential difference. Calculate the total
energy stored in the capacitors.

Salamat Ali

Numerade Educator

Problem 32

A parallel-plate air-filled capacitor having area 40 $\mathrm{cm}^{2}$ and
plate spacing 1.0 $\mathrm{mm}$ is charged to a potential difference of 600 $\mathrm{V}$ .
Find (a) the capacitance, (b) the magnitude of the charge on each
plate, (c) the stored energy, (d) the electric field between the
plates, and (e) the energy density between the plates.

Ze-Han Lee

Numerade Educator

Problem 33

A charged isolated metal sphere of diameter 10 $\mathrm{cm}$ has a po-
tential of 8000 $\mathrm{V}$ relative to $V=0$ at infinity. Calculate the energy
density in the electric field near the surface of the sphere.

Salamat Ali

Numerade Educator

Problem 34

In Fig. $25-28,$ a potential difference $V=100 \mathrm{V}$ is applied
across a capacitor arrangement with capacitances $C_{1}=10.0 \mu \mathrm{F}$ $C_{2}=5.00 \mu \mathrm{F},$ and $C_{3}=4.00 \mu \mathrm{F} .$ What are (a) charge $q_{3},$ (b) poten-
tial difference $V_{3},$ and $(\mathrm{c})$ stored energy $U_{3}$ for capacitor $3,$ (d) $q_{1}$ (e) $V_{1},$ and $(\mathrm{f}) U_{1}$ for capacitor $1,$ and $(\mathrm{g}) q_{2},(\mathrm{h}) V_{2},$ and $(\mathrm{i}) U_{2}$ for
capacitor 2$?$

Ze-Han Lee

Numerade Educator

Problem 35

Assume that a stationary electron is a point of charge. What
is the energy density $u$ of its electric field at radial distances (a) $r=$
$1.00 \mathrm{mm},(\mathrm{b}) r=1.00 \mu \mathrm{m},(\mathrm{c}) r=1.00 \mathrm{nm},$ and (d) $r=1.00 \mathrm{pm}$ ?
(e) What is $u$ in the limit as $r \rightarrow 0 ?$

Salamat Ali

Numerade Educator

Problem 36

As a safety engineer, you must evaluate the practice of storing flammable conducting liquids in nonconducting containers.The company supplying a certain liquid has been using a squat, cylindrical plastic container of radius $r=0.20 \mathrm{m}$ and filling it to height
$h=10 \mathrm{cm},$ which is not the container's full interior height (Fig. $25-44 )$ . Your investigation reveals that during handling at the company, the exterior surface of the
container commonly acquires a negative charge density of magnitude 2.0$\mu \mathrm{C} / \mathrm{m}^{2}$ (approximately uniform). Because the liquid is a
conducting material, the charge on the container induces charge
separation within the liquid. (a) How much negative charge is induced in the center of the liquid's bulk? (b) Assume the capacitance of the central portion of the liquid relative to ground is
35 pF. What is the potential energy associated with the negative
charge in that effective capacitor? (c) If a spark occurs between the ground and the central portion of the liquid (through the vent-
ing port), the potential energy can be fed into the spark. The mini-
mum spark energy needed to ignite the liquid is 10 $\mathrm{mJ} .$ In this
situation, can a spark ignite the liquid?

Ze-Han Lee

Numerade Educator

Problem 37

The parallel plates in a capacitor, with a plate area of 8.50 $\mathrm{cm}^{2}$ and an air-filled separation of $3.00 \mathrm{mm},$ are
charged by a 6.00 $\mathrm{V}$ battery. They are then disconnected from the
battery and pulled apart (without discharge) to a separation of 8.00 $\mathrm{mm} .$ Neglecting fringing, find (a) the potential difference between the plates, (b) the initial stored energy, (c) the final stored energy, and (d) the work required to separate the plates.

