Problem 1

The current in a coil changes from 3.50 A to 2.00 A in the same direction in 0.500 s. If the average emf induced in the coil is 12.0 mV, what is the inductance of the coil?

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Problem 2

A technician wraps wire around a tube of length 36.0 cm having a diameter of 8.00 cm. When the windings are evenly spread over the full length of the tube, the result is a solenoid containing 580 turns of wire. (a) Find the inductance of this solenoid. (b) If the current in this solenoid increases at the rate of 4.00 A/s, find the self-induced emf in the solenoid.

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Problem 3

A 2.00-H inductor carries a steady current of 0.500 A. When the switch in the circuit is opened, the current is effectively A 2.00-H inductor carries a steady current of 0.500 A. When the switch in the circuit is opened, the current is effectively

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Problem 4

A solenoid of radius 2.50 $\mathrm{cm}$ has 400 turns and a length of $20.0 \mathrm{cm} .$ Find $(\mathrm{a})$ its inductance and $(\mathrm{b})$ the rate at which current must change through it to produce an emf of $75.0 \mu \mathrm{V} .$

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Problem 5

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Problem 6

A 40.0-mA current is carried by a uniformly wound air-core solenoid with 450 turns, a 15.0-mm diameter, and 12.0-cm length. Compute (a) the magnetic field inside A 40.0-mA current is carried by a uniformly wound air-core solenoid with 450 turns, a 15.0-mm diameter, and 12.0-cm length. Compute (a) the magnetic field inside

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Problem 7

A 10.0 -mH inductor carries a current $I=I_{\max } \sin \omega t$ with $I_{\max }=5.00 \mathrm{A}$ and $f=\omega / 2 \pi=60.0 \mathrm{Hz}$ . What is the self-induced emf as a function of time?

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Problem 8

The current in a 4.00 $\mathrm{mH}$ -inductor varies in time as shown in Figure $\mathrm{P} 32.8 .$ Construct a graph of the self-induced emf across the inductor over the time interval $t=0$ to $t=$ $12.0 \mathrm{ms} .$

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Problem 9

The current in a 90.0 -mH inductor changes with time as $I=1.00 t^{2}-6.00 t,$ where $I$ is in amperes and $t$ is in seconds. Find the magnitude of the induced emf at (a) $t=1.00 \mathrm{s}$ and $(\mathrm{b}) t=4.00 \mathrm{s} .$ (c) At what time is the emf zero?

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Problem 10

An inductor in the form of a solenoid contains 420 turns and is 16.0 cm in length. A uniform rate of decrease of current through the inductor of 0.421 $\mathrm{A} / \mathrm{s}$ induces an emf of 175$\mu \mathrm{V}$ . What is the radius of the solenoid?

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Problem 11

A self-induced emf in a solenoid of inductance $L$ changes in time as $\mathcal{E}=\mathcal{E}_{0} e^{-k t} .$ Assuming the charge is finite, find the total charge that passes a point in the wire of the solenoid.

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Problem 12

A toroid has a major radius R and a minor radius r and is tightly wound with N turns of wire on a hollow cardboard torus. Figure P32.12 shows half of this toroid, allowing us to see its cross section. If $R > > r,$ the magnetic field in the region enclosed by the wire is essentially the same as the magnetic field of a solenoid that has been bent into a large circle of radius $R$ . Modeling the field as the uniform field of a long solenoid, show that the inductance of such a toroid is approximately

$$L \approx \frac{1}{2} \mu_{0} N^{2} \frac{r^{2}}{R}$$

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Problem 13

A 510 -turn solenoid has a radius of 8.00 $\mathrm{mm}$ and an over- all length of $14.0 \mathrm{cm} .$ (a) What is its inductance? (b) If the solenoid is connected in series with a $2.50-\Omega$ resistor and a

battery, what is the time constant of the circuit?

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Problem 14

A 12.0-V battery is connected into a series circuit containing a $10.0-\Omega$ resistor and a $2.00-\mathrm{H}$ inductor. In what time interval will the current reach (a) 50.0$\%$ and (b) 90.0$\%$ of its final value?

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Problem 15

A series $R L$ circuit with $L=3.00 \mathrm{H}$ and a series $R C$ circuit with $C=3.00 \mu \mathrm{F}$ have equal time constants. If the two circuits contain the same resistance $R,$ (a) what is the value of $R ?(\text { b) What is the time constant? }$

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Problem 16

In the circuit diagrammed in Figure $\mathrm{P} 32.16,$ take $\mathcal{E}=$ 12.0 $\mathrm{V}$ and $R=24.0 \Omega .$ Assume the switch is open for $t < 0$ and is closed at $t=0 .$ On a single set of axes, sketch graphs of the current in the circuit as a function of time for $t \geq$ $0,$ assuming (a) the inductance in the circuit is essentially zero, (b) the inductance has an intermediate value, and (c) the inductance has a very large value. Label the initial and final values of the current.

