University Physics with Modern Physics

Hugh D. Young, Roger A. Freeman

Chapter 30

Inductance - all with Video Answers

Educators

Chapter Questions

Problem 1

Two coils have mutual inductance $M=3.25 \times 10^{-4} \mathrm{H}$ . The current $i_{1}$ in the first coil increases at a uniform rate of 830 $\mathrm{A} / \mathrm{s}$ . (a) What is the magnitude of the induced emf in the second coil? Is it constant? (b) Suppose that the current described is in the second coil rather than the first. What is the magnitude of the induced emf in the first coil?

Abhishek Jana

Abhishek Jana

Numerade Educator

Problem 2

Two coils are wound around the same cylindrical form, like the coils in Example $30.1 .$ When the current in the first coil is decreasing at a rate of $-0.242 \mathrm{A} / \mathrm{s}$ , the induced emf in the second coil has magnitude $1.65 \times 10^{-3} \mathrm{V} .$ (a) What is the mutual inductance of the pair of coils? (b) If the second coil has 25 turns, what is the flux through each turn when the current in the first coil equals 1.20 $\mathrm{A} ?(\mathrm{c})$ If the current in the second coil increases at a rate of 0.360 $\mathrm{A} / \mathrm{s}$ , what is the magnitude of the induced emf in the first coil?

Kevin Hayakawa

University of California - Los Angeles

Problem 3

From Eq. $(30.5) 1 \mathrm{H}=1 \mathrm{Wb} / \mathrm{A},$ and from Eq. $(30.4)$ $1 \mathrm{H}=1 \Omega \cdot \mathrm{s} .$ Show that these two definitions are equivalent.

Shoukat Ali

Other Schools

Problem 4

A solenoidal coil with 25 turns of wire is wound tightly around another coil with 300 turns (see Example 30.1$)$ . The inner solenoid is 25.0 $\mathrm{cm}$ long and has a diameter of $2.00 \mathrm{cm} .$ At a certain time, the current in the inner solenoid is 0.120 $\mathrm{A}$ and is increasing at a rate of $1.75 \times 10^{3} \mathrm{A} / \mathrm{s}$ . For this time, calculate; (a) the average magnetic flux through each turn of the inner solenoid; (b) the mutual inductance of the two solenoids; (c) the emf induced in the outer solenoid by the changing current in the inner solenoid.

Kevin Hayakawa

University of California - Los Angeles

Problem 5

Two toroidal solenoids are wound around the same form so that the magnetic field of one passes through the turns of the other. Solenoid 1 has 700 turns, and solenoid 2 has 400 turns. When the current in solenoid 1 is 6.52 A, the average flux through each turn of solenoid 2 is 0.0320 Wb. (a) What is the mutual inductance of the pair of solenoids? (b) When the current in solenoid 2 is 2.54 $\mathrm{A}$ , what is the average flux through each turn of solenoid 1$?$

Abhishek Jana

Numerade Educator

Problem 6

A toroidal solenoid has 500 turns, cross-sectional area $6.25 \mathrm{cm}^{2},$ and mean radius $4.00 \mathrm{cm} .$ (a) Calcualte the coil's self-inductance. (b) If the current decreases uniformly from 5.00 $\mathrm{A}$ to 2.00 $\mathrm{A}$ in 3.00 $\mathrm{ms}$ , calculate the self-induced emf in the coil.
(c) The current is directed from terminal $a$ of the coil to terminal $b$ . Is the direction of the induced emf from $a$ to $b$ or from $b$ to $a ?$

Shoukat Ali

Other Schools

Problem 7

At the instant when the current in an inductor is increasing at a rate of 0.0640 $\mathrm{A} / \mathrm{s}$ , the magnitude of the self-induced emf is 0.0160 $\mathrm{V}$ . (a) What is the inductance of the inductor? (b) If the inductor is a solenoid with 400 turns, what is the average magnetic flux through each turn when the current is 0.720 $\mathrm{A} ?$

Abhishek Jana

Numerade Educator

Problem 8

When the current in a toroidal solenoid is changing at a rate of 0.0260 $\mathrm{A} / \mathrm{s}$ , the magnitude of the induced emf is 12.6 $\mathrm{mV}$ . When the current equals 1.40 $\mathrm{A}$ , the average flux through each turn of the solenoid is 0.00285 $\mathrm{Wb}$ . How many turns does the solenoid have?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 9

The inductor in Fig. 30.18 has inductance 0.260 $\mathrm{H}$ and carries a current in the direction shown that is decreasing at a uniform rate, $d i / d t=-0.0180 \mathrm{A} / \mathrm{s}$ . (a) Find the self-induced emf. (b) Which end of the inductor, $a$ or $b,$ is at a higher potential?

Keshav Singh

Numerade Educator

Problem 10

The inductor shown in Fig. 30.18 has inductance 0.260 $\mathrm{H}$ and carries a current in the direction shown. The current is changing at a constant rate. (a) The potential between points $a$ and $b$ is $V_{a b}=1.04 \mathrm{V},$ with point $a$ at higher potential. Is the current increasing or decreasing? b) If the current at $t=0$ is 12.0 $\mathrm{A}$ , what is the current at $t=2.00 \mathrm{s} ?$

Vishal Gupta

Numerade Educator

Problem 11

Inductance of a Solenoid. A long, straight solenoid has $N$ turms, uniform cross-sectional area $A,$ and length $l .$ Show that the inductance of this solenoid is given by the equation $L=\mu_{0} A N^{2} / L$ Assume that the magnetic field is uniform inside the solenoid and zero outside. (Your answer is approximate because $B$ is actually smaller at the ends than at the center. For this reason, your answer is actually an upper limit on the inductance.)

Shoukat Ali

Other Schools

Problem 12

An inductor used in a de power supply has an inductance of 12.0 $\mathrm{H}$ and a resistance of $180 \Omega .$ It carries a current of 0.300 $\mathrm{A}$ . (a) What is the energy stored in the magnetic field? (b) At what rate is thermal energy developed in the inductor?(c) Does your answer to part (b) mean that the magnetic-field energy is decreasing with time? Explain.

