University Physics with Modern Physics

Roger A. Freedman, Hugh D. Young

Chapter 24

Capacitance and Dielectrics - all with Video Answers

Educators

Chapter Questions

Problem 1

The plates of a parallel-plate capacitor are $2.50 \mathrm{~mm}$ apart, and each carries a charge of magnitude $80.0 \mathrm{nC}$. The plates are in vacuum. The electric field between the plates has a magnitude of $4.00 \times 10^{6} \mathrm{~V} / \mathrm{m} .$ What is (a) the potential difference between the plates; (b) the area of each plate; (c) the capacitance?

Ze-Han Lee

Ze-Han Lee

Numerade Educator

Problem 2

The plates of a parallel-plate capacitor are $3.28 \mathrm{~mm}$ apart, and each has an area of $9.82 \mathrm{~cm}^{2}$. Each plate carries a charge of magnitude $4.35 \times 10^{-8} \mathrm{C}$. The plates are in vacuum. What is (a) the capacitance;
(b) the potential difference between the plates; (c) the magnitude of the electric field between the plates?

Penny Riley

Numerade Educator

Problem 3

A parallel-plate air capacitor of capacitance $245 \mathrm{pF}$ has a charge of magnitude $0.148 \mu \mathrm{C}$ on each plate. The plates are $0.328 \mathrm{~mm}$ apart. (a) What is the potential difference between the plates? (b) What is the area of each plate? (c) What is the electric-field magnitude between the plates?
(d) What is the surface charge density on each plate?

Sarah Mccrumb

Numerade Educator

Problem 4

A $5.00 \mu \mathrm{F}$ parallel-plate capacitor is connected to a $12.0 \mathrm{~V}$
battery. After the capacitor is fully charged, the battery is disconnected without loss of any of the charge on the plates. (a) A voltmeter is connected across the two plates without discharging them. What does it read? (b) What would the voltmeter read if (i) the plate separation were doubled; (ii) the radius of each plate were doubled but their separation was unchanged?

Shubham Verma

Texas A&M University

Problem 5

A $10.0 \mu \mathrm{F}$ parallel-plate capacitor with circular plates is connected to a $12.0 \mathrm{~V}$ battery. (a) What is the charge on each plate? (b) How much charge would be on the plates if their separation were doubled while the capacitor remained connected to the battery? (c) How much charge would be on the plates if the capacitor were connected to the $12.0 \mathrm{~V}$ battery after the radius of each plate was doubled without changing their separation?

Ze-Han Lee

Numerade Educator

Problem 6

A $5.00 \mathrm{pF},$ parallel-plate, air-filled capacitor with circular to a $12.0 \mathrm{~V}$ battery. (a) What is the charge on each plate? (b) How much charge would be on the plates if their separation were doubled while the capacitor remained connected to the battery? (c) How much charge would be on the plates if the capacitor were connected to the $12.0 \mathrm{~V}$ battery after the radius of each plate was doubled without changing their separation?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 7

A parallel-plate air capacitor is to store charge of magnitude $240.0 \mathrm{pC}$ on each plate when the potential difference between the plates is $42.0 \mathrm{~V}$. (a) If the area of each plate is $6.80 \mathrm{~cm}^{2}$, what is the separation between the plates? (b) If the separation between the two plates is double the value calculated in part (a), what potential difference is required for the capacitor to store charge of magnitude $240.0 \mathrm{pC}$ on each plate?

Ze-Han Lee

Numerade Educator

Problem 8

A cylindrical capacitor consists of a solid inner conducting core with radius $0.250 \mathrm{~cm}$, surrounded by an outer hollow conducting tube. The two conductors are separated by air, and the length of the cylinder is $12.0 \mathrm{~cm}$. The capacitance is $36.7 \mathrm{pF}$. (a) Calculate the inner radius of the hollow tube. (b) When the capacitor is charged to $125 \mathrm{~V},$ what is the charge per unit length $\lambda$ on the capacitor?

Shubham Verma

Texas A&M University

Problem 9

A capacitor is made from two hollow, coaxial, iron cylinders, one inside the other. The inner cylinder is negatively charged and the outer is positively charged; the magnitude of the charge on each is $10.0 \mathrm{pC}$. The inner cylinder has radius $0.50 \mathrm{~mm},$ the outer one has radius $5.00 \mathrm{~mm},$ and the length of each cylinder is $18.0 \mathrm{~cm} .$ (a) What is the capacitance?
(b) What applied potential difference is necessary to produce these charges on the cylinders?

Ze-Han Lee

Numerade Educator

Problem 10

A cylindrical capacitor has an inner conductor of radius $2.2 \mathrm{~mm}$ and an outer conductor of radius $3.5 \mathrm{~mm} .$ The two conductors are separated by vacuum, and the entire capacitor is $2.8 \mathrm{~m}$ long.
(a) What is the capacitance per unit length? (b) The potential of the inner conductor is $350 \mathrm{mV}$ higher than that of the outer conductor. Find the charge (magnitude and sign) on both conductors.

Shubham Verma

Texas A&M University

Problem 11

A spherical capacitor contains a charge of $3.30 \mathrm{nC}$ when connected to a potential difference of $220 \mathrm{~V}$. If its plates are separated by vacuum and the inner radius of the outer shell is $4.00 \mathrm{~cm}$, calculate:
(a) the capacitance; (b) the radius of the inner sphere; (c) the electric field just outside the surface of the inner sphere.

Ze-Han Lee

Numerade Educator

Problem 12

A spherical capacitor is formed from two concentric, spherical, conducting shells separated by vacuum. The inner sphere has radius $15.0 \mathrm{~cm}$ and the capacitance is $116 \mathrm{pF}$. (a) What is the radius of the outer sphere? (b) If the potential difference between the two spheres is $220 \mathrm{~V},$ what is the magnitude of charge on each sphere?