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Problem 17

Consider the circuit shown in Figure $\mathrm{P} 32.17$ . (a) When the switch is in position $a,$ for what value of $R$ will the circuit have a time constant of 15.0$\mu \mathrm{s}$ ? (b) What is the current in the inductor at the instant the switch is thrown to position $b$ ?

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Problem 18

Show that $I=I_{i} e^{-t / \tau}$ is a solution of the differential equation

$$

I R+L \frac{d I}{d t}=0

$$

where $I_{i}$ is the current at $t=0$ and $\tau=L / R$

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Problem 19

In the circuit shown in Figure $\mathrm{P} 32.16,$ let $L=7.00 \mathrm{H}, R=$ $9.00 \Omega,$ and $\mathcal{E}=120 \mathrm{V}$ . What is the self-induced emf 0.200 $\mathrm{s}$ after the switch is closed?

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Problem 20

Consider the circuit in Figure $\mathrm{P} 32.16,$ taking $\mathcal{E}=6.00 \mathrm{V},$ $L=8.00 \mathrm{mH},$ and $R=4.00 \Omega .$ (a) What is the inductive time constant of the circuit? (b) Calculate the current in the circuit 250$\mu$ s after the switch is closed. (c) What is the value of the final steady-state current? (d) After what time interval does the current reach 80.0$\%$ of its maximum value?

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Problem 21

The switch in Figure $\mathrm{P} 32.21$ is open for $t<0$ and is then thrown closed at time $t=0 .$ Assume $R=4.00 \Omega, L=n$ $1.00 \mathrm{H},$ and $\mathcal{E}=10.0 \mathrm{V}$ . Find $(\mathrm{a})$ the current in the inductor and (b) the current in the switch as functions of time

thereafter.

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Problem 22

The switch in Figure $\mathrm{P} 32.21$ is open for $t<0$ and is then thrown closed at time $t=0 .$ Find $(a)$ the current in the inductor and $(b)$ the current in the switch as functions of time thereafter.

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Problem 23

For the $R L$ circuit shown in Figure $\mathrm{P} 32.16$ , let the inductance be $3.00 \mathrm{H},$ the resistance $8.00 \Omega,$ and the battery emf 36.0 $\mathrm{V}$ . (a) Calculate $\Delta V_{R} / \varepsilon_{L},$ that is, the ratio of the potential difference across the resistor to the emf across the inductor when the current is 2.00 $\mathrm{A}$ . (b) Calculate the emf across the inductor when the current is 4.50 $\mathrm{A}$ .

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Problem 24

Consider the current pulse $I(t)$ shown in Figure $\mathrm{P} 32.24 \mathrm{a}$ . The current begins at zero, becomes 10.0 $\mathrm{A}$ between $t=0$ and $t=200 \mu \mathrm{s},$ and then is zero once again. This pulse is applied to the input of the partial circuit shown in Figure P32. 24 $\mathrm{b}$ . Determine the current in the inductor as a function of time.

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Problem 25

An inductor that has an inductance of 15.0 $\mathrm{H}$ and a resistance of 30.0$\Omega$ is connected across a $100-\mathrm{V}$ battery. What is the rate of increase of the current (a) at $t=0$ and $(\mathrm{b})$ at $t=$ 1.50 $\mathrm{s}$ ?

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Problem 26

Two ideal inductors, $L_{1}$ and $L_{2},$ have zero internal resistance and are far apart, so their magnetic fields do not influence each other. (a) Assuming these inductors are connected in series, show that they are equivalent to a single ideal inductor having $L_{\mathrm{eq}}=L_{1}+L_{2} .$ (b) Assuming these same two inductors are connected in parallel, show that they are equivalent to a single ideal inductor having $1 / L_{\mathrm{eq}}=1 / L_{1}+1 / L_{2} .$ (c) What If? Now consider two inductors $L_{1}$ and $L_{2}$ that have nonzero internal resistances $R_{1}$ and $R_{2},$ respectively. Assume they are still far apart, so their mutual inductance is zero, and assume they are connected in series. Show that they are equivalent to a single inductor having $L_{\mathrm{eq}}=L_{1}+L_{2}$ and $R_{\mathrm{eq}}=R_{1}+R_{2} .$ (d) If these same inductors are now connected in parallel, is it necessarily true that they are equivalent to a single ideal inductor having $1 / L_{\mathrm{eq}}=1 / L_{1}+1 / L_{2}$ and $1 / R_{\mathrm{eq}}=1 / R_{1}+1 / R_{2} ?$ Explain your answer.