Shoukat Ali

Other Schools

Problem 13

An air-filled toroidal solenoid has a mean radius of 15.0 $\mathrm{cm}$ and a cross-sectional area of $5.00 \mathrm{cm}^{2} .$ When the current is 12.0 $\mathrm{A}$ , the energy stored is 0.390 $\mathrm{J}$ . How many turns does the winding have?

Fadil Iqbal

Numerade Educator

Problem 14

An air-filled toroidal solenoid has 300 turns of wire, a mean radius of $12.0 \mathrm{cm},$ and a cross-sectional area of $4.00 \mathrm{cm}^{2} .$ If the current is 5.00 $\mathrm{A}$ , calculate: (a) the magnetic field in the solenoid; (b) the self-inductance of the solenoid; (c) the energy stored in the magnetic field; (d) the energy density in the magnetic field. (e) Check your answer for part (d) by dividing your answer to part (c) by the volume of the solenoid.

Kevin Hayakawa

University of California - Los Angeles

Problem 15

A solenoid 25.0 $\mathrm{cm}$ long and with a cross-sectional area of 0.500 $\mathrm{cm}^{2}$ contains 400 turns of wire and carries a current of 80.0 A. Calculate: (a) the magnetic field in the solenoid; (b) the energy density in the magnetic fleld if the solenoid is filled with air; (c) the total energy contained in the coil's magnetic field (assume the field is uniform); (d) the inductance of the solenoid.

Shoukat Ali

Other Schools

Problem 16

It has been proposed to use large inductors as energy storage devices. (a) How much electrical energy is converted to light and thermal energy by a $200-\mathrm{W}$ light bulb in one day? (b) If the amount of energy calculated in part (a) is stored in an inductor in which the current is 80.0 $\mathrm{A}$ , what is the inductance?

Keshav Singh

Numerade Educator

Problem 17

Starting from Eq. $(30.9)$ , derive in detail Eq. $(30.11)$ for the energy density in a toroidal solenoid filled with a magnetic material.

Shoukat Ali

Other Schools

Problem 18

It is proposed to store $1.00 \mathrm{kW} \cdot \mathrm{h}=3.60 \times 10^{6} \mathrm{J}$ of electrical energy in a uniform magnetic field with magnitude 0.600 $\mathrm{T}$ . (a) What volume (in vacuum) must the magnetic field occupy to store this amount of energy? (b) If instead this amount of energy is to be stored in a volume (in vacuum) equivalent to a cube 40.0 $\mathrm{cm}$ on a side, what magnetic field is required?

Kevin Hayakawa

University of California - Los Angeles

Problem 19

An inductor with an inductance of 2.50 $\mathrm{H}$ and a resistance of 8.00$\Omega$ is connected to the terminals of a battery with an emf of 6.00 $\mathrm{V}$ and negligible intermal resistance. Find (a) the initial rate of increase of current in the circuit; (b) the rate of increase of current at the instant when the current is $0.500 \mathrm{A} ;(\mathrm{c})$ the current 0.250 $\mathrm{s}$ after the circuit is closed; (d) the final steady-state current.

Shoukat Ali

Other Schools

Problem 20

A $15.0-\Omega$ resistor and a coil are connected in series with a 6.30-V battery with negligible internal resistance and a closed switch. (a) At 200 ms after the switch is opened the current has decayed to 0.210 A. Calculate the inductance of the coil. (b) Calculate the time constant of the circuit. (c) How long after the switch is closed will the current reach 1.00$\%$ of its original value?

Shoukat Ali

Other Schools

Problem 21

A $35.0-\mathrm{V}$ battery with negligible internal resistance, a 50.0- $\Omega$ resistor, and a $1.25-\mathrm{mH}$ inductor with negligible resistance are all connected in series with an open switch. The switch is suddenly closed. (a) How long after closing the switch will the current through the inductor reach one-half of its maximum value? (b) How long after closing the switch will the energy stored in the inductor reach one-half of its maximum value?

Abhishek Jana

Numerade Educator

Problem 22

In Fig. 30.11, switch $S_{1}$ is closed while switch $S_{2}$ is kept open. The inductance is $L=0.115 \mathrm{H}$ , and the resistance is $R=120 \Omega .$ (a) When the current has reached its final value, the energy stored in the inductor is 0.260 . What is the emf $\mathcal{E}$ of the battery? (b) After the current has reached its final value, $S_{1}$ is opened and $\mathrm{S}_{2}$ is closed. How much time does it take for the energy stored in the inductor to decrease to 0.130 $\mathrm{J}$ , half the original value?

Keshav Singh

Numerade Educator

Problem 23

Show that $L / R$ has units of time.

Shoukat Ali

Other Schools

Problem 24

Write an equation corresponding to Eq. $(30.13)$ for the current shown in Fig. 30.11 just after switch $\mathrm{S}_{2}$ is closed and switch $\mathrm{S}_{1}$ is opened, if the initial current is $I_{0}$ - Use integration methods to verify Eq. $(30.18) .$

Shoukat Ali

Other Schools

Problem 25

In Fig. $30.11,$ suppose that $\mathcal{E}=60.0 \mathrm{V}, R=240 \Omega,$ and $L=0.160 \mathrm{H}$ . With switch $\mathrm{S}_{2}$ open, switch $\mathrm{S}_{1}$ is left closed until a constant current is established. Then $\mathrm{S}_{2}$ is closed and $\mathrm{S}_{1}$ opened,
taking the battery out of the circuit. (a) What is the initial current in the resistor, just after $S_{2}$ is closed and $S_{1}$ is opened? (b) What is the current in the resistor at $t=4.00 \times 10^{-4} \mathrm{s} ?(\mathrm{c})$ What is the potential difference between points $b$ and $c$ at $t=4.00 \times 10^{-4} \mathrm{s} ?$ Which point is at a higher potential? (d) How long does it take the current to decrease to half its initial value?