Ze-Han Lee

Numerade Educator

Problem 13

You measure the capacitance $C_{1}$ of a capacitor by doing the following: First connect capacitors $C_{1}$ and $C_{2}$ in series to a power supply that provides a voltage $V$ that can be varied. The capacitance of $C_{2}$ is known to be $3.00 \mu \mathrm{F}$. Then vary the applied voltage $V,$ and for each value of $V$ measure the voltage $V_{2}$ across $C_{2}$. After plotting your data as $V_{2}$ versus $V,$ you find that the data fall close to a straight line that has slope $0.650 .$ What is the capacitance $C_{1} ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 14

Figure E24.14 shows a system of four capacitors, where the potential difference across $a b$ is $50.0 \mathrm{~V}$.
(a) Find the equivalent capacitance of this system between $a$ and $b$. (b) How much charge is stored by this combination of capacitors? (c) How much charge is stored in each of the $10.0 \mu \mathrm{F}$ and the $9.0 \mu \mathrm{F}$ capacitors?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 15

Electric eels and electric fish generate large potential differences that are used to stun enemies and prey. These potentials are produced by cells that each can generate $0.10 \mathrm{~V}$. We can plausibly model such cells as charged capacitors. (a) How should these cells be connected (in series or in parallel) to produce a total potential of more than $0.10 \mathrm{~V} ?$ (b) Using the connection in part (a), how many cells must be connected together to produce the $500 \mathrm{~V}$ surge of the electric eel?

Ze-Han Lee

Numerade Educator

Problem 16

For the system of capacitors shown in Fig. E24.16, find the equivalent capacitance (a) between $b$ and $c,$ and
(b) between $a$ and $c$.

Shubham Verma

Texas A&M University

Problem 17

In Fig. E24.17, each capacitor has $C=4.00 \mu \mathrm{F}$ and $V_{a b}=+28.0 \mathrm{~V} .$ Calculate (a) the charge on each capacitor; (b) the potential difference across each capacitor; (c) the potential difference between points $a$ and $d$.

Sarah Mccrumb

Numerade Educator

Problem 18

$\begin{array}{lll}\text { In Fig. } & \text { 24.8a, let } C_{1}=3.00 \mu \mathrm{F}, \quad C_{2}=5.00 \mu \mathrm{F}, \text { and }\end{array}$ $V_{a b}=+64.0 \mathrm{~V} .$ Calculate (a) the charge on each capacitor and (b) the potential difference across each capacitor.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 19

$\begin{array}{llll}\text {? In Fig. } & \text { 24.9a, let } & C_{1}=3.00 \mu \mathrm{F}, & C_{2}=5.00 \mu \mathrm{F}, \text { and }\end{array}$
$V_{a b}=+52.0 \mathrm{~V} .$ Calculate (a) the charge on each capacitor and (b) the potential difference across each capacitor.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 20

$\begin{array}{llll}& \text { In } & \text { Fig. } & \text { E24.20, } & C_{1}=\end{array}$
$6.00 \mu \mathrm{F}, C_{2}=3.00 \mu \mathrm{F},$ and $C_{3}=$
$5.00 \mu \mathrm{F}$. The capacitor network is connected to an applied potential $V_{a b}$. After the charges on the capacitors have reached their final values, the charge on $C_{2}$ is $30.0 \mu \mathrm{C}$. (a) What are the charges on capacitors $C_{1}$ and $C_{3} ?$
(b) What is the applied voltage $V_{a b} ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 21

For the system of capacitors shown in Fig. E24.21, a potential difference of $25 \mathrm{~V}$ is maintained across $a b$. (a) What is the equivalent capacitance of this system between $a$ and $b ?$
(b) How much charge is stored by this system?
(c) How much charge does the $6.5 \mathrm{nF}$ capacitor store?
(d) What is the potential difference across the $7.5 \mathrm{nF}$ capacitor?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 22

A capacitor is formed from two concentric spherical conducting shells separated by vacuum. The inner sphere has radius $12.5 \mathrm{~cm},$ and the outer sphere has radius $14.8 \mathrm{~cm} .$ A potential difference of $120 \mathrm{~V}$ is applied to the capacitor. (a) What is the energy density at $r=12.6 \mathrm{~cm}$, just outside the inner sphere?
(b) What is the energy density at $r=14.7 \mathrm{~cm},$ just inside the outer sphere? (c) For a parallel-plate capacitor the energy density is uniform in the region between the plates, except near the edges of the plates. Is this also true for a spherical capacitor?

Shubham Verma

Texas A&M University

Problem 23

A $5.80 \mu \mathrm{F},$ parallel-plate, air capacitor has a plate separation of $5.00 \mathrm{~mm}$ and is charged to a potential difference of $400 \mathrm{~V}$. Calculate the energy density in the region between the plates, in units of $\mathrm{J} / \mathrm{m}^{3}$.

Ze-Han Lee

Numerade Educator

Problem 24

A parallel-plate air capacitor has a capacitance of $920 \mathrm{pF}$. The charge on each plate is $3.90 \mu \mathrm{C}$. (a) What is the potential difference between the plates? (b) If the charge is kept constant, what will be the potential difference if the plate separation is doubled? (c) How much work is required to double the separation?