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Problem 27

A $140-\mathrm{mH}$ inductor and a $4.90-\Omega$ resistor are connected with a switch to a $6.00-\mathrm{V}$ battery as shown in Figure $\mathrm{P} 32.27$ (a) After the switch is first thrown to $a$ (connecting the battery), what time interval elapses before the current reaches 220 $\mathrm{mA}$ ? (b) What is the current in the inductor 10.0 $\mathrm{s}$ after the switch is closed? (c) Now the switch is quickly thrown from $a$ to $b .$ What time interval elapses before the current in the inductor falls to 160 $\mathrm{mA}$ ?

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Problem 28

Calculate the energy associated with the magnetic field of a 200 -turn solenoid in which a current of 1.75 $\mathrm{A}$ produces a magnetic flux of $3.70 \times 10^{-4} \mathrm{T} \cdot \mathrm{m}^{2}$ in each turn.

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Problem 29

An air-core solenoid with 68 turns is 8.00 cm long and has a diameter of 1.20 cm. When the solenoid carries a current of 0.770 A, how much energy is stored in its magnetic field?

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Problem 30

A $10.0-\mathrm{V}$ battery, a $5.00-\Omega$ resistor, and a $10.0-\mathrm{H}$ inductor are connected in series. After the current in the circuit has reached its maximum value, calculate (a) the power being supplied by the battery, (b) the power being delivered to the resistor, (c) the power being delivered to the inductor, and (d) the energy stored in the magnetic field of the inductor.

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Problem 31

On a clear day at a certain location, a $100-\mathrm{V} / \mathrm{m}$ vertical electric field exists near the Earth's surface. At the same place, the Earth's magnetic field has a magnitude of $0.500 \times$

$10^{-4}$ T. Compute the energy densities of (a) the electric field and (b) the magnetic field.

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Problem 32

Complete the calculation in Example 32.3 by proving that

$$\int_{0}^{\infty} e^{-2 R t / L} d t=\frac{L}{2 R}$$

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Problem 33

The magnetic field inside a superconducting solenoid is 4.50 T. The solenoid has an inner diameter of 6.20 $\mathrm{cm}$ and a length of $26.0 \mathrm{cm} .$ Determine (a) the magnetic energy density in the field and (b) the energy stored in the magnetic field within the solenoid.

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Problem 34

A flat coil of wire has an inductance of 40.0 $\mathrm{mH}$ and a resistance of 5.00$\Omega$ . It is connected to a $22.0-\mathrm{V}$ battery at the instant $t=0 .$ Consider the moment when the current is 3.00 A. (a) At what rate is energy being delivered by the battery? (b) What is the power being delivered to the resistance of the coil? (c) At what rate is energy being stored in the magnetic field of the coil? (d) What is the relationship among these three power values? (e) Is the relationship described in part (d) true at other instants as well? (f) Explain the relationship at the moment immediately after $t=0$ and at a moment several seconds later.

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Problem 35

Two coils, held in fixed positions, have a mutual inductance of 100$\mu \mathrm{H}$ . What is the peak emf in one coil when the current in the other coil is $I(t)=10.0 \sin \left(1.00 \times 10^{3} t\right),$ where $I$ is in amperes and $t$ is in seconds?

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Problem 36

An emf of 96.0 $\mathrm{mV}$ is induced in the windings of a coil when the current in a nearby coil is increasing at the rate of 1.20 $\mathrm{A} / \mathrm{s}$ . What is the mutual inductance of the two coils?

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Problem 37

An emf of 96.0 $\mathrm{mV}$ is induced in the windings of a coil when the current in a nearby coil is increasing at the rate of 1.20 $\mathrm{A} / \mathrm{s}$ . What is the mutual inductance of the two coils? average flux of 300$\mu$ Wh through each turn of $\mathrm{A}$ and a flux of 90.0 \muWb through each turn of $\mathrm{B}$ . (a) Calculate the mutual inductance of the two solenoids. (b) What is the

inductance of $\mathrm{A}$ ? (c) What emf is induced in $\mathrm{B}$ when the current in $\mathrm{A}$ changes at the rate of 0.500 $\mathrm{A} / \mathrm{s}$ ?

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Problem 38

Two coils are close to each other. The first coil carries a current given by $I(t)=5.00 e^{-0.0250 t} \sin 120 \pi t,$ where $I$ is in amperes and $t$ is in seconds. At $t=0.800 \mathrm{s}$ , the emf measured across the second coil is $-3.20 \mathrm{V}$ . What is the mutual inductance of the coils?

