Solutions] Mastering Physics HW35

HW35Due: 11:59pm on Thursday, December 3, 2009Note: To understand how points are awarded, read your instructor's Grading Policy.

[Return to Standard Assignment View]

Graphical Analysis of AC Voltage Source Conceptual QuestionThe figure shows a graph of the output from an AC voltage source.

Part A

What is the maximum voltage of the source?

Hint A.1 AC voltage

Hint not displayed

ANSWER: = 3Correct

Part B

What is the average voltage of the source?

Hint B.1 Average voltage

The AC voltage source illustrated symmetrically alternates between positive and negative voltages. Mathematically, it canbe described by

.

ANSWER: = 0Correct

Part C

What is the root-mean-square voltage of the source?

Hint C.1 Root-mean-square voltage

Any physical quantity described by or has an average value of zero when averaged over a complete

cycle. However, if the quantity is first squared and averaged and then the square root taken, the value is positive and givesinformation about the magnitude of the physical quantity. This is referred to as the quantity's root-mean-square (rms) value.The rms value of any quantity described by or is equal to the maximum value of the quantity divided

by .

Express your answer to three significant figures.

ANSWER: = 2.12Correct

Part D

What is the period of the source?

Hint D.1 Period

Hint not displayed

Express your answer in seconds to two significant figures.


Part E

What is the frequency of the source?

http://session.masteringphysics.com/myct/assignmentView?assignmentID=1306072

Hint E.1 Frequency

The frequency of the AC voltage source is the number of cycles of the source per second. The frequency of the source is theinverse of the period of the source.

Express your answer in hertz to three significant figures.


Part F

What is the angular frequency of the source?

Hint F.1 Angular frequency

Hint not displayed

Express your answer in radians per second to three significant figures.


Inductive Reactance

Learning Goal: To understand the concept of reactance (of an inductor) and its frequency dependence.

When an inductor is connected to a voltage source that varies sinusoidally, a sinusoidal current will flow through the inductor,its magnitude depending on the frequency. This is the essence of AC (alternating current) circuits used in radio, TV, and stereos.Circuit elements like inductors, capacitors, and resistors are linear devices, so the amplitude of the current will be

proportional to the amplitude of the voltage. However, the current and voltage may not be in phase with each other. This new

relationship between voltage and current is summarized by the reactance, the ratio of voltage and current amplitudes, , and

: , where the subscript L indicates that this formula applies to an inductor.

Part A

To find the reactance of an inductor, imagine that a current , is flowing through the inductor. What is

the voltage across this inductor?

Hint A.1 Voltage and current for an inductor

Hint not displayed

Express your answer in terms of , , and the inductance .

ANSWER: =

Correct

Part B

What is the reactance of an inductor?

Hint B.1 The definition of reactance

Hint not displayed

Express your answer in terms of and the inductance .

ANSWER: =

Correct

Part C

In thinking of an inductor as a circuit element, it is helpful to consider its limiting behavior at high and low frequencies. Atone extreme, the inductor might behave like a short circuit, that is, like a resistor with almost no resistance (an ideal wire)having essentially no voltage drop across it no matter what the current. Alternatively, the inductor might behave like an opencircuit, that is, like a resistor with large resistance so that essentially no current will flow no matter what the applied voltage.Based on the formula you obtained for the reactance, how does an inductor behave at high and low frequencies?

ANSWER: like a short circuit at both high frequencies and low frequencieslike an open circuit at both high frequencies and low frequencieslike an open circuit at high frequencies and a short circuit at low frequencieslike an open circuit at low frequencies and a short circuit at high frequencies

Correct

Determining Inductance from Voltage and CurrentAn inductor is hooked up to an AC voltage source. The voltage source has EMF and frequency . The current amplitude in

the inductor is .

Part A

What is the reactance of the inductor?

Hint A.1 Definition of reactance

Hint not displayed

Express your answer in terms of and .

ANSWER: =

Correct

Part B

What is the inductance of the inductor?

Hint B.1 Reactance in terms of and

Hint not displayed

Express your answer in terms of , , and .

ANSWER: =

Correct

Part C

What would happen to the amplitude of the current in the inductor if the inductance were doubled?

Hint C.1 How to approach the problem

Hint not displayed

ANSWER: There would be no change in the amplitude of the current.The amplitude of the current would be doubled.The amplitude of the current would be halved.The amplitude of the current would be quadrupled.

Correct

Note that, in "A Resistor and a Capacitor in a Series AC Circuit" youdo NOT know the amplitude of the driving EMF. It is instructive tocalculate that driving amplitude once you have successfully completedthe problem. You will find the answer in the "Answers to HomeworkProblems" link on our course webpage. GBA

A Resistor and a Capacitor in a Series AC CircuitA resistor with resistance and a capacitor with capacitance are connected in series to an AC voltage source. The time-

dependent voltage across the capacitor is given by .