Shubham Verma

Texas A&M University

Problem 25

An air capacitor is made from two flat parallel plates $1.50 \mathrm{~mm}$ apart. The magnitude of charge on each plate is $0.0180 \mu \mathrm{C}$ when the potential difference is $200 \mathrm{~V}$. (a) What is the capacitance? (b) What is the area of each plate? (c) What maximum voltage can be applied without dielectric breakdown? (Dielectric breakdown for air occurs at an electric-field strength of $3.0 \times 10^{6} \mathrm{~V} / \mathrm{m} .$ ) (d) When the charge is $0.0180 \mu \mathrm{C},$ what total energy is stored?

Prabhat Tyagi

Numerade Educator

Problem 26

- A parallel-plate vacuum capacitor has $8.38 \mathrm{~J}$ of energy stored in it. The separation between the plates is $2.30 \mathrm{~mm}$. If the separation is decreased to $1.15 \mathrm{~mm},$ what is the energy stored (a) if the capacitor is disconnected from the potential source so the charge on the plates remains constant, and (b) if the capacitor remains connected to the potential source so the potential difference between the plates remains constant?

Katie Mcalpine

Numerade Educator

Problem 27

You have two identical capacitors and an external potential source. (a) Compare the total energy stored in the capacitors when they are connected to the applied potential in series and in parallel. (b) Compare the maximum amount of charge stored in each case.
(c) Energy storage in a capacitor can be limited by the maximum electric field between the plates. What is the ratio of the electric field for the series and parallel combinations?

Ze-Han Lee

Numerade Educator

Problem 28

For the capacitor network shown in Fig. $\mathbf{E} 24.28,$ the potential difference across $a b$ is 48 V.
Find (a) the total charge stored in this network; (b) the charge on each

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 29

For the capacitor network shown in Fig. $\mathbf{E} 24.29,$ the potential difference across $a b$ is $220 \mathrm{~V}$. Find
(a) the total charge stored in this network; (b) the charge on each capacitor; (c) the total energy stored in the network; (d) the energy stored in each capacitor; (e) the potential difference across each capacitor.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 30

Two air-filled parallel-plate capacitors with capacitances $C_{1}$ and $C_{2}$ are connected in parallel to a battery that has a voltage of $36.0 \mathrm{~V}$ $C_{1}=4.00 \mu \mathrm{F}$ and $C_{2}=6.00 \mu \mathrm{F}$. (a) What is the total positive charge stored in the two capacitors? (b) While the capacitors remain connected to the battery, a dielectric with dielectric constant 5.00 is inserted between the plates of capacitor $C_{1}$, completely filling the space between them. Then what is the total positive charge stored on the two capacitors? Does the insertion of the dielectric cause total charge stored to increase or decrease?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 31

$\mathrm{A} 12.5 \mu \mathrm{F}$ capacitor is connected to a power supply that keeps a constant potential difference of $24.0 \mathrm{~V}$ across the plates. A piece of material having a dielectric constant of 3.75 is placed between the plates, completely filling the space between them. (a) How much energy is stored in the capacitor before and after the dielectric is inserted?
(b) By how much did the energy change during the insertion? Did it increase or decrease?

Ze-Han Lee

Numerade Educator

Problem 32

A parallel-plate capacitor has capacitance $C_{0}=8.00 \mathrm{pF}$ when there is air between the plates. The separation between the plates is $1.50 \mathrm{~mm}$. (a) What is the maximum magnitude of charge $Q$ that can be placed on each plate if the electric field in the region between the plates is not to exceed $3.00 \times 10^{4} \mathrm{~V} / \mathrm{m} ?$ (b) A dielectric with $K=2.70$ is inserted between the plates of the capacitor, completely filling the volume between the plates. Now what is the maximum magnitude of charge on each plate if the electric field between the plates is not to exceed $3.00 \times 10^{4} \mathrm{~V} / \mathrm{m} ?$

Shubham Verma

Texas A&M University

Problem 33

Two parallel plates have equal and opposite charges. When the space between the plates is evacuated, the electric field is $E=3.20 \times 10^{5} \mathrm{~V} / \mathrm{m} .$ When the space is filled with dielectric, the electric field is $E=2.50 \times 10^{5} \mathrm{~V} / \mathrm{m} .$ (a) What is the charge density on each surface of the dielectric? (b) What is the dielectric constant?

Ze-Han Lee

Numerade Educator

Problem 34

Two identical air-filled parallel-plate capacitors $C_{1}$ and $C_{2}$, each with capacitance $C,$ are connected in series to a battery that has voltage $V$. While the two capacitors remain connected to the battery, a dielectric with dielectric constant $K>1$ is inserted between the plates of one of the capacitors, completely filling the space between them. Let $U_{0}$ be the total energy stored in the two capacitors without the dielectric and $U$ be the total energy stored after the dielectric is inserted. In terms of $K,$ what is the ratio $U / U_{0} ?$ Does the total stored energy increase, decrease, or stay the same after the dielectric is inserted?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 35

Two identical air-filled parallel-plate capacitors $C_{1}$ and $C_{2}$ are connected in series to a battery that has voltage $V .$ The charge on each capacitor is $Q_{0}$. While the two capacitors remain connected to the battery, a dielectric with dielectric constant $K>1$ is inserted between the plates of capacitor $C_{1},$ completely filling the space between them. In terms of $K$ and $Q_{0},$ what is the charge on capacitor $C_{1}$ after the dielectric is inserted? Does the charge on $C_{1}$ increase, decrease, or stay the same?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 36

Two air-filled parallel-plate capacitors with capacitances $C_{1}$ and $C_{2}$ are connected in series to a battery that has voltage $V$ $C_{1}=3.00 \mu \mathrm{F}$ and $C_{2}=6.00 \mu \mathrm{F}$. The electric field between the plates of capacitor $C_{2}$ is $E_{02}$. While the two capacitors remain connected to the battery, a dielectric with dielectric constant $K=4$ is inserted between the plates of capacitor $C_{1}$, completely filling the space between them. After the dielectric is inserted in $C_{1},$ the electric field between the plates of capacitor $C_{2}$ is $E_{2}$. (a) What is the ratio $E_{2} / E_{02}$ ? When the dielectric is inserted into $C_{1},$ does the electric field in $C_{2}$ increase, decrease, or remain the same? (b) Repeat the calculation in part (a) for the two capacitors connected to the battery in parallel.