Shoukat Ali

Other Schools

Problem 26

In Fig. $30.11,$ suppose that $\mathcal{E}=60.0 \mathrm{V}, R=240 \Omega,$ and $L=0.160 \mathrm{H}$ . Initially there is no current in the circuit. Switch $\mathrm{S}_{2}$ is left open, and switch $\mathrm{S}_{1}$ is closed. (a) Just after $\mathrm{S}_{1}$ is closed, what are the potential differences $v_{a b}$ and $v_{b c} ?(b)$ A long time (many time constants) after $S_{1}$ is closed, what are $v_{a b}$ and $v_{b c} ?(c)$ What are $v_{a b}$ and $v_{b c}$ an intermediate time when $i=0.150 \mathrm{A} ?$

Keshav Singh

Numerade Educator

Problem 27

Refer to Exercise $30.19 .$ (a) What is the power input to the inductor from the battery as a function of time if the circuit is completed at $t=0 ?$ (b) What is the rate of dissipation of energy in the resistance of the inductor as a function of time?(c) What is the rate at which the energy of the magnetic field in the inductor is increasing, as a function of time? (d) Compare the results of parts (a), and (c).

Shoukat Ali

Other Schools

Problem 28

A 20.0 -\muF capacitor is charged by a $150.0-\mathrm{V}$ power supply, then disconnected from the power and connected in series with a $0.280-\mathrm{mH}$ inductor. Calculate: (a) the oscillation frequency of the circuit; (b) the energy stored in the capacitor at time $t=0 \mathrm{ms}$ (the moment of connection with the inductor); (c) the energy stored in the inductor at $t=1.30 \mathrm{ms}$ .

Shoukat Ali

Other Schools

Problem 29

A $7.50-\mathrm{nF}$ capacitor is charged up to $12.0 \mathrm{V},$ then disconnected from the power supply and connected in series through a coil. The period of oscillation of the circuit is then measured to be $8.60 \times 10^{-5} \mathrm{s}$ . Calculate: (a) the inductance of the coil; $(\mathrm{b})$ the maximum charge on the capacitor; (c) the total energy of the circuit; (d) the maximum current in the circuit.

Abhishek Jana

Numerade Educator

Problem 30

A $18.0-\mu F$ capacitor is placed across a $22.5-\mathrm{V}$ battery for several seconds and is then connected across a 12.0 -mH inductor that has no appreciable resistance. (a) After the capacitor and
inductor are connected together, find the maximum current in the circuit. When the current is a maximum, what is the charge on the capacitor? (b) How long after the capacitor and inductor are connected together does it take for the capacitor to be completely discharged for the first time? For the second time? (c) Sketch graphs of the charge on the capacitor plates and the current through the inductor as functions of time.

Shoukat Ali

Other Schools

Problem 31

L-C Oscillations. A capacitor with capacitance $6.00 \times$ $10^{-5} \mathrm{F}$ is charged by connecting it to a $12.0-\mathrm{V}$ battery. The capacitor is disconnected from the battery and connected across an inductor with $L=1.50 \mathrm{H}$ (a) What are the angular frequency $\omega$ of the electrical oscillations and the period of these oscillations (the time for one oscillation $) ?(\text { b) What is the initial charge on the }$ capacitor? (c) How much energy is initially stored in the capacitor? (d) What is the charge on the capacitor 0.0230 $\mathrm{s}$ after the connection to the inductor is made? Interpret the sign of your answer.(e) At the time given in part (d), what is the current in the inductor? Interpret the sign of your answer. (f) At the time given in part (d), how much electrical energy is stored in the capacitor and how much is stored in the inductor?

Shoukat Ali

Other Schools

Problem 32

A Radio Tuning Circuit. The minimum capacitance of a variable capacitor in a radio is 4.18 $\mathrm{pF}$ . (a) What is the inductance of a coil connected to this capacitor if the oscillation frequency of the $L-C$ circuit is $1600 \times 10^{3} \mathrm{Hz}$ , corresponding to one end of the AM radio broadcast band, when the capacitor is set to its minimum capacitance? (b) The frequency at the other end of the broadcast band is $540 \times 10^{3} \mathrm{Hz}$ . What is the maximum capacitance of the capacitor if the oscillation frequency is adjustable over the range of the broadcast band?

Shoukat Ali

Other Schools

Problem 33

An $L C$ cirruit containing an $80.0-\mathrm{mH}$ inductor and a 1.25-nF capacitor oscillates with a maximum current of 0.750 $\mathrm{A}$ . Calculate: (a) the maximum charge on the capacitor and (b) the oscillation frequency of the circuit. (c) Assuming the capacitor had its maximum charge at time $t=0$ , calculate the energy stored in the inductor after 2.50 $\mathrm{ms}$ of oscillation.

Shoukat Ali

Other Schools

Problem 34

In an $L-$ circuit, $L=85.0 \mathrm{mH}$ and $C=3.20 \mu \mathrm{F}$ . During the oseillations the maximum current in the inductor is 0.850 $\mathrm{mA}$ . (a) What is the maximum charge on the capacitor? (b) What is the magnitude of the charge on the capacitor at an instant when the current in the inductor has magnitude 0.500 $\mathrm{mA} ?$

Shoukat Ali

Other Schools

Problem 35

(a) Using Eqs. $(30.21)$ and $(30.23)$ for an $L-C$ circuit, write expressions for the energy stored in the capacitor as a function of time and for the energy stored in the inductor as a function of time. (b) Using Eq. $(30.22)$ and the trigonometric identity $\sin ^{2} x+$ $\cos ^{2} x=1,$ show that the total energy in the $L-C$ circuit is constant and equal to $Q^{2} / 2 C$ .

Shoukat Ali

Other Schools

Problem 36

Show that the differential equation of Eq. $(30.20)$ is satisfied by the function $q=Q \cos (\omega t+\phi),$ with $\omega$ given by 1$/ \sqrt{L C}$

NM

Nicholas Mesmer

Numerade Educator

Problem 37

Show that $\sqrt{L C}$ has units of time.