Part A

What is the amplitude of the total current in the circuit?

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Applying Ohm's law to a capacitor

Hint not displayed

Hint A.3 The reactance of a capacitor

Hint not displayed

Express your answer in terms of any or all of , , , and .

ANSWER: =

Correct

Part B

What is the amplitude of the voltage across the resistor?

Hint B.1 Relating to

Hint not displayed

Express your answer in terms of , , , and .

ANSWER:

ANSWER: =

Correct

Part C

If , , , and , what is ?

Hint C.1 Calculating the answer

Use the equation obtained in Part B to work out the answer. Be careful of powers of ten in your calculation.

Express your answer numerically, in millivolts, to the nearest integer.

ANSWER: = 3Correct mV

Five Voltmeters in an AC CircuitFive infinite-impedance voltmeters, calibrated to read rms values, are connected as shown in figure . Let 200 ,

0.400 , 6.00 , and 30.0 .

Part A

What is the reading of the voltmeter if 200 ?


Part B



Part C



Part D



Part E



Part F



Part G



Part H



Part I



Part J



The following two problems are a reminder that AC circuits have awide range of applications beyond the 60-Hz circuits we use todeliver power to our homes. The alternating current in audio circuitsis not current of a single frequency; instead, the audio "signals"consist of current of a mixture of frequencies, from 20 Hz up to20 kHz. The following two circuit designs are used to suppressthe low frequencies (high-pass filter) or to suppress the highfrequencies (low-pass filter); these designs frequently appear inaudio circuits. In "High-Pass Filter", I suggest that you ignore MP's commentfollowing a correct answer to Part A. In my opinion, Parts B and Care much easier to answer by considering the answer to A in the form sqrt(R^2+X_L^2)/Z (i.e. just think about adding the three curvesfor the resistor, capacitor, and inductor voltages). GBA

A High-Pass FilterA series L-R-C circuit consisting of a voltage source, a capacitor of capacitance , an inductor of inductance , and a resistor

of resistance is driven with an AC voltage of amplitude and frequency . Define to be the amplitude of the voltage

across the resistor and the inductor.

Part A

Find the ratio .

Hint A.1 Find

Hint not displayed

Hint A.2 Find

Hint not displayed

Express your answer in terms of either , , and , or , , and .

ANSWER: =

Correct

For the following questions it will be useful to write the voltage ratio in the following form:

.

Part B

Which of the following statements is true in the limit of large ( )?

Hint B.1 Implications of large

Hint not displayed

ANSWER: is proportional to .

is proportional to .


is close to 1.

Correct

Part C

Which of the following statements is true in the limit of small ( )?

Hint C.1 Implications of small

Hint not displayed




is close to 1.

Correct

When is large, , and when is small, . Therefore, this circuit has the property that only the

amplitude of the low-frequency inputs will be attenuated (reduced in value) at the output, while the amplitude of thehigh-frequency inputs will pass through relatively unchanged. This is why such a circuit is called a high-pass filter.

In "Constructing a Low-Pass Filter", I suggest that you ignore MP'scomment following a correct answer to Part A. In my opinion, PartsB and C are much easier to answer by considering the answer to A inthe form X_C/Z (i.e. just think about adding the three curves for theresistor, capacitor, and inductor voltages). GBA

Constructing a Low-Pass FilterA series L-R-C circuit is driven with AC voltage of amplitude and frequency . Define to be the amplitude of the

voltage across the capacitor. The resistance of the resistor is , the capacitance of the capacitor is , and the inductance of the

inductor is .

Part A

What is the ratio ?

Hint A.1 Find

Hint not displayed

Hint A.2 Find

Hint not displayed

Express your answer in terms of either , , , and or , , and .

ANSWER: =

Correct

For the following questions it will be useful to write the voltage ratio in the following form:

.

Part B

Which of the following statements is true in the large limit (that is, for )?

Hint B.1 Implications of large

In the large limit, , and , so becomes the only significant term in the denominator.




is close to 1.

Correct

Part C

Which of the following statements is true in the small limit (that is, for )?

Hint C.1 Implications of small

In the small limit, and , so all terms in the denominator involving can be neglected.




is close to 1.

Correct

When omega is large, ; and when omega is small, . Therefore, the circuit of this problem has the

property that only the high-frequency inputs will be attenuated (reduced in value) at the output, while low-frequencyinputs will pass through relatively unchanged. That is why such a circuit is called a low-pass filter.

Score Summary:Your score on this assignment is 99.6%.You received 49.8 out of a possible total of 50 points.

Documents

Solutions] Mastering Physics HW35