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Exercise 24.49 A parallel-plate parallel-pla te capacitor has has the volume volume betw between een its plates filled with plastic with dielectric constant magnitude magnitude of the charge on each plate is

. Each plate has has area

. The The

, and and the distance between the plates is

. Part A Use Gauss's law to calculate the magnitude of the electric field in the dielectric. Express your answer in terms of the given quantities and appropriate constants.

ANSWER: = Correct

Part B Use the electric field determined in part A to calculate the potential difference between the two plates. Express your answer in terms of the given quantities and appropriate constants.

ANSWER: = Correct

Part C Use the result of part B to determine the capacitance of the capacitor. Express your answer in terms of the given quantities and appropriate constants.

ANSWER: = Correct

Problem 24.73 Capacitors in networks cannot always be grouped into simple series or parallel combinations. As an example, the figure (a) shows three capacitors capacitor s , and in a delta network , so called because of its triangular

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shape. This network has three terminals three terminals a, b, and c and hence cannot be transformed into a single equivalent capacitor. It can be shown that as far as any effect on the external circuit is concerned, a delta network is equivalent to what is called a Y network . For example, the delta network of the figure (a) can be replaced by the Y network of the figure (b). (The name "Y network" also refers r efers to the shape shape of the network.) The transformation transfor mation equations that give , , and in terms of

, and

, and

are

. Part A For the network shown in the figure (c), determine the equivalent capacitance between the terminals at the left end of the network. ( Hint: Use Hint: Use the delta-Y transformation derived in the introduction. Use points a, b, and c to form the delta, and transform the delta into a Y. The capacitors can then be combined using the relationships for series and parallel combinations of capacitors.) ANSWER:

= 14.0 Correct

Part B Determine the charges of, and the potential differences across, each capacitor in the figure (c). ANSWER:

4 = 5.04×10 Correct −

Part C

ANSWER:

= 7.00 Correct

Part D

ANSWER:

Part E

4 = 2.30×10 Correct −

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ANSWER:

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= 13.0 All attempts used; correct answer displayed

Part F

ANSWER:

4 = 2.80×10 Correct −

Part G

ANSWER:

= 10.0 Correct

Part H

ANSWER:

4 = 2.60×10 Correct −

Part I

ANSWER:

= 9.00 Correct

Part J

ANSWER:

4 = 2.50×10 Correct −

Part K

ANSWER:

Part L

= 12.0 Correct

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ANSWER:

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5 = 1.50×10 Correct −

Part M

ANSWER:

= 2.50 Correct

Problem 24.75 Two square conducting plates with sides of length constant

with dimensions dimensions

are separated separat ed by a distance

is inserted a distance

. A dielectric dielectr ic slab with

into the space betw between een the plates, as

shown in the figure .

Part A Find the capacitance C of C of this system . Express your answer in terms of the given quantities and appropriate constants.

ANSWER: = Correct

Part B Suppose that the capacitor is connected to a battery that maintains a constant potential difference between betw een the plates. If the dielectric dielectr ic slab is inserted an additional distance plates, show that the change in stored energy is

into the space betw between een the

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Essay answers are limited to about 500 words (3800 characters maximum, including spaces).

ANSWER:

My Answer:

Part C Suppose that before the slab is moved by

, the the plates are disconnected from the battery, batte ry, so that the the

charges on the plates remain constant. Determine the magnitude of the charge on each plate. Express your answer in terms of the given quantities and appropriate constants.

ANSWER: = Correct

Part D Show that when the slab is moved

farther fart her into the space between the plates, the stored store d energy changes

by an amount that is the negative of negative of the expression for

given in part B.

Essay answers are limited to about 500 words (3800 characters maximum, including spaces).

ANSWER:

My Answer:

Part E If

is the force exerted on the slab by the charges on the plates, then then

against this against this force to move the slab a distance

. Thus Thus

should should equal the work done

. Show that applying this expression to

the result of part B suggests that the electric force on the slab pushes it out out of of the capacitor, while the result of part D suggests that the force pulls the slab into the into the capacitor. Essay answers are limited to about 500 words (3800 characters maximum, including spaces).

ANSWER:

My Answer:

Part F Figure24.16 in the textbook 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. Essay answers are limited to about 500 words (3800 characters maximum, including spaces).

ANSWER:

Part G

My Answer:

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Calculate the magnitude of the force. (This method does not require knowledge of the nature of the fringing field.) Express your answer in terms of the given quantities and appropriate constants.

ANSWER: = Correct

Exercise 24.11 A cylindrical capacitor has an inner inner conductor with a radius of 1.3 3.3

and an outer conductor with a radius of

. The two conductors are separated separat ed by vacuum, vacuum, and the entire capacitor has has a length of 3.0

.