Salamat Ali

Numerade Educator

Problem 38

In Fig. $25-29$ , a potential difference $V=100 \mathrm{V}$ is applied across a capacitor arrangement with capacitances $C_{1}=10.0 \mu \mathrm{F}$ $C_{2}=5.00 \mu \mathrm{F},$ and $C_{3}=15.0 \mu \mathrm{F} .$ What are (a) charge $q_{3},$ (b) potential difference $V_{3},$ and ( c) stored cnergy $U_{3}$ for capacitor $3,$ (d) $q_{1}$ (e) $V_{1},$ and $(\mathrm{f}) U_{1}$ for capacitor $1,$ and $(\mathrm{g}) q_{2},$ (h) $V_{2},$ and $(\mathrm{i}) U_{2}$ for capacitor 2$?$

Ze-Han Lee

Numerade Educator

Problem 39

In Fig. $25-45, C_{1}=10.0$ $\mu \mathrm{F}, \quad C_{2}=20.0 \mu \mathrm{F},$ and $C_{3}=$
25.0$\mu \mathrm{F} .$ If no capacitor $\mathrm{can}$ withstand a potential difference of more than 100 $\mathrm{V}$ without failure, what are (a) the magnitude of the maximum potential difference that can exist between points $A$ and $B$ and (b) the maximum energy that can be stored in
the three-capacitor arrangement?

Salamat Ali

Numerade Educator

Problem 40

An air-filled parallel-plate capacitor has a capacitance of
1.3 pF.The separation of the plates is doubled, and wax is inserted
between them. The new capacitance is 2.6 pF. Find the dielectric
constant of the wax.

Ze-Han Lee

Numerade Educator

Problem 41

A coaxial cable used in a transmission line has an inner
radius of 0.10 $\mathrm{mm}$ and an outer radius of 0.60 $\mathrm{mm}$ . Calculate the
capacitance per meter for the cable. Assume that the space
between the conductors is filled with polystyrene.

Salamat Ali

Numerade Educator

Problem 42

A parallel-plate air-filled capacitor has a capacitance of
50 pF. (a) If each of its plates has an area of 0.35 $\mathrm{m}^{2}$ , what is the
separation? (b) If the region between the plates is now filled with
material having $\kappa=5.6,$ what is the capacitance?

Ze-Han Lee

Numerade Educator

Problem 43

Given a 7.4 $\mathrm{pF}$ air-filled capacitor, you are asked to convert it to
a capacitor that can store up to 7.4$\mu \mathrm{J}$ with a maximum potential difference of 652 $\mathrm{V} .$ Which dielectric in Table $25-1$ should you use to fill
the gap in the capacitor if you do not allow for a margin of error?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 44

You are asked to construct a capacitor having a capacitance
near 1 $\mathrm{nF}$ and a breakdown potential in excess of 10$~ 000 \mathrm{V}$ . You
think of using the sides of a tall Pyrex drinking glass as a dielectric,
lining the inside and outside curved surfaces with aluminum foil to act as the plates. The glass is 15 $\mathrm{cm}$ tall with an inner radius of
3.6 $\mathrm{cm}$ and an outer radius of 3.8 $\mathrm{cm} .$ What are the (a) capacitance
and (b) breakdown potential of this capacitor?

Ze-Han Lee

Numerade Educator

Problem 45

A certain parallel-plate capacitor is filled with a dielectric
for which $\kappa=5.5 .$ The area of each plate is $0.034 \mathrm{m}^{2},$ and the plates
are separated by 2.0 $\mathrm{mm}$ . The capacitor will fail (short out and burn up) if the electric field between the plates exceeds 200 $\mathrm{kN} / \mathrm{C}$ . What
is the maximum energy that can be stored in the capacitor?

