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Fundamentals of Physics, Volume 2

David Halliday & Robert Resnick & Jearl Walker

Chapter 25

Capacitance - all with Video Answers

Educators

Chapter Questions

02:16

Problem 1

The two metal objects in Fig. 25.7 have net charges of $+70 \mathrm{pC}$ 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 capacitance become? (c) What does the potential difference become?
FIGURE CANT COPY
Fig. 25.7

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
01:09

Problem 2

The capacitor in Fig. 25.8 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?
FIGURE CANT COPY
Fig. 25.8

Prabhu Ramji
Prabhu Ramji
Numerade Educator
01:46

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
Salamat Ali
Numerade Educator
02:40

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
Ze-Han Lee
Numerade Educator
01:35

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
Salamat Ali
Numerade Educator
02:01

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
Ze-Han Lee
Numerade Educator
03:53

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. 25.9, 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 $/ \mathrm{m}^3$. 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 $C / A$ ?
FIGURE CANT COPY
Fig. 25.9

Sunita Kumari
Sunita Kumari
Numerade Educator
01:53

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?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
03:23

Problem 9

Each of the uncharged capacitors in Fig. 25.10 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 A?
FIGURE CANT COPY
Fig. 25.10

David Morabito
David Morabito
Numerade Educator
02:12

Problem 10

In Fig. 25.11, 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 $4.00 \mu \mathrm{F}$.
FIGURE CANT COPY
Fig. 25.11

Sunita Kumari
Sunita Kumari
Numerade Educator
02:12

Problem 11

In Fig. 25.12, find the equivalent capacitance of the combination. Assume that $C_1=10.0 \mu \mathrm{F}, C_2=5.00 \mu \mathrm{F}$, and $C_3=4.00 \mu \mathrm{F}$.
FIGURE CANT COPY
Fig. 25.12

Sunita Kumari
Sunita Kumari
Numerade Educator
01:41

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 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?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
01:55

Problem 13

A $100 \mathrm{pF}$ 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
Salamat Ali
Numerade Educator
06:43

Problem 14

In Fig. 25.13 , the battery has a potential difference of $V=10.0 \mathrm{~V}$ and the five capacitors each have a capacitance of $10.0 \mu \mathrm{F}$. What is the charge on (a) capacitor 1 and (b) capacitor 2 ?
FIGURE CANT COPY
Fig. 25.14

Sunita Kumari
Sunita Kumari
Numerade Educator
03:51

Problem 15

In Fig. 25.14, 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 (c) $V_1$ and (d) $q_1$ of capacitor 1 , (e) $V_2$ and (f) $q_2$ of capacitor 2 , and (g) $V_3$ and (h) $q_3$ of capacitor 3 ?

Salamat Ali
Salamat Ali
Numerade Educator
03:25

Problem 16

Plot 1 in Fig. $25.15 a$ 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 $V_s=2.0 \mathrm{~V}$. Plots 2 and 3 are similar plots for capacitors 2 and 3, respectively. Figure $25.15 b$ 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?
FIGURE CANT COPY
Fig. 25.15

Keshav Singh
Keshav Singh
Numerade Educator
02:55

Problem 17

In Fig. 25.12, 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 ?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
02:50

Problem 18

Figure 25.16 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 ?
FIGURE CANT COPY
Fig. 25.16

Ze-Han Lee
Ze-Han Lee
Numerade Educator
05:45

Problem 19

In Fig. 25.17, 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 (b) $C_3$ ?
FIGURE CANT COPY
Fig. 25.17

Sunita Kumari
Sunita Kumari
Numerade Educator
03:04

Problem 20

Figure 25.18 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, each 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?
FIGURE CANT COPY
Fig. 25.18

Sunita Kumari
Sunita Kumari
Numerade Educator
02:12

Problem 21

In Fig. 25.19, the capacitances are $C_1=1.0 \mu \mathrm{F}$ and $C_2=3.0 \mu \mathrm{F}$; both capacitors are charged to a potential difference of $V=100 \mathrm{~V}$ but with opposite polarity as shown. Switches $S_1$ 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 ?
FIGURE CANT COPY
Fig. 25.19

Salamat Ali
Salamat Ali
Numerade Educator
04:33

Problem 22

In Fig. 25.20, $V=10 \mathrm{~V}$, $C_1=10 \mu \mathrm{F}$, and $C_2=C_3=20 \mu \mathrm{F}$. Switch $S$ 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 ?
FIGURE CANT COPY
Fig. 25.20

