Physics for Scientists and Engineers, Technology Update (No access codes included)
Physics for Scientists and Engineers, Technology Update (No access codes included)
9th Edition
ISBN: 9781305116399
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 33, Problem 33.66AP

A capacitor, a coil, and two resistors of equal resistance are arranged in an AC circuit as shown in Figure P33.66 (page 1028). An AC source provides an emf of ΔVrms = 20.0 V at a frequency of 60.0 Hz. When the double throw switch S is open as shown in the figure, the rms current is 183 mA. When the switch is closed in position a, the rms current is 298 mA. When the switch is closed in position b, the rms current is 137 mA. Determine the values of (a) R, (b) C, and (c) L. (d) Is more than one set of values possible? Explain.

Chapter 33, Problem 33.66AP, A capacitor, a coil, and two resistors of equal resistance are arranged in an AC circuit as shown in

(a)

Expert Solution
Check Mark
To determine

 The value of resistance.

Answer to Problem 33.66AP

 The value of resistance is 99.6Ω .

Explanation of Solution

Given info: The value of the source emf is 20.0V and frequency of the source emf is 60.0Hz . The current through the circuit when switch is open is 183mA , the current through the circuit when switch is at a position is 298mA and the current through the circuit when the switch is at b position is 137mA .

The expression for inductive reactance is,

XL=2πfL

Here,

f is the source frequency.

L is the inductance.

The expression for capacitive reactance is,

XC=12πfC

Here,

C is the capacitance.

The expression for the impedance of the circuit is.

Z=Req2+(XLXC)2

Here,

Req is the equivalent resistance of the circuit.

The expression of the impedance in terms of voltage and current is,

Z=ΔVrmsIrms

Here,

ΔVrms is the source voltage.

Irms is the current.

The figure below depicts the circuit when the switch is in open condition.

Physics for Scientists and Engineers, Technology Update (No access codes included), Chapter 33, Problem 33.66AP , additional homework tip  1

Figure (1)

From figure(1), for the throw switch in open condition:

Substitute R for Req and ΔVrmsIrms for Z in the expression of the impedance of the circuit.

ΔVrmsIrms=R2+(XLXC)2R2+(XLXC)2=(ΔVrmsIrms)2

Substitute 20.0V for ΔVrms and 183mA for Irms in the above expression.

R2+(XLXC)2=(20.0V183mA)2R2+(XLXC)2=11944.22Ω2 (1)

For the throw switch at a position:

The two resistances are in parallel.

The expression of the equivalent resistance is.

Req=(1R+1R)1=R2

Substitute R2 for Req and ΔVrmsIrms for Z in the expression of the impedance of the circuit.

ΔVrmsIrms=R42+(XLXC)2R42+(XLXC)2=(ΔVrmsIrms)2

Substitute 20.0V for ΔVrms and 298mA for Irms in the above expression.

R42+(XLXC)2=(20.0V298mA)2R42+(XLXC)2=4504.30Ω2 (2)

Subtract equation (2) from equation (1).

(R2+(XLXC)2)(R42+(XLXC)2)=11944.22Ω24504.30Ω234R2=7439.92Ω2R=99.6Ω

Conclusion:

Therefore, the value of resistance is 99.6Ω .

(b)

Expert Solution
Check Mark
To determine

 The value of the capacitance.

Answer to Problem 33.66AP

 The value of the capacitance is 24.8μF .

Explanation of Solution

Given info: The value of the source emf is 20.0V and frequency of the source emf is 60.0Hz . The current through the circuit when switch is open is 183mA , the current through the circuit when switch is at a position is 298mA and the current through the circuit when the switch is at b position is 137mA .

The figure below depicts the circuit when switch is at position b.

Physics for Scientists and Engineers, Technology Update (No access codes included), Chapter 33, Problem 33.66AP , additional homework tip  2

Figure (2)

From figure (2), for the throw switch at b position:

When the switch is at position b the inductor gets short circuited.