Salamat Ali

Numerade Educator

Problem 46

In Fig. $25-46,$ how much charge is
stored on the parallel-plate capacitors
by the 12.0 $\mathrm{V}$ battery? One is filled with air, and the other is filled with a di-
electric for which $\kappa=3.00 ;$ both capaci-
tors have a plate area of $5.00 \times 10^{-3} \mathrm{m}^{2}$
and a plate separation of 2.00 $\mathrm{mm} .$

Ze-Han Lee

Numerade Educator

Problem 47

A certain substance has a dielectric constant of 2.8 and a dielectric strength of 18 $\mathrm{MV} / \mathrm{m}$ . If it is used as the dielectric material in a parallel-plate capacitor, what minimum area
should the plates of the capacitor have to obtain a capacitance of $7.0 \times 10^{-2} \mu \mathrm{F}$ and to ensure that the capacitor will be able to with-
stand a potential difference of 4.0 $\mathrm{kV} ?$

Salamat Ali

Numerade Educator

Problem 48

Figure $25-47$ shows a parallelplate capacitor with a plate area $A$
$=5.56 \mathrm{cm}^{2}$ and separation $d=5.56$ mm. The left half of the gap is filled
with material of dielectric constant
$\kappa_{1}=7.00 ;$ the right half is filled with
material of dielectric constant $\kappa_{2}=$
$12.0 .$ What is the capacitance?

Ze-Han Lee

Numerade Educator

Problem 49

Figure $25-48$ shows a parallel-plate ca-
pacitor with a plate area $A=7.89 \mathrm{cm}^{2}$ and
plate separation $d=4.62 \mathrm{mm}$ . The top half of the gap is filled with material of dielectric
constant $\kappa_{1}=11.0 ;$ the bottom half is filled
with material of dielectric constant $\kappa_{2}=12.0 .$
What is the capacitance?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 50

Figure $25-49$ shows a parallelplate capacitor of plate area $A=10.5$
$\mathrm{cm}^{2}$ and plate separation $2 d=7.12 \mathrm{mm} .$
The left half of the gap is filled with material of dielectric constant $\kappa_{1}=21.0$
the top of the right half is filled with material of diclectric constant $\kappa_{2}=42.0$ ; the bottom of the right half is filled
with material of dielectric constant $\kappa_{3}=$
$58.0 .$ What is the capacitance?

Ze-Han Lee

Numerade Educator

Problem 51

A parallel-plate capacitor has a capacitance of $100 \mathrm{pF},$ a plate area of $100 \mathrm{cm}^{2},$ and a mica dielectric $(\kappa=5.4)$
completely filling the space between the plates. At 50 $\mathrm{V}$ potential
difference, calculate (a) the electric field magnitude $E$ in the mica, (b) the magnitude of the free charge on the plates, and (c) the magnitude of the induced surface charge on the mica.

Salamat Ali

Numerade Educator

Problem 52

For the arrangement of Fig. $25-17,$ suppose that the battery remains connected while the dielectric slab is being introduced.
Calculate (a) the capacitance, (b) the charge on the capacitor
plates, (c) the electric field in the gap, and (d) the electric field in
the slab, after the slab is in place.

Ze-Han Lee

Numerade Educator

Problem 53

A parallel-plate capacitor has plates of area 0.12 $\mathrm{m}^{2}$ and a
separation of 1.2 $\mathrm{cm}$ . A battery charges the plates to a potential difference of 120 $\mathrm{V}$ and is then disconnected. A dielectric slab of thickness 4.0 $\mathrm{mm}$ and dielectric constant 4.8 is then placed symmetrically
between the plates (a) What is the capacitance before the slab is inserted? (b) What is the capacitance with the slab in place? What is the free charge $q(\mathrm{cc})$ before and (d) after the slab is inserted? What is the magnitude of the electric field (e) in the space between the plates and dielectric and (f) in the dielectric itself? (g) With the slab in place, what is the potential difference across the plates? (h) How much external work is involved in inserting the slab?

Linda Winkler

Numerade Educator

Problem 54

Two parallel plates of area 100 $\mathrm{cm}^{2}$ are given charges of equal magnitudes $8.9 \times 10^{-7} \mathrm{C}$ but opposite signs. The electric
field within the diclectric material filling the space between the plates is $1.4 \times 10^{6} \mathrm{V} / \mathrm{m} .$ (a) Calculate the dielectric constant of the material. (b) Determine the magnitude of the charge induced on each dielectric surface.