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Problem 39

On a printed circuit board, a relatively long, straight conductor and a conducting rectangular loop lie in the same plane as shown in Figure $\mathrm{P} 32.39$ . Taking $h=0.400 \mathrm{mm}, w=$ $1.30 \mathrm{mm},$ and $\ell=2.70 \mathrm{mm},$ find their mutual inductance.

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Problem 40

Solenoid $S_{1}$ has $N_{1}$ turns, radius $R_{1},$ and length $\ell$ It is so long that its magnetic field is uniform nearly everywhere inside it and is nearly zero outside. Solenoid $\mathrm{S}_{2}$ has $N_{2}$ turns, radius $R_{2}<R_{1},$ and the same length as $\mathrm{S}_{1} .$ It lies inside $\mathrm{S}_{1},$ with their axes parallel. (a) Assume $\mathrm{S}_{1}$ carries variable current $I$ . Compute the mutual inductance characterizing the emf induced in $\mathrm{S}_{2}$ . (b) Now assume $\mathrm{S}_{2}$ carries current $I$ . Compute the mutual inductance to which the emf in $\mathrm{S}_{1}$ is proportional. (c) State how the results of parts (a) and (b) compare with each other.

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Problem 41

Two single-turn circular loops of wire have radii $R$ and $r,$ with $R > > r$ . The loops lie in the same plane and are concentric. (a) Show that the mutual inductance of the pair is approximately $M=\mu_{0} \pi r^{2} / 2 R .$ (b) Evaluate $M$ for $r=$ 2.00 $\mathrm{cm}$ and $R=20.0 \mathrm{cm} .$

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Problem 42

A $1.05-\mu \mathrm{H}$ inductor is connected in series with a variable capacitor in the tuning section of a short wave radio set. What capacitance tunes the circuit to the signal from a transmitter broadcasting at 6.30 $\mathrm{MHz}$ ?

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Problem 43

In the circuit of Figure $\mathrm{P} 32.43$ , the battery emf is 50.0 $\mathrm{V}$ , the resistance is $250 \Omega,$ and the capacitance is 0.500$\mu \mathrm{F}$ . The switch $\mathrm{S}$ is closed for a long time interval, and zero potential difference is measured across the capacitor. After the switch is opened, the potential difference across the capacitor reaches a maximum value of 150 $\mathrm{V}$ . What is the

value of the inductance?

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Problem 44

Calculate the inductance of an $L C$ circuit that oscillates at 120 $\mathrm{Hz}$ when the capacitance is 8.00$\mu \mathrm{F}$ .

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Problem 45

A $1.00-\mu F$ capacitor is charged by a $40.0-\mathrm{V}$ power supply. The fully charged capacitor is then discharged through a $10.0-\mathrm{mH}$ inductor. Find the maximum current in the resulting oscillations.

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Problem 46

Why is the following situation impossible? The $L C$ circuit shown in Figure $\mathrm{CQ} 32.8$ has $L=30.0 \mathrm{mH}$ and $C=50.0 \mu \mathrm{F}$ . The capacitor has an initial charge of 200$\mu \mathrm{C}$ . The switch is closed, and the circuit undergoes undamped $L C$ oscillations. At periodic instants, the energies stored by the capacitor and the inductor are equal, with each of the two components storing 250$\mu \mathrm{J}$ .

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Problem 47

An $L C$ circuit consists of a $20.0-\mathrm{mH}$ inductor and a $0.500-\mu \mathrm{F}$ capacitor. If the maximum instantaneous current is $0.100 \mathrm{A},$ what is the greatest potential difference across the capacitor?

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Problem 48

An $L C$ circuit like that in Figure $\mathrm{CQ}^{32.8}$ consists of a $3.30-\mathrm{H}$ inductor and an $840-\mathrm{pF}$ capacitor that initially carries a $105-\mu \mathrm{C}$ charge. The switch is open for $t<0$ and is then thrown closed at $t=0 .$ Compute the following quantities at $t=2.00 \mathrm{ms} :(\mathrm{a})$ the energy stored in the capacitor, $(\mathrm{b})$ the energy stored in the inductor, and $(\mathrm{c})$ the total energy in the circuit.

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Problem 49

The switch in Figure $\mathrm{P} 32.49$ is connected to position $a$ for a long time interval. At $t=0,$ the switch is thrown to position b. After this time, what are (a) the frequency of oscillation of the $L C$ circuit, $(b)$ the maximum charge that appears on the capacitor, (c) the maximum current in the inductor,

and (d) the total energy the circuit possesses at $t=3.00 \mathrm{s}$ ?