Sunita Kumari
Sunita Kumari
Numerade Educator
07:33

Problem 23

The capacitors in Fig. 25.21 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 $b$, (c) point $c$, and (d) point $d$ ? In the figure, do the electrons travel up or down through (e) point $b$ and (f) point $c$ ?
FIGURE CANT COPY
Fig. 25.21

Sunita Kumari
Sunita Kumari
Numerade Educator
05:59

Problem 24

Figure 25.22 represents two air-filled cylindrical capacitors connected in series across a battery with potential $V=10 \mathrm{~V}$. Capacitor 1 has an inner plate radius of $5.0 \mathrm{~mm}$, an outer plate radius 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?
FIGURE CANT COPY
Fig. 25.22

Morgan Cheatham
Morgan Cheatham
Numerade Educator
02:24

Problem 25

In Fig. 25.23, two parallelplate 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?
FIGURE CANT COPY
Fig. 25.23

Eric Mockensturm
Eric Mockensturm
Numerade Educator
06:08

Problem 26

Capacitor 3 in Fig. $25.24 a$ is a variable capacitor (its capacitance $C_3$ can be varied). Figure $25.24 b$ 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, (b) $C_1$, and (c) $C_2$ ?
FIGURE CANT COPY
Fig. 25.24

Sunita Kumari
Sunita Kumari
Numerade Educator
05:40

Problem 27

Figure 25.25 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 $S_1$ 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, (g) capacitor 3, and (h) capacitor 4?
FIGURE CANT COPY
Fig. 25.25

Keshav Singh
Keshav Singh
Numerade Educator
04:38

Problem 28

Figure 25.26 displays a $12.0 \mathrm{~V}$ 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 ?
FIGURE CANT COPY
Fig. 25.26

Keshav Singh
Keshav Singh
Numerade Educator
01:06

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
Salamat Ali
Numerade Educator
01:37

Problem 30

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

Sunita Kumari
Sunita Kumari
Numerade Educator
00:54

Problem 31

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.

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
02:22

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
Ze-Han Lee
Numerade Educator
01:23

Problem 33

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

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
10:47

Problem 34

In Fig. 25.11, 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) potential difference $V_3$, and (c) stored energy $U_3$ for capacitor 3, (d) $q_1$, (e) $V_1$, and (f) $U_1$ for capacitor 1 , and (g) $q_2$, (h) $V_2$, and (i) $U_2$ for capacitor 2 ?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
02:35

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) $r=1.00 \mathrm{~mm}$, (b) $r=1.00 \mu \mathrm{m}$, (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
Salamat Ali
Numerade Educator
03:28

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. 25.27). 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 $35 \mathrm{pF}$. 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 venting port), the potential energy can be fed into the spark. The minimum spark energy needed to ignite the liquid is $10 \mathrm{~mJ}$. In this situation, can a spark ignite the liquid?
FIGURE CANT COPY
Fig. 25.27

Keshav Singh
Keshav Singh
Numerade Educator
03:45

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
Salamat Ali
Numerade Educator
15:22

Problem 38

In Fig. 25.12, 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 energy $U_3$ for capacitor 3 , (d) $q_1$, (e) $V_1$, and (f) $U_1$ for capacitor 1 , and (g) $q_2$, (h) $V_2$, and (i) $U_2$ for capacitor 2 ?

Rachel B.
Rachel B.
Numerade Educator
04:37

Problem 39

In Fig. 25.28, $C_1=10.0 \mu \mathrm{F}, C_2=20.0 \mu \mathrm{F}$, and $C_3=25.0 \mu \mathrm{F}$. If no capacitor 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?
FIGURE CANT COPY
Fig. 25.28

Sunita Kumari
Sunita Kumari
Numerade Educator
01:07

Problem 40

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

Ze-Han Lee
Ze-Han Lee
Numerade Educator
01:19

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
Salamat Ali
Numerade Educator
01:46

Problem 42

A parallel-plate air-filled capacitor has a capacitance of $50 \mathrm{pF}$. (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
Ze-Han Lee
Numerade Educator
02:24

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 25.5.1 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
03:26

Problem 44

You are asked to construct a capacitor having a capacitance near $1 \mathrm{nF}$ and a breakdown potential in excess of $10000 \mathrm{~V}$. 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
Ze-Han Lee
Numerade Educator
00:50

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
Salamat Ali
Numerade Educator
02:49

Problem 46

In Fig. 25.29, how much charge is stored on the parallelplate capacitors by the $12.0 \mathrm{~V}$ battery? One is filled with air, and the other is filled with a dielectric for which $\kappa=3.00$; both capacitors have a plate area of $5.00 \times 10^{-3} \mathrm{~m}^2$ and a plate separation of $2.00 \mathrm{~mm}$.
FIGURE CANT COPY
Fig. 25.29

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
03:16

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 withstand a potential difference of $4.0 \mathrm{kV}$ ?