Eduard Sanchez

Numerade Educator

Problem 37

The dielectric to be used in a parallel-plate capacitor has a dielectric constant of 3.60 and a dielectric strength of $1.60 \times 10^{7} \mathrm{~V} / \mathrm{m} .$ The capacitor is to have a capacitance of $1.25 \times 10^{-9} \mathrm{~F}$ and must be able to withstand a maximum potential difference of $5500 \mathrm{~V}$. What is the minimum area the plates of the capacitor may have?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 38

BIO Potential in Human Cells. Some cell walls in the human body have a layer of negative charge on the inside surface and a layer of positive charge of equal magnitude on the outside surface. Suppose that the charge density on either surface is $\pm 0.50 \times 10^{-3} \mathrm{C} / \mathrm{m}^{2},$ the cell wall is $5.0 \mathrm{nm}$ thick, and the cell-wall material is air.
(a) Find the magnitude of $\vec{E}$ in the wall between the two layers of charge. (b) Find the potential difference between the inside and the outside of the cell. Which is at the higher potential? (c) A typical cell in the human body has a volume of $10^{-16} \mathrm{~m}^{3}$. Estimate the total electric-field energy stored in the wall of a cell of this size. (Hint: Assume that the cell is spherical, and calculate the volume of the cell wall.) (d) In reality, the cell wall is made up, not of air, but of tissue with a dielectric constant of 5.4 . Repeat parts (a) and (b) in this case.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 39

A constant potential difference of $12 \mathrm{~V}$ is maintained between the terminals of a $0.25 \mu \mathrm{F},$ parallel-plate, air capacitor. (a) $\mathrm{A}$ sheet of Mylar is inserted between the plates of the capacitor, completely filling the space between the plates. When this is done, how much additional charge flows onto the positive plate of the capacitor (see Table 24.1 )? (b) What is the total induced charge on either face of the Mylar sheet? (c) What effect does the Mylar sheet have on the electric field between the plates? Explain how you can reconcile this with the increase in charge on the plates, which acts to increase the electric field.

Ze-Han Lee

Numerade Educator

Problem 40

- Polystyrene has dielectric constant 2.6 and dielectric strength $2.0 \times 10^{7} \mathrm{~V} / \mathrm{m} .$ A piece of polystyrene is used as a dielectric in a parallel-plate capacitor, filling the volume between the plates.
(a) When the electric field between the plates is $80 \%$ of the dielectric strength, what is the energy density of the stored energy? (b) When the capacitor is connected to a battery with voltage $500.0 \mathrm{~V},$ the electric field between the plates is $80 \%$ of the dielectric strength. What is the area of each plate if the capacitor stores $0.200 \mathrm{~mJ}$ of energy under these conditions?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 41

When a $360 \mathrm{nF}$ air capacitor $\left(1 \mathrm{nF}=10^{-9} \mathrm{~F}\right)$ is connected to a power supply, the energy stored in the capacitor is $1.85 \times 10^{-5} \mathrm{~J}$. While the capacitor is kept connected to the power supply, a slab of dielectric is inserted that completely fills the space between the plates. This increases the stored energy by $2.32 \times 10^{-5} \mathrm{~J}$. (a) What is the potential difference between the capacitor plates? (b) What is the dielectric constant of the slab?

Sarah Mccrumb

Numerade Educator

Problem 42

A parallel-plate capacitor has capacitance $C=12.5 \mathrm{pF}$ when the volume between the plates is filled with air. The plates are circular, with radius $3.00 \mathrm{~cm}$. The capacitor is connected to a battery, and a charge of magnitude $25.0 \mathrm{pC}$ goes onto each plate. With the capacitor still connected to the battery, a slab of dielectric is inserted between the plates, completely filling the space between the plates. After the dielectric has been inserted, the charge on each plate has magnitude $45.0 \mathrm{pC}$. (a) What is the dielectric constant $K$ of the dielectric?
(b) What is the potential difference between the plates before and after the dielectric has been inserted? (c) What is the electric field at a point midway between the plates before and after the dielectric has been inserted?

Salamat Ali

Numerade Educator

Problem 43

A parallel-plate capacitor has the volume between its plates filled with plastic with dielectric constant $K .$ The magnitude of the charge on each plate is $Q .$ Each plate has area $A,$ and the distance between the plates is $d$. (a) Use Gauss's law as stated in Eq. (24.23) to calculate the magnitude of the electric field in the dielectric. (b) Use the electric field determined in part (a) to calculate the potential difference between the two plates. (c) Use the result of part (b) to determine the capacitance of the capacitor. Compare your result to Eq. (24.12)

Ze-Han Lee

Numerade Educator

Problem 44

A parallel-plate capacitor has plates with area $0.0225 \mathrm{~m}^{2}$ separated by $1.00 \mathrm{~mm}$ of Teflon. (a) Calculate the charge on the plates when they are charged to a potential difference of $12.0 \mathrm{~V}$. (b) Use Gauss's law [Eq. (24.23)$]$ to calculate the electric field inside the Teflon. (c) Use Gauss's law to calculate the electric field if the voltage source is disconnected and the Teflon is removed.