Shoukat Ali

Other Schools

Problem 38

For the circuit of Fig. $30.17,$ let $C=15.0 \mathrm{nF}, L=22 \mathrm{mH}$ and $R=75.0 \Omega(\text { a) Calculate the oscillation frequency of the circuit }$ once the capacitor has been charged and the switch has been connected to point $a$ (b) How long will it take for the amplitude of the oscillation to decay to 10.0$\%$ of its original value? (c) What value of $R$ would result in a critically damped circuit?

Keshav Singh

Numerade Educator

Problem 39

(a) In Eq. $(13.41),$ substitute $q$ for $x, L$ for $m, 1 / C$ for $k,$ and R for the damping constant $b$ . Show that the result is Eq. $(30.27)$ . (b) Make these same substitutions in Eq. $(13.43)$ and show that Eq. $(30.29)$ results. (c) Make these substitutions in Eq. $(13.42)$ and show that Eq. $(30.28)$ results.

Keshav Singh

Numerade Educator

Problem 40

(a) Take first and second derivatives with respect to time of q given in Eq. $(30.28)$ , and show that it is a solution of Eq. $(30.27)$ .
(b) At $t=0$ the switch shown in Fig. 30.17 is thrown so that it connects points $d$ and $a ;$ at this time, $q=Q$ and $i=d q / d t=0$ . Show that the constants $\phi$ and $A$ in Eq. $(30.28)$ are given by
$$
\tan \phi=-\frac{R}{2 L \sqrt{(1 / L C)-\left(R^{2} / 4 L^{2}\right)}} \text { and } A=\frac{Q}{\cos \phi}
$$

Shoukat Ali

Other Schools

Problem 41

An $L-R-C$ circuit has $L=0.450 \mathrm{H}, C=2.50 \times 10^{-5} \mathrm{F}$ and resistance $R$ (a) What is the angular frequency of the circuit when $R=0 ?$ (b) What value must $R$ have to give a 5.0$\%$
decrease in angular frequency compared to the value calculated in part (a)?

Shoukat Ali

Other Schools

Problem 42

Show that the quantity $\sqrt{L} / C$ has units of resistance (ohms).

Shoukat Ali

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Problem 43

One solenoid is centered inside another. The outer one has a length of 50.0 $\mathrm{cm}$ and contains 6750 coils, while the coarial inner solenoid is 3.0 $\mathrm{cm}$ long and 0.120 $\mathrm{cm}$ in diameter and contains 15 coils. The current in the outer solenoid is changing at 37.5 $\mathrm{A} / \mathrm{s}$ . (a) what is the mutual inductance of these solenoids? (b) Find the emf induced in the innner solenoid.

Shoukat Ali

Other Schools

Problem 44

A coil has 400 turns and self-inductance 3.50 $\mathrm{mH}$ . The current in the coil varies with time according to $i=$ $(680 \mathrm{mA}) \cos (\pi t)(0.0250 \mathrm{s}) \cdot(\mathrm{a})$ What is the maximum emf induced in the coil? (b) What is the maximum average flux through each turn of the coil?(c) At $t=0.0180 \mathrm{s}$ , what is the magnitude of the induced emf?

Shoukat Ali

Other Schools

Problem 45

A Differentiating Circuit. The current in a resistanceless inductor is caused to vary with time as shown in the graph of Fig. 30.19 . (a) Sketch the pattern that would be observed on the screen of an oscilloscope connected to the terminals of the inductor. (The oscilloscope spot sweeps horizontally across the screen at a constant speed, and its vertical deflection is proportional to the potential difference between the inductor terminals.) (b) Explain why a circuit with an inductor can be described as a "differentiating circuit".

Shoukat Ali

Other Schools

Problem 46

A $0.250-\mathrm{H}$ inductor carries a time-varying current given by the expression $i=(124 \mathrm{mA}) \cos [(240 \pi / \mathrm{s}) t] .$ (a) Find an expression for the induced emf as a function of time. Graph the current and induced emf as functions of time for $t=0$ to $t=\frac{1}{60} \mathrm{s}$ . (b) What is the maximum emf? What is the current when the induced emf is a maximum? (c) What is the maximum current? What is the induced emf when the current is a maximum?

Shoukat Ali

Other Schools

Problem 47

Inductors in Series and Parallel. You are given two inductors, one of self-inductance $L_{1}$ and the other of self-inductance $L_{2} \cdot(\text { a) You connect the two inductors in series and arrange them so }$
that their mutual inductance is negligible. Show that the equivalent inductance of the combination is $L_{\mathrm{oq}}=\left(1 / L_{1}+1 / L_{2}\right)^{-1}$ . (Hint: For either a series or a parallel combination, the potential difference across the combination is $L_{\mathrm{eq}}(d i / d t),$ where $i$ is the current
through the combination. For a parallel combination, $i$ is the sum of the currents through the two inductors.)

Shoukat Ali

Other Schools

Problem 48

A Coasial Cable. A small solid conductor with radius a is supported by insulating, nonmagnetic disks on the axis of a thin-walled tube with inner radius $b$ . The inner and outer conductors carry equal currents $i$ in opposite directions. (a) Use Ampere's law to find the magnetic field at any point in the volume between the conductors, (b) Write the expression for the flux $d \Phi_{B}$ through a narrow strip of length $l$ parallel to the axis, of width $d r,$ at a distance $r$ from the axis of the cable and lying in a plane containing the axis. (c) Integrate your expression from part (b) over the volume between the two conductors to find the total flux produced by a current i in the central conductor. (d) Show that the inductance of a length $l$ of the cable is
$$
L=l \frac{\mu_{0}}{2 \pi} h u\left(\frac{b}{a}\right)
$$
(e) Use Eq. $(30.9)$ to calculate the energy stored in the magnetic field for a length $l$ of the cable.