Substitute R for Req , 0 for XL   and ΔVrmsIrms for Z in the expression of the impedance of the circuit.

ΔVrmsIrms=R2+(0XC)2

Rearrange the above expression for value of XC

XC=((ΔVrmsIrms)2R2)12

Substitute 20.0V for ΔVrms and 137mA for Irms in the above expression.

XC2=((20.0V137mA)2(99.6Ω)2)12XC=106.73Ω

Substitute 12πfC for XC in the above expression.

12πfC=106.73Ω

Rearrange the above equation for value of value of C .

C=12πf(106.73Ω)

Substitute 60.0Hz for f in the above expression.

C=12π(60.0Hz)(106.73Ω)=24.8×106F=24.8μF

Conclusion:

Therefore, the value of the capacitance is 24.8μF .

(c)

Expert Solution
Check Mark
To determine

 The value of the inductance.

Answer to Problem 33.66AP

 The values of the inductance are 0.164H , 0.402H .

Explanation of Solution

Given info: The value of the source emf is 20.0V and frequency of the source emf is 60.0Hz . The current through the circuit when switch is open is 183mA , the current through the circuit when switch is at a position is 298mA and the current through the circuit when the switch is at b position is 137mA .

Substitute 99.6Ω for R and 106.73Ω for XC in the equation (1).

R2+(XL106.73Ω)2=11944.22Ω2XL=106.73Ω±11944.22Ω2(99.6Ω)2XL=106.73Ω±44.98ΩXL=61.75Ω,151.71Ω (3)

For first value of the inductor,

Substitute 2πfL for XL in the above expression.

2πfL=61.75ΩL=61.75Ω2πf

Substitute 60.0Hz for f in the above expression.

L=61.75Ω2π(60.0Hz)=0.164H

Thus, the first value of the inductor is 0.164H

For second value of the inductor,

Substitute 2πfL for XL in equation (3).

2πfL=151.71ΩL=151.71Ω2πf

Substitute Physics for Scientists and Engineers, Technology Update (No access codes included), Chapter 33, Problem 33.66AP , additional homework tip  3 for f in the above expression.

L=151.71Ω2π(60.0Hz)=0.402H

Thus, the second value of the inductor is 0.402H .

Conclusion:

Therefore, the values of the inductance are 0.164H , 0.402H .

(d)

Expert Solution
Check Mark
To determine

 Whether more than one set of value possible.

Answer to Problem 33.66AP

 The resistance and capacitance has one set of values, the inductor has in two set of values.

Explanation of Solution

Given info: : The value of the source emf is 20.0V and frequency of the source emf is 60.0Hz . The current through the circuit when switch is open is 183mA , the current through the circuit when switch is at a position is 298mA and the current through the circuit when the switch is at b position is 137mA .

From the calculation in part (a),(b) and (c),

The value of resistance is 99.6Ω .

The value of capacitance is 24.8μF .

The values of the inductor are 0.164H , 0.402H .

Hence the resistance and the capacitance has one value while the inductor has two set of value in the circuit.

Conclusion:

Therefore, the resistance and capacitance has one set of values, the inductor has in two set of values.

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Chapter 33 Solutions

Physics for Scientists and Engineers, Technology Update (No access codes included)