Ze-Han Lee

Numerade Educator

Problem 55

The space between two concentric conducting spherical shells of radii $b=1.70 \mathrm{cm}$ and $a=1.20 \mathrm{cm}$ is filled with
a substance of dielectric constant $\kappa=23.5 . \mathrm{A}$
potential difference $V=73.0 \mathrm{V}$ is applied across the inner and outer shells. Determine (a) the capacitance of the device, (b) the free charge $q$ on
the inner shell, and (c) the charge $q^{\prime}$ induced
along the surface of the inner shell.

Salamat Ali

Numerade Educator

Problem 56

In Fig. $25-50$ , the battery potential
difference $V$ is 10.0 $\mathrm{V}$ and each of the seven
capacitors has capacitance 10.0$\mu \mathrm{F} .$ What is the charge on (a) capacitor 1 and $(b)$ capacitor 2$?$

Ze-Han Lee

Numerade Educator

Problem 57

In Fig. $25-51, V$ $=9.0 \mathrm{V}, C_{1}=C_{2}=30$ $\mu \mathrm{F},$ and $C_{3}=C_{4}=15 \mu \mathrm{F}$ . What is the charge on capacitor 4$?$

Salamat Ali

Numerade Educator

Problem 58

(a) If $C=50 \mu \mathrm{F}$ in Fig. $25-52,$ what
is the equivalent capacitance between
points $A$ and $B ?$ (Hint: First imagine
that a battery is connected between
those two points.) (b) Repeat for points
$A$ and $D .$

Ze-Han Lee

Numerade Educator

Problem 59

In Fig. $25-53, V=12 \mathrm{V}, C_{1}=C_{4}=$
$2.0 \mu \mathrm{F}, C_{2}=4.0 \mu \mathrm{F},$ and $C_{3}=1.0 \mu \mathrm{F}$
What is the charge on capacitor 4 ?

Salamat Ali

Numerade Educator

Problem 60

The chocolate crumb mystery. This story begins with Problem 60 in
Chapter $23 .$ As part of the investigation
of the biscuit factory explosion, the electric potentials of the workers were
measured as they emptied sacks of
chocolate crumb powder into the loading bin, stirring up a cloud of the powder
around themselves. Each worker had an electric potential of about 7.0 $\mathrm{kV}$ relative to the ground, which was
taken as zero potential. (a) Assuming that each worker was effectively a capacitor with a typical capacitance of 200 $\mathrm{pF}$ , find the energy
stored in that effective capacitor. If a single spark between the worker and any conducting object connected to the ground neutral-
ized the worker, that energy would be transferred to the spark.
According to measurements, a spark that could ignite a cloud of chocolate crumb powder, and thus set off an explosion, had to have
an energy of at least 150 $\mathrm{mJ} .$ (b) Could a spark from a worker have
set off an explosion in the cloud of powder in the loading bin? (The
story continues with Problem 60 in
Chapter $26 .$ )

Ze-Han Lee

Numerade Educator

Problem 61

Figure $25-54$ shows capacitor
$1\left(C_{1}=8.00 \mu \mathrm{F}\right),$ capacitor 2$\left(C_{2}\right.$
$=6.00 \mu \mathrm{F} ),$ and capacitor 3$\left(C_{3}=\right.$ 8.00$\mu \mathrm{F}$ ) connected to a 12.0 $\mathrm{V}$ battery. When switch $\mathrm{S}$ is closed so as
to connect uncharged capacitor $4\left(C_{4}=6.00 \mu \mathrm{F}\right),$ (a) how much charge passes through point $P$ from the battery and (b) how much
charge shows up on capacitor 4$?$ (c) Explain the discrepancy in
those two results.

Salamat Ali

Numerade Educator

Problem 62

Two air-filled, parallel-plate capacitors are to be connected to a
10 $\mathrm{V}$ battery, first individually, then in series, and then in parallel. In
those arrangements, the energy stored in the capacitors turns out to be, listed least to greatest: $75 \mu \mathrm{J}, 100 \mu \mathrm{J}, 300 \mu \mathrm{J},$ and 400$\mu \mathrm{J}$ Of the
two capacitors, what is the (a) smaller and (b) greater capacitance?