Keshav Singh

Numerade Educator

Problem 49

Consider the coarial cable of Problem 30.48 . The conductors carry equal currents $i$ in opposite directions. (a) Use Ampere's law to find the magnetic field at any point in the volume between
the conductors. (b) Use the energy density for a magnetic field, Eq. $(30.10),$ to calculate the energy stored in a thin, cylindrical shell between the two conductors. Let the cylindrical shell have inner radius $r,$ outer radius $r+d r,$ and length $L$ (c) integrate your result in part (b) over the volume between the two conductors to find the total energy stored in the magnetic field for a length $l$ of the cable. (d) Use your result in part (c) and Eq. (30.9) to calculate the inductance $L$ of a length $I$ of the cable. Compare your result to $L$ calculated in part (d) of Problem 30.48 .

Shoukat Ali

Other Schools

Problem 50

A toroidal solenoid has a mean radius $r$ and a cross-sectional area $A$ and is wound uniformly with $N_{1}$ turns. A second toroidal solenoid with $N_{2}$ turns is wound uniformly around the first. The
two coils are wound in the same direction. (a) Derive an expression for the inductance $L_{1}$ when only the first coil is used and an expression for $L_{2}$ when only the second coil is used. (b) Show that
$M^{2}=L_{1} L_{2} .$

Shoukat Ali

Other Schools

Problem 51

(a) What would have to be the self-inductance of a solenoid for it to store 10.0 $\mathrm{J}$ of energy when a $1.50-\mathrm{A}$ current runs throught it? $(\mathrm{b})$ If this solenoid's cross-sectional diameter is $4.00 \mathrm{cm},$ and if you could wrap its coils to a density of 10 coils/mm, how long
would the solenoid be? (See Exercise $30.11 . )$ Is this a realistic length for ordinary laboratory use?

Shoukat Ali

Other Schools

Problem 52

An inductor is connected to the terminals of a battery that has an emf of $12.0 \mathrm{~V}$ and negligible internal resistance. The current is $4.86 \mathrm{~mA}$ at $0.725 \mathrm{~ms}$ after the connection is completed. After a long time the current is $6.45 \mathrm{~mA}$. What are (a) the resistance $R$ of the inductor and (b) the inductance $L$ of the inductor?

Shoukat Ali

Other Schools

Problem 53

Continuation of Exercises 30.19 and $30.27 .$ (a) How much energy is stored in the magnetic field of the inductor one time constant after the battery has been connected? Compute this both by integrating the expression in Exercise 30.27$(\mathrm{c})$ and by using Eq. $(30.9),$ and compare the results. (b) Integrate the expression obtained in Exercise 30.27$(a)$ to find the total energy supplied by the battery during the time interval considered in part (a). (c) Integrate the expression obtained in Exercise 30.27$(\mathrm{b})$ to find the total energy dissipated in the resistance of the inductor during the same time period. (d) Compare the results obtained in parts (a), (b), and (c).

Shoukat Ali

Other Schools

Problem 54

Continuation of Exercise $30.25 .$ (a) What is the total energy initially stored in the inductor?(b) At $t=4.00 \times 10^{-4} \mathrm{s}$ , at what rate is the energy stored in the inductor decreasing? (c) Att
$t=4.00 \times 10^{-4} \mathrm{s},$ at what rate is electrical energy being converted into thermal energy in the resistor? (d) Obtain an expression for the rate at which electrical energy is being converted into thermal energy in the resistor as a function of time. Integrate this expression from $t=0$ to $t=\infty$ to obtain the total electrical energy dissipated in the resistor. Compare your result to that of part (a).

Shoukat Ali

Other Schools

Problem 55

The equation preceding Eq. $(30.27)$ may be converted into an energy relationship. Multiply both sides of this equation by $-i=-d q / d t .$ The first term then becomes $i^{2} R$ . Show that the second term can be written as $d\left(\frac{1}{2} L i^{2}\right) / d t,$ and that the third term can be written as $d\left(q^{2} / 2 C\right) / d t .$ What does the resulting equation say about energy conservation in the circuit?

Keshav Singh

Numerade Educator

Problem 56

A $5.00-\mu F$ capacitor is initially charged to a potential of 16.0 $\mathrm{V}$ . It is then connected in series with a $3.75-\mathrm{mH}$ inductor. (a) What is the total energy stored in this circuit? (b) What is the maximum current in the inductor? What is the charge on the capacitor plates at the instant the current in the inductor is maximal?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 57

An Electromagnetic CarAlarm. Your latest invention is a car alarm that produces sound at a particularly annoying frequency of 3500 $\mathrm{Hz}$ . To do this, the car-alarm circuitry must produce an altermating electric current of the same frequency. That's why your design includes an inductor and a capacitor in series. The maximum voltage across the capacitor is to be 12.0 $\mathrm{V}$ (the same voltage as the car battery). To produce a sufficiently loud sound, the capacitor must store 0.0160 $\mathrm{J}$ of energy. What values of capacitance and inductance should you choose for your car-alarm circuit?

Vishal Gupta

Numerade Educator

Problem 58

An $L-C$ circuit consists of a $60.0-\mathrm{mH}$ inductor and a $250-\mu F$ capacitor. The initial charge on the capacitor is 6.00$\mu \mathrm{C}$ , and the initial current in the inductor is zero. (a) What is the maximum voltage across the capacitor? (b) What is the maximum current in the inductor? (c) What is the maximum energy stored in the inductor? (d) When the current in the inductor has half its maximum value, what is the charge on the capacitor and what is the energy stored in the inductor?