Salamat Ali

Numerade Educator

Problem 45

Electronic flash units for cameras contain a capacitor for storing the energy used to produce the flash. In one such unit, the flash lasts for $\frac{1}{675} \mathrm{~s}$ with an average light power output of $2.70 \times 10^{5} \mathrm{~W}$. (a) If the conversion of electrical energy to light is $95 \%$ efficient (the rest of the energy goes to thermal energy), how much energy must be stored in the capacitor for one flash? (b) The capacitor has a potential difference between its plates of $125 \mathrm{~V}$ when the stored energy equals the value calculated in part (a). What is the capacitance?

Kai Chen

Princeton University

Problem 46

A parallel-plate air capacitor is made by using two plates $12 \mathrm{~cm}$ square, spaced $3.7 \mathrm{~mm}$ apart. It is connected to a $12 \mathrm{~V}$ battery. (a) What is the capacitance? (b) What is the charge on each plate? (c) What is the electric field between the plates? (d) What is the energy stored in the capacitor? (e) If the battery is disconnected and then the plates are pulled apart to a separation of $7.4 \mathrm{~mm},$ what are the answers to parts (a)-(d)?

Vidhi Bhatt

Numerade Educator

Problem 47

In one type of computer keyboard, each key holds a small
metal plate that serves as one plate of a parallel-plate, air-filled capacitor. When the key is depressed, the plate separation decreases and the capacitance increases. Electronic circuitry detects the change in capacitance and thus detects that the key has been pressed. In one particular keyboard, the area of each metal plate is $42.0 \mathrm{~mm}^{2},$ and the separation between the plates is $0.700 \mathrm{~mm}$ before the key is depressed.
(a) Calculate the capacitance before the key is depressed. (b) If the cir-
key be depressed before the circuitry detects its depression?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 48

BIO Cell Membranes. Cell membranes (the walled enclosure around
a cell) are typically about $7.5 \mathrm{nm}$ thick. They are partially permeable to allow charged material to pass in and out, as needed. Equal but opposite charge densities build up on the inside and outside faces of such a membrane, and these charges prevent additional charges from passing through the cell wall. We can model a cell membrane as a parallel-plate capacitor, with the membrane itself containing proteins embedded in an organic material to give the membrane a dielectric constant of about $10 .$ (See Fig. $\mathbf{P 2 4 . 4 8}$.)
(a) What is the capacitance per square centimeter of such a cell wall?
(b) In its normal resting state, a cell has a potential difference of $85 \mathrm{mV}$ across its membrane. What is the electric field inside this membrane?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 49

A $20.0 \mu \mathrm{F}$ capacitor is charged to a potential difference of $800 \mathrm{~V}$. The terminals of the charged capacitor are then connected to those of an uncharged $10.0 \mu \mathrm{F}$ capacitor. Compute (a) the original charge of the system, (b) the final potential difference across each capacitor, (c) the final energy of the system, and (d) the decrease in energy when the capacitors are connected.

Ze-Han Lee

Numerade Educator

Problem 50

When lightning strikes a car, the metallic outer shell, which is insulated from the ground by its rubber tires, attains a high voltage. We can estimate how much charge is deposited by roughly modeling the car as a spherical capacitor with the outer radius taken to infinity. (a) Determine the capacitance of a sphere in terms of its radius, either by considering the potential on a sphere relative to infinity as a function of its charge, or by considering a spherical capacitor as the outer shell becomes very large. (These methods provide the same result.) (b) Estimate the radius of a sphere that corresponds to the size of a car. (c) Determine the corresponding capacitance. (d) A typical lighting strike could deliver a $100 \mathrm{MV}$ potential to a car. Then what net charge would be deposited on the car?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 51

For the capacitor network shown in Fig. $\mathbf{P} 24.51,$ the potential difference across $a b$ is $12.0 \mathrm{~V}$. Find (a) the total energy stored in this network and (b) the energy stored in the $4.80 \mu \mathrm{F}$ capacitor.

Kai Chen

Princeton University

Problem 52

In Fig. E24.17, $C_{1}=6.00 \mu \mathrm{F}, C_{2}=3.00 \mu \mathrm{F}, C_{3}=4.00 \mu \mathrm{F}$
and $C_{4}=8.00 \mu \mathrm{F} .$ The capacitor network is connected to an applied potential difference $V_{a b} .$ After the charges on the capacitors have reached their final values, the voltage across $C_{3}$ is $40.0 \mathrm{~V} .$ What are
(a) the voltages across $C_{1}$ and $C_{2},$ (b) the voltage across $C_{4},$ and (c) the voltage $V_{a b}$ applied to the network?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 53

In Fig. $\mathbf{P 2 4 . 5 3}, C_{1}=C_{5}=8.4 \mu \mathrm{F}$ and $C_{2}=C_{3}=C_{4}=$ 4.2 $\mu$ F. The applied potential is $V_{a b}=220 \mathrm{~V}$. (a) What is the equivalent capacitance of the network between points $a$ and $b ?$ (b) Calculate the charge on each capacitor and the potential difference across each capacitor.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 54

After combing your hair on a dry day, some of your hair stands up, forced away from your head by electrostatic repulsion.
(a) Estimate the length $L$ of your hair. (b) Using the average linear mass density of hair, which is $65 \mu \mathrm{g} / \mathrm{cm},$ estimate the mass $m$ of one of your hairs. (c) Estimate the number $N$ of hairs that stand after combing.
(d) Assume that the comb has taken away a charge $-2 Q,$ and that your hair has therefore gained an amount of charge $2 Q .$ Assume further that half of this charge resides next to your head and the other half is distributed at the ends of the $N$ strands that stand up. Assume the electrostatic force that lifted a hair was twice its weight. Show that this leads to
$2 m g=\frac{1}{4 \pi \epsilon_{0}} \frac{Q^{2} / N}{L^{2}}$ Use this equation to estimate the charge $Q$ that resides on your head.
(e) If your head were a sphere with radius $R$, it would have a capacitance of $4 \pi \epsilon_{0} R .$ Estimate the radius of your head; then use your estimate to determine your head's capacitance. (f) Use your result to estimate the potential attained due to combing. (The surprising result illustrates an interesting feature of static electricity.)