Ze-Han Lee

Numerade Educator

Problem 63

Two parallel-plate capacitors, 6.0$\mu \mathrm{F}$ each, are connected
in series to a 10 $\mathrm{V}$ battery. One of the capacitors is then squeezed
so that its plate separation is halved. Because of the squeezing,
(a) how much additional charge is transferred to the capacitors by the battery and (b) what is the increase in the total charge stored
on the capacitors (the charge on the positive plate of one capacitor
plus the charge on the positive plate of the other capacitor)?

Salamat Ali

Numerade Educator

Problem 64

In Fig. $25-55, V=12 \mathrm{V}, C_{1}=$ $C_{5}=C_{6}=6.0 \mu \mathrm{F},$ and $C_{2}=C_{3}=C_{4}=$
4.0$\mu \mathrm{F} .$ What are (a) the net charge stored on the capacitors and (b) the
charge on capacitor 4$?$

Ze-Han Lee

Numerade Educator

Problem 65

In Fig. $25-56,$ the parallel-plate
capacitor of plate area $2.00 \times 10^{-2} \mathrm{m}^{2}$
is filled with two diclectric slabs, each with thickness 2.00 $\mathrm{mm}$ . One slab has di-
electric constant 3.00 , and the other,
$4.00 .$ How much charge does the 7.00 $\mathrm{V}$
battery store on the capacitor?

Salamat Ali

Numerade Educator

Problem 66

A cylindrical capacitor has radii $a$
and $b$ as in Fig. $25-6.6$ . Show that half the
stored electric potential energy lies
within a cylinder whose radius is
$r=\sqrt{a b} .$

Ze-Han Lee

Numerade Educator

Problem 67

A capacitor of capacitance $C_{1}=$ 6.00$\mu \mathrm{F}$ is connected in series with a capacitor of capacitance $C_{2}=$
$4.00 \mu \mathrm{F},$ and a potential difference of 200 $\mathrm{V}$ is applied across the pair. (a) Calculate the equivalent capacitance. What are (b) charge
$q_{1}$ and $(\mathrm{c})$ potential difference $V_{1}$ on capacitor 1 and $(\mathrm{d}) q_{2}$ and $(\mathrm{e})$
$V_{2}$ on capacitor 2$?$

Salamat Ali

Numerade Educator

Problem 68

Repeat Problem 67 for the same two capacitors but with them
now connected in parallel.

Vishal Gupta

Numerade Educator

Problem 69

A certain capacitor is charged to a potential difference $V .$ If
you wish to increase its stored energy by $10 \%,$ by what percentage
should you increase $V$ ?

Salamat Ali

Numerade Educator

Problem 70

A slab of copper of thickness $b=2.00 \mathrm{mm}$ is thrust into a parallelplate capacitor of plate area $A=2.40$ $\mathrm{cm}^{2}$ and plate separation $d=5.00$ (a) What is the capacitance after the slab is introduced? (b) If a charge
$q=3.40 \mu \mathrm{C}$ is maintained on the plates, what is the ratio of the stored energy before to that after the
slab is inserted? (c) How much work is done on the slab as it is inserted? (d) Is the slab sucked in or must it be pushed in?

Ze-Han Lee

Numerade Educator

Problem 71

Repeat Problem $70,$ assuming that a potential difference $V=$
$85.0 \mathrm{V},$ rather than the charge, is held constant.

Salamat Ali

Numerade Educator

Problem 72

A potential difference of 300 $\mathrm{V}$ is applied to a series
connection of two capacitors of capacitances $C_{1}=2.00 \mu \mathrm{F}$ and
$C_{2}=8.00 \mu \mathrm{F} .$ What are (a) charge $q_{1}$ and (b) potential difference $V_{1}$ on capacitor 1 and $(\mathrm{c}) q_{2}$ and $(\mathrm{d}) \mathrm{V}_{2}$ on capacitor 2$?$ The charged capacitors are then disconnected from each other and from the
battery. Then the capacitors are reconnected with plates of the same signs wired together (the battery is not used). What now are
(e) $q_{1},(\mathrm{f}) V_{1},(\mathrm{g}) q_{2},$ and $(\mathrm{h}) V_{2} ?$ Suppose, instead, the capacitors charged in part (a) are reconnected with plates of opposite signs
wired together. What now are (i) $q_{1},(\mathrm{j}) V_{1},(\mathrm{k}) q_{2},$ and $(1) V_{2} ?$