Abhishek Jana

Numerade Educator

Problem 59

Solar Magnetic Energy. Magnetic fields within a sunspot can be as strong as 0.4 $\mathrm{T}$ . (By comparison, the earth's magnetic field is about $1 / 10,000$ as strong.) Sunspots can be as large as
$25,000 \mathrm{km}$ in radius. The material in a sunspot has a density of about $3 \times 10^{-4} \mathrm{kg} / \mathrm{m}^{3}$ . Assume $\mu$ for the sunspot material is $\mu_{0}$ . If 100$\%$ of the magnetic-field energy stored in a sunspot could be used to eject the sunspot's material away from the sun's surface, at what speed would that material be ejected? Compare to the sun's escape speed, which is about $6 \times 10^{3} \mathrm{m} / \mathrm{s}$ . (Hint . Calcualte the kinetic energy the magnetic field could supply to 1 $\mathrm{m}^{3}$ of sunspot material.)

Keshav Singh

Numerade Educator

Problem 60

While studying a coil of unknown inductance and internal resistance, you connect it in series with a $25.0-\mathrm{V}$ battery and a $150-\Omega$ resistor. You then place an oscilloscope across one of these circuit elements and use the oscilloscope to measure the voltage across the circuit element as a function of time. The result is shown in Fig. 30.20 . ( a) Across which circuit element (coil or resistor) is the oscilloscope connected? How do you know this? (b) Find the inductance and the intermal resistance of the coil. (c) Carefully make a quantitative sketch showing the voltage versus time you would observe if you put the oscilloscope across the other circuit element (resistor or coil).

Shoukat Ali

Other Schools

Problem 61

In the lab, you are trying to find the inductance and internal resistance of a solenoid. You place it in series with a battery of negligible internal resistance, a $10.0-\Omega$ resistor, and a switch.
You then put an oscilloscope across one of these circuit elements to measure the voltage across that circuit element as a function of time. You close the switch, and the oscilloscope shows voltage versus time as shown in Fig, $30.21 .$ ( a) Across which circuit element (solenoid or resistor) is the oscilloscope connected? How do you know this? (b) Why doesn't the graph approach zero as $t \rightarrow \infty$ ? (c) What is the emf of the battery? (d) Find the maximum current in the circuit. (e) What are the internal resistance and self-inductance of the solenoid?

Shoukat Ali

Other Schools

Problem 62

In the circuit shown in Fig. 30.22 , find the reading in each ammeter and voltmeter (a) just after switch $\mathrm{S}$ is closed and $(\mathrm{b})$ after $\mathrm{S}$ has been closed a very long time.

Shoukat Ali

Other Schools

Problem 63

In the circuit shown in Fig. 30.23 , switch $\mathrm{S}$ is closed at time $t=0$ with no charge initially on the capacitor. (a) Find the reading of each ammeter and each voltmeter just after $\mathrm{S}$ is closed. (b) Find the reading of each meter after a long time has elapsed. (c) Find the maximum charge on the capacitor. (d) Draw a qualitative graph of the reading of voltmeter $V_{2}$ as a function of time.

Ajay Singhal

Numerade Educator

Problem 64

In the circuit shown in Fig. 30.24 the bartery and the inductor have no appreciable internal resistance and there is no current in the circuit. After the switch is closed, find the readings of the ammeter (A) and voltmeters $\left(V_{1} \text { and } V_{2}\right)$ (a) the instant after the switch is closed and (b) after
the switch has been closed for a very long time. (c) Which answers in parts $(a)$ and $(b)$ would change if the inductance were 24.0 $\mathrm{mH}$ instead?

Shoukat Ali

Other Schools

Problem 65

In the circuit shown in Fig. $30.25,$ switch $S$ is closed at time $t=0 .(\mathrm{a})$ Find the reading of each meter just after $\mathrm{S}$ is closed. (b) What does each meter read long after $\mathrm{S}$ is closed?

Keshav Singh

Numerade Educator

Problem 66

In the circuit shown in Fig. $30.26,$ switch $S$ has been closed for a long enough time so that the current reads a steady 3.50 $\mathrm{A}$ . Suddenly, switch $\mathrm{S}_{2}$ is closed and $\mathrm{S}_{1}$ is opened at the same instant. (a) What is the maximum charge that the capacitor will receive? (b) What is the current in the inductor at this time?

Shoukat Ali

Other Schools

Problem 67

In the circuit shown in Fig. $30.27, \quad \mathcal{E}=60.0 \mathrm{V}, R_{1}=$ $40.0 \Omega, R_{2}=25.0 \Omega,$ and $L=$ $0.300 \mathrm{H} .$ Switch $\mathrm{S}$ is closed at $t=0 .$ Just after the switch is closed, (a) what is the potential difference $v_{a b}$ across the resistor $R_{1} ;(b)$ which point, $a$ or $b,$ is at a higher potential; (c) what is the potential difference $v_{c d}$ across the inductor $L ;$ (d) which point, $c$ or $d,$ is at a higher potential? The switch is left closed a long time and then opened. Just after the switch is opened, (e) what is the potential difference $v_{a b}$ across the resistor $R_{1} ;(f)$ which point, $a$ or $b,$ is at a higher potential; (g) what is the potential difference $v_{c d}$ across the inductor $L_{i}$ (h) which point, $c$ or $d,$ is at a higher potential?

Shoukat Ali

Other Schools

Problem 68

In the circuit shown in Fig. $30.27, \quad \varepsilon=60.0 \mathrm{V}, R_{1}=$ $40.0 \Omega, R_{2}=25.0 \Omega,$ and $L=0.300 \mathrm{H} .$ (a) Switch $\mathrm{S}$ is closed. At some time $t$ afterward the current in the inductor is increasing at a rate of $d i / d t=50.0 \mathrm{A} / \mathrm{s}$ . At this instant, what are the current $i_{1}$ through $R_{1}$ and the current $i_{2}$ through $R_{2} ?$ (Hint: Analyze two separate loops: one containing $\mathcal{E}$ and $R_{1}$ and the other containing $\mathcal{E}, R_{2},$ and $L, )$ (b) After the switch has been closed a long time, it is opened again. Just after it is opened, what is the current through $R_{1} ?$