Vishal Gupta
Vishal Gupta
Numerade Educator
03:44

Problem 48

Figure 25.30 shows a parallel-plate capacitor with a plate area $A=5.56 \mathrm{~cm}^2$ and separation $d=5.56 \mathrm{~mm}$. 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?
FIGURE CANT COPY
Fig. 25.30

Sunita Kumari
Sunita Kumari
Numerade Educator
03:36

Problem 49

Figure 25.31 shows a parallel-plate capacitor 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?
FIGURE CANT COPY
Fig. 25.31

Keshav Singh
Keshav Singh
Numerade Educator
09:04

Problem 50

Figure 25.32 shows a parallel-plate 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 dielectric 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?
FIGURE CANT COPY
Fig. 25.32

Eduard Sanchez
Eduard Sanchez
Numerade Educator
04:11

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
Salamat Ali
Numerade Educator
05:18

Problem 52

For the arrangement of Fig. 25.6.2, 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
Ze-Han Lee
Numerade Educator
07:08

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 $q$ (c) 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?

Keshav Singh
Keshav Singh
Numerade Educator
02:09

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 dielectric 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.

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
02:24

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 $\kappa=23.5$. 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
Salamat Ali
Numerade Educator
03:58

Problem 56

In Fig. 25.33, 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 ?
FIGURE CANT COPY
Fig. 25.33

Ze-Han Lee
Ze-Han Lee
Numerade Educator
01:08

Problem 57

In Fig. 25.34, $V=9.0 \mathrm{~V}, C_1=C_2=30 \mu \mathrm{F}$, and $C_3=C_4=$ $15 \mu \mathrm{F}$. What is the charge on capacitor 4 ?
FIGURE CANT COPY
Fig. 25.34

Salamat Ali
Salamat Ali
Numerade Educator
03:05

Problem 58

(a) If $C=50 \mu \mathrm{F}$ in Fig. 25.35, what is the equivalent capacitance between points $A$ and $B$ ? (b) Repeat for points $A$ and $D$.
FIGURE CANT COPY
Fig. 25.35

Ze-Han Lee
Ze-Han Lee
Numerade Educator
01:24

Problem 59

In Fig. 25.36, $V=12 \mathrm{~V}, C_1=$ $C_4=2.0 \mu \mathrm{F}, C_2=4.0 \mu \mathrm{F}$, and $C_3=$ $1.0 \mu \mathrm{F}$. What is the charge on capacitor 4 ?
FIGURE CANT COPY
Fig. 25.36

Salamat Ali
Salamat Ali
Numerade Educator
01:41

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 neutralized 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.)

Keshav Singh
Keshav Singh
Numerade Educator
03:22

Problem 61

Figure 25.37 shows capacitor $1\left(C_1=8.00 \mu \mathrm{F}\right)$, capacitor $2\left(C_2=6.00 \mu \mathrm{F}\right)$, and capacitor $3\left(C_3=8.00\right.$ $\mu \mathrm{F})$ 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
Salamat Ali
Numerade Educator
03:00

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
Ze-Han Lee
Numerade Educator
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Problem 63

The inner and outer cylindrical conductors of a long coaxial cable have diameters $a=0.15 \mathrm{~mm}$ and $b=2.1$ $\mathrm{mm}$. What is the capacitance per unit length?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
01:11

Problem 64

What is the capacitance of Earth, viewed as an isolated conducting sphere of radius $6370 \mathrm{~km}$ ?

Shahab Ullah
Shahab Ullah
Numerade Educator
05:27

Problem 65

An isolated conducting sphere has radius $R=6.85 \mathrm{~cm}$ and charge $q=1.25 \mathrm{nC}$. (a) How much potential energy is stored in the electric field? (b) What is the energy density at the surface of the sphere? (c) What is the radius $R_0$ of an imaginary spherical surface such that one-half of the stored potential energy lies within it?