Eduard Sanchez

Numerade Educator

Problem 55

In Fig. E24.20, $C_{1}=3.00 \mu \mathrm{F}$ and $V_{a b}=150 \mathrm{~V}$. The charge on capacitor $C_{1}$ is $150 \mu \mathrm{C}$ and the charge on $C_{3}$ is $450 \mu \mathrm{C}$. What are the values of the capacitances of $C_{2}$ and $C_{3} ?$

Sarah Mccrumb

Numerade Educator

Problem 56

The capacitors in Fig. $\mathbf{P} 24.56$ are initially uncharged and are connected, as in the diagram, with switch $S$ open. The applied potential difference is $V_{a b}=+210 \mathrm{~V}$.
(a) What is the potential difference $V_{c d} ?$ (b) What is the potential difference across each capacitor after switch $S$ is closed?
(c) How much charge flowed through the switch when it was closed?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 57

Three capacitors having capacitances of $8.4,8.4,$ and $4.2 \mu \mathrm{F}$ are connected in series across a $36 \mathrm{~V}$ potential difference. (a) What is the charge on the $4.2 \mu \mathrm{F}$ capacitor? (b) What is the total energy stored in all three capacitors? (c) The capacitors are disconnected from the potential difference without allowing them to discharge. They are then reconnected in parallel with each other, with the positively charged plates connected together. What is the voltage across each capacitor in the parallel combination? (d) What is the total energy now stored in the capacitors?

Sarah Mccrumb

Numerade Educator

Problem 58

Capacitance of a Thundercloud. The charge center of a thundercloud, drifting $3.0 \mathrm{~km}$ above the earth's surface, contains $20 \mathrm{C}$ of negative charge. Assuming the charge center has a radius of $1.0 \mathrm{~km}$, and modeling the charge center and the earth's surface as parallel plates, calculate: (a) the capacitance of the system; (b) the potential difference between charge center and ground; (c) the average strength of the electric field between cloud and ground; (d) the electrical energy stored in the system.

Salamat Ali

Numerade Educator

Problem 59

$\begin{array}{lll}\text { In } & \text { Fig. } & \text { P24.59, } & \text { each }\end{array}$ capacitance $C_{1}$ is $6.9 \mu \mathrm{F},$ and each capacitance $C_{2}$ is $4.6 \mu \mathrm{F}$. (a) Compute the equivalent capacitance of the network between points $a$ and $b$.
(b) Compute the charge on each of the three capacitors nearest $a$ and $b$ when $V_{a b}=420 \mathrm{~V} .$ (c) With $420 \mathrm{~V}$ across $a$
and $b,$ compute $V_{c d}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 60

An air capacitor is made by using two flat plates, each with area $A,$ separated by a distance $d .$ Then a metal slab having thickness $a$ (less than $d$ ) and the same shape and size as the plates is inserted between them, parallel to the plates and not touching either plate (Fig. $\mathbf{P 2 4 . 6 0}$ ).
(a) What is the capacitance of this arrangement?
(b) Express the capacitance as a multiple of the capacitance $C_{0}$ when the metal slab is not present. (c) Discuss what happens to the capacitance in the limits $a \rightarrow 0$ and $a \rightarrow d$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 61

A potential difference $V_{a b}=48.0 \mathrm{~V}$ is applied across the capacitor network of Fig. E24.17. If $C_{1}=C_{2}=4.00 \mu \mathrm{F}$ and $C_{4}=8.00 \mu \mathrm{F},$ what must the capacitance $C_{3}$ be if the network is to store $2.90 \times 10^{-3} \mathrm{~J}$ of electrical energy?

Sarah Mccrumb

Numerade Educator

Problem 62

CALC The inner cylinder of a long, cylindrical capacitor has radius $r_{a}$ and linear charge density $+\lambda$. It is surrounded by a coaxial cylindrical conducting shell with inner radius $r_{b}$ and linear charge density $-\lambda$ (see Fig. 24.6). (a) What is the energy density in the region between the conductors at a distance $r$ from the axis? (b) Integrate the energy density calculated in part (a) over the volume between the conductors in a length $L$ of the capacitor to obtain the total electric-field energy per unit length. (c) Use Eq. ( 24.9 ) and the capacitance per unit length calculated in Example 24.4 (Section 24.1 ) to calculate $U / L$. Does your result agree with that obtained in part (b)?

Sarah Mccrumb

Numerade Educator

Problem 63

CP Dielectric elastomers are used to create voltage-dependent capacitors. These elastomers contract when subject to the stresses induced by Coulomb forces between capacitor plates. Consider a pair of parallel plates with area $A$ separated by distance $d_{0}$ when no voltage is applied. A material with dielectric constant $K$ and Young's modulus $Y$ fills the space between the plates.
(a) What compressive force is applied to the dielectric when the plates have opposite charges $Q$ and $-Q ?$
(b) The distance between the plates is $d=d_{0}-s Q^{2},$ where $s$ is a squeezing coefficient. Determine $s$ in terms of the parameters specified. (c) A capacitor of this sort has plate area $1.00 \mathrm{~cm}^{2}$ and noncharged separation distance $d_{0}=0.400 \mathrm{~mm}$. The plates are separated by a silicone dielectric elastomer with dielectric constant $K=3.00$ and Young's modulus $0.0100 \mathrm{GPa}$. What is the squeezing coefficient in this case? (d) If each plate has charge of magnitude $0.700 \mu \mathrm{C}$, what is the applied voltage?
(e) What new voltage would double the amount of charge on the plates?