Ze-Han Lee

Numerade Educator

Problem 73

Figure $25-58$ shows a four capacitor arrangement that is connected to a larger circuit at points $A$
and $B .$ The capacitances are $C_{1}==$
10$\mu \mathrm{F}$ and $C_{2}=C_{3}=C_{4}=20 \mu \mathrm{F}$
The charge on capacitor 1 is 30$\mu \mathrm{C}$ .
What is the magnitude of the potential difference $V_{A}-V_{B} ?$

Sheh Lit Chang

University of Washington

Problem 74

You have two plates of copper, a sheet of mica (thickness =
$0.10 \mathrm{mm}, \kappa=5.4 ),$ a sheet of glass (thickness $=2.0 \mathrm{mm}, \kappa=7.0 )$
and a slab of paraffin (thickness $=1.0 \mathrm{cm}, \kappa=2.0 ) .$ To make a parallel-plate capacitor with the largest $C,$ which sheet should you
place between the copper plates?

Ze-Han Lee

Numerade Educator

Problem 75

A capacitor of unknown capacitance $C$ is charged to 100 $\mathrm{V}$ and
connected across an initially uncharged 60 $\mu$ F capacitor. If the final
potential difference across the 60$\mu \mathrm{F}$ capacitor is $40 \mathrm{V},$ what is $C$ ?

Salamat Ali

Numerade Educator

Problem 76

A 10 $\mathrm{V}$ battery is connected to a series of $n$ capacitors, each of
capacitance 2.0$\mu \mathrm{F}$ . If the total stored energy is $25 \mu \mathrm{J},$ what is $n$ ?

Ze-Han Lee

Numerade Educator

Problem 77

In Fig. $25-59$ , two parallel-plate capacitors $A$ and $B$ are connected in parallel across a 600 $\mathrm{V}$ battery. Each plate has area 80.0 $\mathrm{cm}^{2}$ ;
the plate separations are 3.00 $\mathrm{mm}$ ;
Capacitor $A$ is filled with air; capacitor $B$ is filled with a dielectric of dielectric constant $\kappa=2.60$ .
Find the magnitude of the electric field within (a) the dielectric of
capacitor $B$ and (b) the air of capacitor $A .$ What are the free charge densities $\sigma$ on the higher-potential plate of (c) capacitor $A$ and
(d) capacitor $B ?$ (e) What is the induced charge density $\sigma^{\prime}$ on the
top surface of the dielectric?

Salamat Ali

Numerade Educator

Problem 78

You have many 2.0$\mu \mathrm{F}$ capacitors, each capable of with-
standing 200 $\mathrm{V}$ without undergoing electrical breakdown (in which
they conduct charge instead of storing it). How would you assemble a combination having an equivalent capacitance of (a) 0.40$\mu \mathrm{F}$
and (b) 1.2$\mu \mathrm{F}$ , each combination capable of withstanding 1000 $\mathrm{v} ?$

Ze-Han Lee

Numerade Educator

Problem 79

A parallel-plate capacitor has charge $q$ and plate area $A .$
(a) By finding the work needed to increase the plate separation
from $x$ to $x+d x,$ determine the force between the plates. (Hint: See Eq. $8-22 .$ ) (b) Then show that the force per unit area (the electrostatic stress) acting on either plate is equal to the energy density $\varepsilon_{0} E^{2} / 2$ between the plates.

Salamat Ali

Numerade Educator

Problem 80

A capacitor is charged until its stored energy is 4.00 $\mathrm{J} . \mathrm{A}$ second capacitor is then connected to it in parallel. (a) If the charge
distributes equally, what is the total energy stored in the electric
fields? (b) Where did the missing energy go?

Ze-Han Lee

Numerade Educator