Keshav Singh

Numerade Educator

Problem 69

Consider the circuit shown in Fig. 30.28 . Let $\mathcal{E}=36.0 \mathrm{V}, R_{0}=$ $50.0 \Omega, R=150 \Omega,$ and $L=4.00 \mathrm{H} .$ (a) Switch $S_{1}$ is closed and switch $S_{2}$ is
left open. Just after $S_{1}$ is closed, what are the current $i_{0}$ through $R_{0}$ and the potential differences $v_{a c}$ and $v_{c b}$ ? (b) After $S_{1}$ has been closed a long time $\left(\mathrm{S}_{2} \text { is still open) so that the cur- }\right.$ rent has reached its final, steady value, what are $i_{0}, v_{a c},$ and $v_{c b} ?$ (c) Find the expressions for $i_{0}, v_{\alpha c},$ and $v_{c b}$ as functions of the time $t$ since $S_{1}$ was closed. Your results should agree with part (a) when $t=0$ and with part $(b)$ when $t \rightarrow \infty$ . Graph $i_{0}, v_{a c},$ and $v_{c b}$ versus time.

Shoukat Ali

Other Schools

Problem 70

After the current in the circuit of Fig. 30.28 has reached its final, steady value with switch $S_{1}$ closed and $S_{2}$ open, switch $S_{2}$ is closed, thus short circuiting the inductor. (Switch $S_{1}$ remains closed. See Problem 30.69 for numerical values of the circuit elements. (a) Just after $S_{2}$ is closed, what are $v_{a c}$ and $v_{c b}$ and what are the currents through $R_{0}, R,$ and $S_{2} ?\left(\text { b ) A long time after } S_{2} \text { is }\right.$ closed, what are $v_{a c}$ and $v_{c b}$ and what are the currents through $R_{0}$ . $R,$ and $S_{2} ?(c)$ Derive expressions for the currents through $R_{0}, R,$
and $S_{2}$ as functions of the time $t$ that has elapsed since $S_{2}$ was closed. Your results should agree with part (a) when $t=0$ and with part (b) when $t \rightarrow \infty$ . Graph these three currents versus time.

Shoukat Ali

Other Schools

Problem 71

In the circuit shown in Fig. 30.29 , the switch has been open for a long time and is suddenly closed. Neither the battery nor the inductors have any appreciable resistance. Review the results of
Problem 30.47 . (a) What do the ammeter and voltmeter read just after $S$ is closed? (b) What do the ammeter and the voltmeter read after $S$ has been closed a very long time? (c) What do the ammeter and the voltmeter read 0.115 $\mathrm{ms}$ after $S$ is closed?

Dading Chen

Numerade Educator

Problem 72

In the circuit shown in Fig 30.30 , neither the battery nor the inductors have any appreciable resistance, the capacitors are initially uncharged, and the switch $S$ has been in position 1 for a very long time. Review the results of Problem 30.47 . (a) What is the current in the circuit? (b) The switch is now suddenly fipped to position 2 . Find the maximum charge that each capacitor will receive, and how much time after the switch is flipped it will take them to acquire this charge.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 73

We have ignored the variation of the magnetic field across the cross section of a toroidal solenoid. Let's now examine the validity of that approximation. A certain toroidal solenoxid has a rectangular cross section (Fig. 30.31$)$ . It has $N$ uniformly spaced turns, with air inside. The magnetic field at a point inside the toroid is given by the equation derived in Example 28.11 (Section $28.7 ) .$ Do not assume the field is uniform over the cross section. (a) Show that the magnetic flux through a cross section of the toroid is
$$
\Phi_{B}=\frac{\mu_{0} N i h}{2 \pi} \ln \left(\frac{b}{a}\right)
$$
(b) Show that the inductance of the toroidal solenoid is given by
$$
L=\frac{\mu_{0} N^{2} h}{2 \pi} \ln \left(\frac{b}{a}\right)
$$
(c) The fraction $b / a$ may be written as
$$
\frac{b}{a}=\frac{a+b-a}{a}=1+\frac{b-a}{a}
$$
Use the power series expansion $\ln (1+z)=z+z^{2} / 2+\cdots$ valid for $|z|<1,$ to show that when $b-a$ is much less than $a,$ the inductance is approximately equal to
$$
L=\frac{\mu_{0} N^{2} h(b-a)}{2 \pi a}
$$
Compare this result with the result given in Example $30.3(\text { Section } 30.2) .$

Shoukat Ali

Other Schools

Problem 74

In Fig. 30.32 the switch is closed, with the capacitor having the polarity shown. Find the direction (clockwise or counter-clockwise) of the current induced in the rectangular wire loop $A$ .

Shoukat Ali

Other Schools

Problem 75

Demonstrating Inductance. A common demonstration of inductance employs a circuit such as the one shown in Fig. 30.27 . Switch $S$ is closed, and the light bulb (represented by resistance $R_{1} )$ just barely glows. After a period of time, switch $S$ is opened, and the bulb lights up brightly for a short period of time. To understand this effect, think of an inductor as a device that imparts an "inertia" to the current, preventing a discontinuous change in the current through it. (a) Derive, as explicit functions of time, expressions for $i_{1}$ (the current through the light bulb) and $i_{2}$ (the current through the inductor) after switch $S$ is closed. (b) After a long period of time, the currents $i_{1}$ and $i_{2}$ reach their steady-state values. Obtain expressions for these steady-state currents. (c) Switch $S$ is now opened. Obtain an expression for the current through the inductor and light bulb as an explicit function of time. (d) You have been asked to design a demonstration apparatus using the circuit shown in Fig. 30.27 with a $22.0-\mathrm{H}$ inductor and a $40.0-\mathrm{W}$ light bulb. You are to connect a resistor in series with the inductor, and $R_{2}$ represents the sum of that resistance plus the internal resistance of the inductor. When switch $\mathrm{S}$ is opened, a transient current is to be set up that starts at 0.600 $\mathrm{A}$ and is not to fall below 0.150 $\mathrm{A}$ until after 0.0800 s. For simplicity, assume that the resistance of the light bulb is constant and equals the resistance the bulb must have to dissipate 40.0 $\mathrm{W}$ at 120 $\mathrm{V}$ . Determine $R_{2}$ and $\mathcal{E}$ for the given design considerations. (e) With
the numerical values determined in part (d), what is the current through the light bulb just before the switch is opened? Does this result confirm the qualitative description of what is observed in the demonstration?