Eduard Sanchez

Numerade Educator

Problem 64

A parallel-plate capacitor is made from two plates $12.0 \mathrm{~cm}$ on each side and $4.50 \mathrm{~mm}$ apart. Half of the space between these plates contains only air, but the other half is filled with Plexiglas of dielectric constant 3.40 (Fig. $\mathbf{P} 24.64)$. An $18.0 \mathrm{~V}$ battery is connected across the plates. (a) What is the capacitance of this combination? (Hint: Can you think of this capacitor as equivalent to two capacitors in parallel?) (b) How much energy is stored in the capacitor? (c) If we remove the Plexiglas but change nothing else, how much energy will be stored in the capacitor?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 65

CP CALC A metallic circular plate with radius $r$ is fixed to a tabletop. An identical circular plate supported from above by a cable is fixed in place a distance $d$ above the first plate. Assume that $d$ is much smaller than $r$. The two plates are attached by wires to a battery that supplies voltage $V$. (a) What is the tension in the cable? Neglect the weight of the plate.
(b) The upper plate is slowly raised to a new height $2 d .$ Determine the work done by the cable by integrating $\int_{d}^{2 d} F(z) d z$ where $F(z)$ is the cable tension when the plates are separated by a distance z. (c) Compute the energy stored in the electric field before the top plate was raised. (d) Compute the energy stored in the electric field after the top plate was raised. (e) Is the work done by the cable equal to the change in the stored electrical energy? If not, why not?

Lainey Roebuck

Numerade Educator

Problem 66

A fuel gauge uses a capacitor to determine the height of the fuel in a tank. The effective dielectric con-
stant $K_{\text {eff }}$ changes from a value of 1 when the tank is empty to a value of $K$, the dielectric constant of the fuel, when the tank is full. The appropriate electronic circuitry can determine the effective dielectric constant of the combined air and fuel between the capacitor plates. Each of the two rectangular plates has a width $w$ and a length $L$ (Fig. $\mathbf{P 2 4 . 6 6}$ ). The height of the fuel between the plates is $h$. You can ignore any fringing effects. (a) Derive an expression for $K_{\text {eff }}$ as a function of $h$. (b) What is the effective dielectric constant for a tank $\frac{1}{4}$ full, $\frac{1}{2}$ full, and $\frac{3}{4}$ full if the fuel is gasoline $(K=1.95) ?$
(c) Repeat part (b) for methanol $(K=33.0)$. (d) For which fuel is this fuel gauge more practical?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 67

DATA Your electronics company has several identical capacitors with capacitance $C_{1}$ and several others with capacitance $C_{2}$. You must determine the values of $C_{1}$ and $C_{2}$ but don't have access to $C_{1}$ and $C_{2}$ individually. Instead, you have a network with $C_{1}$ and $C_{2}$ connected in series and a network with $C_{1}$ and $C_{2}$ connected in parallel. You have a $200.0 \mathrm{~V}$ battery and instrumentation that measures the total $\mathrm{en}$ ergy supplied by the battery when it is connected to the network. When the parallel combination is connected to the battery, $0.180 \mathrm{~J}$ of energyis stored in the network. When the series combination is connected. $0.0400 \mathrm{~J}$ of energy is stored. You are told that $C_{1}$ is greater than $C_{2}$
(a) Calculate $C_{1}$ and $C_{2}$.
(b) For the series combination, does $C_{1}$ or $C_{2}$ store more charge, or are the values equal? Does $C_{1}$ or $C_{2}$ store more energy, or are the values equal? (c) Repeat part (b) for the parallel combination.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 68

DATA You are designing capacitors for various applications. For one application, you want the maximum possible stored energy. For another, you want the maximum stored charge. For a third application, you want the capacitor to withstand a large applied voltage without dielectric breakdown. You start with an air-filled parallel-plate capacitor that has $C_{0}=6.00 \mathrm{pF}$ and a plate separation of $2.50 \mathrm{~mm}$. You then consider the use of each of the dielectric materials listed in Table $24.2 .$ In each application, the dielectric will fill the volume between the plates, and the electric field between the plates will be $50 \%$ of the dielectric strength given in the table.
(a) For each of the five materials given in the table, calculate the energy stored in the capacitor. Which dielectric allows the maximum stored energy?
(b) For each material, what is the charge $Q$ stored on each plate of the capacitor?
(c) For each material, what is the voltage applied across the capacitor?
(d) Is one dielectric material in the table your best choice for all three applications?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 69

DATA You are conducting experiments with an air-filled parallel-plate capacitor. You connect the capacitor to a battery with voltage $24.0 \mathrm{~V}$. Initially the separation $d$ between the plates is $0.0500 \mathrm{~cm}$. In one experiment, you leave the battery connected to the capacitor, increase the separation between the plates, and measure the energy stored in the capacitor for each value of $d$. In a second experiment, you make the same measurements but disconnect the battery before you change the plate separation. One set of your data is given in Fig. $\mathrm{P} 24.69,$ where you have plotted the stored energy $U$ versus $1 / d$. (a) For which experiment does this data set apply: the first (battery remains connected) or the second (battery disconnected before $d$ is changed)? Explain.
(b) Use the data plotted in Fig. $\mathrm{P} 24.69$ to calculate the area $A$ of each plate. (c) For which case, the battery connected or the battery disconnected, is there more energy stored in the capacitor when $d=0.400 \mathrm{~cm}$ ? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 70