Narayan Hari
Narayan Hari
Numerade Educator
00:56

Problem 66

On a day with low humidity, you can become charged by walking over certain carpets (there is charge transfer between the carpet and your shoes). If a spark jumps between your hand and a metal doorknob when the separation is about $5.0 \mathrm{~mm}$, you were probably at a potential of $15 \mathrm{kV}$ relative to the doorknob. To determine your accumulated charge $q$, make the rough approximation your body can be represented by a uniformly charged conducting sphere with radius $R=25 \mathrm{~cm}$ in radius and isolated from its surroundings. What is $q$ ?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:07

Problem 67

Module 8.3 relates force to potential energy: $|F|=d U / d x$. (a) Apply that relationship to a parallel-plate capacitor with charge $q$, plate area $A$, and plate separation $x$ to find an expression for the magnitude of the force between the plates. (b) Evaluate the magnitude of that force for $q=6.00 \mu \mathrm{C}$ and $A=2.50 \mathrm{~cm}^2$. (c) Electrostatic stress is the force per unit area $|F / A|$ on either plate. Find an expression for the stress in terms of $\varepsilon_0$ and the magnitude $E$ of the electric field between the plates. (d) Evaluate the stress for a potential difference of $110 \mathrm{~V}$ and a plate separation of $x=2.00 \mathrm{~mm}$.

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
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Problem 68

A capacitor is to be designed to operate, with constant capacitance, in an environment of fluctuating temperature. As shown in Fig. 25.38, the capacitor is a parallel-plate type with thin plastic "spacers" to keep the plates aligned. (a) Show that the rate of change of capacitance $C$ with temperature $T$ is given by
$$
\frac{d C}{d T}=C\left(\frac{1}{A} \frac{d A}{d T}-\frac{1}{x} \frac{d x}{d T}\right)
$$
where $A$ is the plate area and $x$ the plate separation, both of which vary with a temperature change. (b) If the plates are aluminum, what should be the coefficient of thermal expansion of the spacers in order that the capacitance not vary with temperature? (Neglect the effect of the spacers on the capacitance.)
FIGURE CANT COPY
Fig. 25.38

Victor Salazar
Victor Salazar
Numerade Educator
01:58

Problem 69

An ideal diode allows negative charge (electrons) to move through it only in the direction opposite the schematic arrow in a circuit diagram. Figure 25.39 shows a circuit with two such ideal diodes and two identical capacitors $C$. A $100 \mathrm{~V}$ battery is connected across the input terminals $a$ and $b$ (the potential difference between them is $100 \mathrm{~V}$ ). What is the output voltage $V_{\text {out }}$ if the battery's positive terminal is connected to (a) $a$ and then (b) $b$ ?
FIGURE CANT COPY
Fig. 25.39

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:19

Problem 70

The ability of a capacitor to store potential energy is the basis of defibrillator devices, which are used by emergency teams to stop the fibrillation of heart attack victims (Fig. 25.40). In the portable version, a battery charges a capacitor to a high potential difference, storing a large amount of energy in less than a minute. The battery maintains only a modest potential difference; an electronic circuit repeatedly uses that potential difference to greatly increase the potential difference of the capacitor. The power, or rate of energy transfer, during this process is also modest. Conducting leads ("paddles") are placed on the victim's chest. When a control switch is closed, the capacitor sends a portion of its stored energy from paddle to paddle through the victim. (a) If a $70 \mu \mathrm{F}$ capacitor in a defibrillator is charged to $5.0 \mathrm{kV}$, what is the stored potential energy? (b) If $23 \%$ of that energy is sent through the chest in $2.0 \mathrm{~ms}$, what is the power of the pulse?
FIGURE CANT COPY
Fig. 25.40

Narayan Hari
Narayan Hari
Numerade Educator
04:00

Problem 71

Figure 25.41 shows two capacitors in series, with a rigid center section that can be moved vertically, either upward or downward. (a) The plate area $A$ is the same for the capacitors. In terms of $A, a, b$, and $\varepsilon_0$, what is the equivalent capacitance $C$ ? (b) Evaluate $C$ for $A=2.0 \mathrm{~cm}^2, a=7.0 \mathrm{~mm}$, and $b=4.0 \mathrm{~mm}$. (c) If the center section is moved downward (without touching the bottom plate), does $C$ increase, decrease, or stay the same?
FIGURE CANT COPY
Fig. 25.41

Vishal Gupta
Vishal Gupta
Numerade Educator
01:02

Problem 72

A person walking through airborne dust in, say, a cosmetic plant can possibly become dangerously charged by contact with the floor and various objects that are touched. Safety engineers often calculate the danger threshold for the electric potential on a person by modeling the person as a spherical capacitor of radius $R=1.8 \mathrm{~m}$. What electric potential corresponds to the threshold value $U_t=150 \mathrm{~mJ}$ of stored energy at which a spark could ignite the dust and set off an explosion?

Dominador Tan
Dominador Tan
Numerade Educator