Ch. 33 - Prob. 33.4OQCh. 33 - Prob. 33.5OQCh. 33 - A sinusoidally varying potential difference has...Ch. 33 - A series RLCcircuit contains a 20.0- resistor, a...Ch. 33 - A resistor, a capacitor, and an inductor are...Ch. 33 - (a) Why does a capacitor act as a short circuit at...Ch. 33 - What is the plia.se angle in a series RLC circuit...Ch. 33 - Prob. 33.11OQCh. 33 - A 6.00-V battery is connected across the primary...Ch. 33 - Do AC ammeters and voltmeters read (a)...Ch. 33 - (a) Explain how the quality factor is related to...Ch. 33 - (a) Explain how the mnemonic ELI the ICE man can...Ch. 33 - Why is the sum of the maximum voltages across each...Ch. 33 - (a) Does the phase angle in an RLC series circuit...Ch. 33 - Prob. 33.5CQCh. 33 - As shown in Figure CQ33.6, a person pulls a vacuum...Ch. 33 - Prob. 33.7CQCh. 33 - Will a transformer operate if a battery is used...Ch. 33 - Prob. 33.9CQCh. 33 - Prob. 33.10CQCh. 33 - When an AC source is connected across a 12.0-...Ch. 33 - (a) What is the resistance of a lightbulb that...Ch. 33 - An AC power supply produces a maximum voltage Vmax...Ch. 33 - A certain lightbulb is rated at 60.0 W when...Ch. 33 - The current in the circuit shown in Figure P32.3...Ch. 33 - In the AC circuit shown in Figure P32.3, R = 70.0 ...Ch. 33 - An audio amplifier, represented by the AC I source...Ch. 33 - Figure P32.4 shows three lightbulbs connected to a...Ch. 33 - An inductor has a .54.0- reactance when connected...Ch. 33 - In a purely inductive AC circuit as shown in...Ch. 33 - Prob. 33.11PCh. 33 - An inductor is connected to an AC power supply...Ch. 33 - An AC source has an output rms voltage of 78.0 V...Ch. 33 - A 20.0-mH inductor is connected to a North...Ch. 33 - Review. 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In an RLC series circuit that includes a...Ch. 33 - Prob. 33.34PCh. 33 - A series RLC circuit has a resistance of 45.0 and...Ch. 33 - An AC voltage of the form = 100 sin 1 000t, where...Ch. 33 - A series RLC circuit has a resistance of 22.0 and...Ch. 33 - An AC voltage of the form v = 90.0 sin 350t, where...Ch. 33 - ln a certain series RLC circuit, Irms = 9.00 A,...Ch. 33 - Prob. 33.40PCh. 33 - Prob. 33.41PCh. 33 - A series RLC circuit has components with the...Ch. 33 - An RLC circuit is used in a radio to tune into an...Ch. 33 - The LC circuit of a radar transmitter oscillates...Ch. 33 - A 10.0- resistor, 10.0-mH inductor, and 100-F...Ch. 33 - A resistor R, inductor L, and capacitor C are...Ch. 33 - Review. 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The voltage phasor diagram for a certain...Ch. 33 - Prob. 33.60APCh. 33 - Energy is to be transmitted over a pair of copper...Ch. 33 - Energy is to be transmitted over a pair of copper...Ch. 33 - A 400- resistor, an inductor, and a capacitor are...Ch. 33 - Show that the rms value for the sawtooth voltage...Ch. 33 - A transformer may be used to provide maximum power...Ch. 33 - A capacitor, a coil, and two resistors of equal...Ch. 33 - Marie Cornu, a physicist at the Polytechnic...Ch. 33 - A series RLC circuit has resonance angular...Ch. 33 - Review. One insulated conductor from a household...Ch. 33 - (a) Sketch a graph of the phase angle for an RLC...Ch. 33 - In Figure P33.71, find the rms current delivered...Ch. 33 - Review. In the circuit shown in Figure P32.44,...Ch. 33 - Prob. 33.73APCh. 33 - A series RLC circuit is operating at 2.00 103 Hz....Ch. 33 - A series RLC circuit consists of an 8.00-...Ch. 33 - A series RLC circuit in which R = l.00 , L = 1.00...Ch. 33 - The resistor in Figure P32.49 represents the...Ch. 33 - An 80.0- resistor and a 200-mH inductor are...Ch. 33 - Prob. 33.79CPCh. 33 - P33.80a shows a parallel RLC circuit. The...Ch. 33 - Prob. 33.81CP
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