A thin-walled, hollow, conducting cylinder with radius $r_{b}$ is concentric with a solid conducting cylinder with radius $r_{a}<n_{b}$. Each has length $L$. The two cylinders are attached by conducting wires to the terminals of a battery that supplies potential $V$. A solid cylindrical shell, with inner radius $r_{a}$ and outer radius $R<r_{b},$ made of a material with dielectric constant $K,$ slides between the conducting cylinders, as shown in Fig. $\mathrm{P} 24.70 .$ By changing the insertion distance $x,$ we can alter the capacitance seen by the battery and therefore alter the amount of charge stored in this device. (a) Determine the capacitance as a function of $x$. (b) If $L=10.0 \mathrm{~cm}, r_{a}=1.00 \mathrm{~cm}, r_{b}=4.00 \mathrm{~cm}, R=3.00 \mathrm{~cm},$ and
$K=3.21,$ what is the capacitance when $x=0 ?$ (c) What is the capacitance when $x=L ?$ (d) What value of $x$ results in $6.00 \mathrm{nC}$ of charge on the positively charged cylinder plate when $V=1.00 \mathrm{kV} ?$

Dominador Tan

Numerade Educator

Problem 71

CALC Two conducting plates with area $A$ are separated by distance $d$. Between the plates is a material with a dielectric constant that varies linearly from a value of unity next to one plate to $K$ next to the other plate. (a) What is the capacitance of this device? [Hint: We can envision this as a continuum of capacitors with differential plate separation connected in series. The reciprocal of the capacitance of a differential slice is then $d x /\left(K(x) \epsilon_{0} A\right),$ where $K(x)$ is the dielectric constant specific to that locale. $]$ (b) Show that this result matches the expected result when $K \rightarrow 1$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 72

Two square conducting plates with sides of length $L$ are separated by a distance $D$. A diclectric slab with constant $K$ with dimensions $L \times L \times D$ is inserted a distance $x$ into the space between the plates, as shown in Fig. $P 24.72$.
(a) Find the capacitance $C$ of this system.
(b) Suppose that the capacitor is connected to a battery that maintains a constant potential difference $V$ between the plates. If the dielectric slab is inserted an additional distance $d x$ into the space between the plates, show that the change in stored energy is $d U=+\frac{(K-1) \epsilon_{0} V^{2} L}{2 D} d x$ (c) Suppose that before the slab is moved by $d x$, the plates are disconnected from the battery, so that the charges on the plates remain constant. Determine the magnitude of the charge on each plate, and then show that when the slab is moved $d x$ farther into the space between the plates, the stored energy changes by an amount that is the negative of the expression for $d U$ given in part (b). (d) If $F$ is the force exerted on the slab by the charges on the plates, then $d U$ should equal the work done against this force to move the slab a distance $d x$. Thus $d U=-F d x$. Show that applying this expression to the result of part (b) suggests that the electric force on the slab pushes it out of the capacitor, while the result of part (c) suggests that the force pulls the slab into the capacitor. (e) Figure 24.16 shows that the force in fact pulls the slab into the capacitor. Explain why the result of part (b) gives an incorrect answer for the direction of this force, and calculate the magnitude of the force. (This method does not require knowledge of the nature of the fringing field.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 73

How many moles of $\mathrm{Na}^{+}$ must move per unit area of membrane to change $V_{\mathrm{m}}$ from $-70 \mathrm{mV}$ to $+30 \mathrm{mV}$, if we assume that the membrane behaves purely as a capacitor?
(a) $10^{-4} \mathrm{~mol} / \mathrm{cm}^{2}$
(b) $10^{-9} \mathrm{~mol} / \mathrm{cm}^{2}$
(c) $10^{-12} \mathrm{~mol} / \mathrm{cm}^{2} ;$
(d) $10^{-14} \mathrm{~mol} / \mathrm{cm}^{2}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 74

Suppose that the egg has a diameter of $200 \mu \mathrm{m}$. What fractional change in the internal $\mathrm{Na}^{+}$ concentration results from the fertilizationinduced change in $V_{\mathrm{m}} ?$ Assume that $\mathrm{Na}^{+}$ ions are distributed throughout the cell volume. The concentration increases by (a) 1 part in $10^{4} ;$ (b) 1 part in $10^{5} ;$ (c) 1 part in $10^{6} ;$ (d) 1 part in $10^{7}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 75

Suppose that the change in $V_{\mathrm{m}}$ was caused by the entry of $\mathrm{Ca}^{2+}$ instead of $\mathrm{Na}^{+}$. How many $\mathrm{Ca}^{2+}$ ions would have to enter the cell per unit membrane to produce the change?
(a) Half as many as for $\mathrm{Na}^{+}$;
(b) the same as for $\mathrm{Na}^{+} ;$
(c) twice as many as for $\mathrm{Na}^{+} ;$
(d) cannot say without knowing the inside and outside concentrations of $\mathrm{Ca}^{2+}$

Ze-Han Lee

Numerade Educator

Problem 76

What is the minimum amount of work that must be done by the cell to restore $V_{\mathrm{m}}$ to $-70 \mathrm{mV} ?$
(a) $3 \mathrm{~mJ} ;$ (b) $3 \mu \mathrm{J}$
(c) $3 \mathrm{~nJ}$
(d) $3 \mathrm{pJ}$.

Vidhi Bhatt

Numerade Educator