Part A What is the capacitance per unit length? Express your answer using two significant figures.

ANSWER:

11 = 6.0×10 Correct −

Part B The potential of the inner inner conductor relative to th that at of the outer conductor is 350

. Find Find the charge

(magnitude and sign) on both conductors. Enter your answers numerically separated by a comma. Express your answers using two significant figures.

ANSWER:

11 = 6.3×10 , 6.3×10 Correct −

11

−

−

Exercise 24.13 A spherical capacitor capacito r contains a charge of 3.50

when connected to a potential difference differ ence of 230

plates are separated separat ed by vacuum vacuum and the inner inner radius of the outer shell shell is 5.00 Part A Calculate the capacitance. ANSWER:

= 15.2 Correct

.

. If its

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Part B Calculate the radius of the inner sphere. ANSWER:

= 3.66 All attempts used; correct answer displayed

Part C Calculate the electric field just outside the surface of the inner sphere. ANSWER:

4 = 2.35×10 Correct

Exercise 24.23 Suppose the 3

capacitor in Figure 24.10a in the textbook were removed and replaced by a different differ ent one,

and that this changed the equivalent equivalent capacitance capacita nce betw between een points

and

to 8

.

Part A What would be the capacitance of the replacement capacitor? ANSWER:

= 57.0 Correct

Exercise 24.30 A parallel-plate parallel-pla te vacuu vacuum m capacitor has 8.10 2.90

of energy energy stored in it. The separation separat ion betw between een the plates is

. If the separation is decreased to 1.85

,

Part A what is the energy now stored if the capacitor was disconnected from the potential source before the separation of the plates was changed? ANSWER:

= 5.17 Correct

Part B What is the energy now stored if the capacitor remained connected to the potential source while the separation of the plates was changed?

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ANSWER:

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= 12.7 All attempts used; correct answer displayed

Problem 24.56 Two capacitors are connected connected parallel to each other and and connected connected to the batt battery ery with voltage 28 = 9.1

,

= 4.9

. Let

be the their ir capacitan capacitances. ces. Suppos Suppose e the the charged charged capacitors capacitors are discon disconne nected cted from the

source and from each other, and then reconnected to each other with plates of opposite sign opposite sign together. Part A By how much does the energy of the system decrease? Express your answer using two significant figures.

ANSWER:

3 = 5.0×10 Correct −

Problem 26.86 An

circuit has a time constant

.

Part A If the the circuit is discharging, how long will it take for its stored stor ed energy to be reduced to

of its initial

value? Express your answer in terms of the variables

,

, and constant .

ANSWER: = Correct

Part B If it is charging, how long will it take for the stored energy to reach Express your answer in terms of the variables

ANSWER: = Correct

,

of its maximum maximum value? value?

, and constant .

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Problem 26.88 Part A Using Using the equation

for the current in a discharging capacitor, capacito r, derive an expression for the

instantaneous instantaneous power

dissipated in the resistor. resist or.

Express your answer in terms of the variables

,

, and

.

ANSWER: = Correct

Part B Integrate Integrat e the expression for

to find the tota totall energy dissipated dissipate d in the resistor. resisto r.

Express your answer in terms of the variables

,

, and

.

ANSWER: = Correct

Part C Show that the total energy dissipated in the resistor is equal to the total energy initially stored in the capacitor. Essay answers are limited to about 500 words (3800 characters maximum, including spaces).

ANSWER:

My Answer:

Problem 26.90: A Capacitor Burglar Alarm The capacitance of a capacitor can be affected by dielectric material that, although not inside the capacitor, is near enough to the capacitor to be polarized by the fringing electric field that exists near a charged capacitor. This This effect is usu usuall ally y of the order order of picofarads picofarads ( ), but it can be used used with appropriate appropriate electro electron nic circui circuitry try to detect a change in the dielectric material surrounding the capacitor. Such a dielectric material might be the human body, and the effect described above might be used in the design of a burglar alarm. Consider the simplified circuit shown in the figure . The voltage source has emf = 1000 , and the capacitor capacito r has capacitance

10.0

. The electronic electro nic circuitry for

detecting the current, represented as an ammeter in the diagram, has negligible resistance and is capable of detecting a current current that persists at a level of at least 1.00 for at least 200 after the capacitance capacitance has

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changed abruptly from

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to

. The burglar alarm is

designed to be activated if the capacitance changes by 10%.

Part A Determin Determine e the the charge charge on on the the 10.0-

ANSWER:

capacitor capacitor when when it is fully fully charged charged..