Shoukat Ali

Other Schools

Problem 76

Consider the circuit shown in Fig. 30.33 . The circuit elements are as follows: $\mathcal{E}=32.0 \mathrm{V}$ , $L=0.640 \mathrm{H}, C=2.00 \mu \mathrm{F},$ and $R=400 \Omega .$ At time $t=0$ , switch $S$ is closed. The current through the inductor is $i_{1}$ , the current through the capacitor branch is $i_{2},$ and the charge on the capacitor is $q_{2}$ (a) Using Kirchhoff's rules, verify the circuit equations
$$
\begin{array}{c}{R\left(i_{1}+i_{2}\right)+L\left(\frac{d i_{1}}{d t}\right)=\mathcal{E}} \\ {R\left(i_{2}+i_{2}\right)+\frac{q_{2}}{C}=\mathcal{E}}\end{array}
$$
(b) What are the initial values of $i_{1}, i_{2},$ and $q_{2} ?(\mathrm{c})$ Show by direct substitution that the following solutions for $i_{1}$ and $q_{2}$ satisfy the circuit equations from part (a). Also, show that they satisfy the initial conditions
$$
\begin{aligned} i_{1} &=\left(\frac{\mathcal{E}}{R}\right)\left[1-e^{-\beta t}(2 \omega R C)^{-1} \sin (\omega t)+\cos (\omega t)\right] \\ q_{2} &=\left(\frac{\mathcal{E}}{\omega R}\right) e^{-\beta t} \sin (\omega t) \end{aligned}
$$
where $\beta=(2 R C)^{-1}$ and $\omega=\left[(L C)^{-1}-(2 R C)^{-2}\right]^{1 / 2} .$ (d) Determine the time $t_{1}$ at which $i_{2}$ first becomes zero.

Emily Anderson

Numerade Educator

Problem 77

A Volume Gauge. A tank containing a liquid has turns of wire wrapped around it, causing it to act like an inductor. The hiquid content of the tank can be measured by using its inductance to determine the height of the liquid in the tank. The inductance of the tank. The from a value of $L_{0}$ corresponding to a relative permeability of 1 when the tank is empty to a value of $L_{f}$ corresponding to a relative permeability of $K_{m}(\text { the relative permeability of the liquid) }$ when the tank is full. The appropriate electronic circuitry can determine the inductance to five significant figures and thus the
effective relative permeability of the combined air and liquid within the rectangular cavity of the tank. The four sides of the tank each have width $W$ and height $D$ (Fig. 30.34$)$ . The height of the
liquid in the tank is $d$ . You can ignore any fringing effects and assume that the relative permeability of the material of which the tank is made can be ignored. (a) Derive an expression for $d$ as a function of $L$ , the inductance corresponding to a certain fluid height, $L_{0}, L_{f},$ and $D .$ (b) What is the inductance (to five significant figures) for a $\operatorname{tank} \frac{1}{4}$ full, $\frac{1}{2}$ full, $\frac{3}{4}$ full, and completely full if the tank contains liquid oxygen? Take $L_{0}=0.63000 \mathrm{H}$ . The magnetic susceptibility of liquid oxygen is $\chi_{m}=1.52 \times 10^{-3} .$ (c) Repeat part (b) for mercury. The magnetic susceptibility of mercury is given in Table $28.1 .$ (d) For which material is this volume gauge more practical?

Shoukat Ali

Other Schools

Problem 78

Two coils are wrapped around each other as shown in Fig. 30.3 . The current travels in the same sense around each coil. One coil has self-inductance $L_{1},$ and the other coil has self-inductance $L_{2}$ . The mutual inductance of the two coils is $M$ . (a) Show that if the two coils are connected in series, the equivalent inductance of the combination is $L_{\infty}=L_{1}+L_{2}+2 M$ .(b) Show that if the two coils are connected in parallel, the equivalent inductance of the combination is
$$
L_{\mathrm{eq}}=\frac{L_{1} L_{2}-M^{2}}{L_{1}+L_{2}-2 M}
$$

Shoukat Ali

Other Schools

Problem 79

Consider the circuit shown in Fig. $30.35 .$ Switch $S$ is closed at time $t=0,$ causing a current $i_{1}$ through the inductive branch and a current $i_{2}$ through the capacitive branch. The initial charge on the capacitor is zero, and the charge at time $t$ is $q_{2}$ (a) Derive expressions for $i_{1}, i_{2},$ and $q_{2}$ as functions of time. Express your answers in terms of $\mathcal{E}, L, C, R_{1}, R_{2},$ and $t$ . For the remainder of the problem let the circuit elements have the following values: $\mathcal{E}=48 \mathrm{V}, L=8.0 \mathrm{H},$ $C=20 \mu \mathrm{F}, R_{1}=25 \Omega,$ and $R_{2}=5000 \Omega .$ (b) What is the initial current through the inductive branch? What is the initial current through the capacitive branch?(c) What are the currents through the inductive and capacitive branches a long time after the switch has been closed? How long is a "long time"? Explain. (d) At what
time $t_{1}$ (accurate to two significant figures) will the currents $i_{1}$ and $i_{2}$ be equal? (Hint: You might consider using serpansions for the exponentials.) (e) For the conditions given in part (d), determine $i_{1}$ . (f) The total current through the battery is $i=i_{1}+i_{2}$ . At what time $t_{2}$ (accurate to two significant figures) will $i$ equal one-half of its final value? (Hint The numerical work is greatly simplified if one makes suitable approximations. A sketch of $i_{1}$ and $i_{2}$ versus $t$ may help you decide what approximations are valid.)

Lainey Roebuck

Numerade Educator