8 = 1.00×10 Correct −

Part B If the capacitor is fully charged before the intruder is detected, assuming that the time taken for the capacitance to change by 10% is short enough to be ignored, derive an equation that expresses the current through the resistor resist or as a function of the time since the capacitance capacita nce has changed. Expre xpress ss you your answer swer in term terms s of the the vari ariabl ables

,

,

,

(

) and and .

ANSWER: = All attempts used; correct answer displayed

Part C Determine the range of values values of the resistance resista nce

that will meet the the design specifications specifica tions of the burglar

alarm. (Hint: ( Hint: You You will not be able to solve this part analytically but must use numerical methods. Express as a logarithmic function of

plus known quantities quantities.. Use a trial value value of

and calculate from the

expression a new value. value. Continue Continue to do this until until the input input and output values values of

agree to within three

significant figures.) Enter your answers numerically separated by a comma.

ANSWER:

,

6 7 = 7.15×10 ,7.01×10 Correct

Part D What happens if

is too small? Too large?

Essay answers are limited to about 500 words (3800 characters maximum, including spaces).

ANSWER:

My Answer:

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Exercise 26.40 A 12.8

capacitor is connected through a 0.890

resistor resist or to a constant potential difference differ ence of 60.0

. Part A Compute the the charge on the capacitor at the following times after the connections connections are made: 0, 5.0 , 10.0 , 20.0 , and 100.0 . Express your answers using two significant figures. Enter your answers numerically separated by commas.

ANSWER:

,

,

,

,

4 4 4 = 0,2.7×10 ,4.5×10 ,6.4×10 ,7.7×10 Correct −

−

−

4

−

Part B Compute the charging currents at the same instants. Express your answers using two significant figures. Enter your answers numerically separated by commas.

ANSWER:

,

,

,

,

5 5 5 5 = 6.7×10 ,4.3×10 ,2.8×10 ,1.2×10 ,1.0×10 Correct −

−

−

−

8

−

Exercise 26.41 In the circuit shown in the figure both capacitors are initially charged to 50.0 .

Part A How long after closing the the switch S will the potential across each capacitor be reduced to 10.0

ANSWER:

= 4.51 Correct

?

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Part B What will be the current at that time? ANSWER:

= 0.125 Correct

Exercise 26.45 In the circuit shown in the figure each capacitor initially has a charge of magnitude 3.20 on its plates.

Part A After the switch

is closed, what will be the current current in the circuit at the the instant instant that the capacitors capacit ors have have lost

80.0% of their initial stored energy? ANSWER:

= 12.4 Correct

Exercise 31.7 In each circuit described next, next, an ac voltage source producing a current

is connected to an

additional circuit element. Part A The ac source is connected connected across a resistor resist or

. Sketch graphs of the current current in the circuit and the

potential difference across the resistor as functions of time, covering one cycle of oscillation. Put both graphs on the same set same set of axes so you can compare them. ANSWER:

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View Answer Requested

Part B Do the same as in part A, except suppose the resistor is replaced by an inductor inductor

. Sketch the same

graphs as in part A, but this time across the inductor instead of the resistor. ANSWER:

View Answer Requested

Part C Do th the e same as in part A, except suppose the resistor resisto r is replaced by an capacitor

. Sketch the same same

graphs as in part A, but this time across the capacitor instead of the resistor. ANSWER:

View Answer Requested

Part D Sketch phasor phasor diagram for resistor. Draw the phasors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded.

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ANSWER:

View Correct

Part E Sketch phasor diagram for inductor. Draw the phasors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded.

ANSWER:

View Correct

Part F Sketch phasor diagram for capacitor. Draw the phasors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded.

ANSWER:

View Correct

Exercise 31.18 You have have a 200200-

resistor resistor and and a 6.00-

capacitor. capacitor. Suppos Suppose e yo you u take the resistor resistor and and ca capaci pacitor tor and and make make

a series circuit circuit with a voltage source that has has a voltage amplitude amplitude of 30.0

and an angular angular frequency of 250

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. The current is given by

, so

Part A Graph , versus versus

for

0 to

27.0

.

ANSWER:

View Answer Requested

Part B Graph

, versus versus

for

0 to

27.0

.

ANSWER:

View Answer Requested

Part C Graph

, versus versus

for

0 to

27.0

ANSWER:

View Answer Requested

Part D

.

.

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What is

at

ANSWER:

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?

= -25.1 Correct

Part E What is

at

ANSWER:

?

= 2.45 Correct

Part F What is

at

ANSWER:

?

=

-27.5 Correct

Part G What is

at

ANSWER:

?

= -22.9 Correct

Part H What is

at

ANSWER:

?

= -7.23 Correct

Part I What is

ANSWER:

at

?

= -15.6 Correct

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Score Summary: Your score on this assignment is 82%. You received 81.97 out of a possible total of 100 points.

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