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Problem 50

An $L C$ circuit like the one in Figure $\mathrm{CQ}^{32} .8$ contains an $82.0-\mathrm{mH}$ inductor and a $17.0-\mu \mathrm{F}$ capacitor that initially carries a $180-\mu \mathrm{C}$ charge. The switch i open for $t < 0$ and is then thrown closed at $t=0 .$ (a) Find the frequency (in hertz) of the resulting oscillations. At $t=1.00 \mathrm{ms}$ , find (b) the charge on the capacitor and $(\mathrm{c})$ the current in the circuit.

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Problem 51

In Figure $\mathrm{P} 32.51,$ let $R=7.60 \Omega, L=2.20 \mathrm{mH},$ and $C=$ 1.80$\mu \mathrm{F}$ (a) Calculate the frequency of the damped oscillation of the circuit when the switch is thrown to position $b$ . (b) What is the critical resistance for damped oscillations?

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Problem 52

Show that Equation 32.28 in the text is Kirchhoff’s loop rule as applied to the circuit in Figure $\mathrm{P} 32.51$ with the switch thrown to position $b$ .

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Problem 53

Consider an $L C$ circuit in which $L=500 \mathrm{mH}$ and $C=$ 0.100$\mu \mathrm{F}$ . (a) What is the resonance frequency $\omega_{0} ?$ (b) If a resistance of 1.00 $\mathrm{k} \Omega$ is introduced into this circuit, what is the frequency of the damped oscillations? (c) By what percentage does the frequency of the damped oscillations differ from the resonance frequency?

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Problem 54

Electrical oscillations are initiated in a series circuit containing a capacitance C, inductance L, and resistance $R .$ (a) If $R < < \sqrt{4 L / C}$ (weak damping), what time interval elapses before the amplitude of the current oscillation falls to 50.0$\%$ of its initial value? (b) Over what time interval does the energy decrease to 50.0$\%$ of its initial value?

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Problem 55

A capacitor in a series LC circuit has an initial charge Q and is being discharged. When the charge on the capacitor is $Q / 2,$ find the flux through each of the $N$ turns in the coil of the inductor in terms of $Q, N, L,$ and $C .$

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Problem 56

Review. This problem extends the reasoning of Section 26.4 , Problem 36 in Chapter $26,$ Problem 38 in Chapter $30,$ and Section 32.3 . (a) Consider a capacitor with vacuum between its large, closely spaced, oppositely charged parallel plates. Show that the force on one plate can be accounted for by thinking of the electric field between the plates as exerting a "negative pressure" equal to the energy

density of the electric field. (b) Consider two infinite plane sheets carrying electric currents in opposite directions with equal linear current densities $J_{s}$ . Calculate the force per area acting on one sheet due to the magnetic field, of magnitude $\mu_{0} J_{s} / 2,$ created by the other sheet. (c) Calculate the net magnetic field between the sheets and the field outside of the volume between them. (d) Calculate the energy density in the magnetic field between the sheets. (e) Show that the force on one sheet can be accounted for by thinking of the magnetic field between the sheets as exerting a positive pressure equal to its energy density. This result for magnetic pressure applies to all current configurations, not only to sheets of current.

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Problem 57

A 1.00 -mH inductor and a $1.00-\mu \mathrm{F}$ capacitor are connected in series. The current in the circuit increases linearly in time as $I=20.0 t,$ where $I$ is in amperes and $t$ is in seconds. The capacitor initially has no charge. Determine (a) the voltage across the inductor as a function of time, (b) the voltage across the capacitor as a function of time, and (c) the time when the energy stored in the capacitor first exceeds that in the inductor.

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Problem 58

An inductor having inductance $L$ and a capacitor having capacitance $C$ are connected in series. The current in the circuit increases linearly in time as described by $I=K t,$ where $K$ is a constant. The capacitor is initially uncharged. Determine (a) the voltage across the inductor as a function of time, (b) the voltage across the capacitor as a function of time, and (c) the time when the energy stored in the capacitor first exceeds that in the inductor.

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Problem 59

When the current in the portion of the circuit shown in Figure P32.59 is 2.00 A and increases at a rate of

0.500 $\mathrm{A} / \mathrm{s}$ , the measured voltage is $\Delta V_{a b}=9.00 \mathrm{V}$ . When the current is 2.00 A and decreases at the rate of $0.500 \mathrm{A} / \mathrm{s},$ the measured voltage is $\Delta V_{a b}=5.00 \mathrm{V} .$ Calculate the values of (a) $L$ and (b) $R$

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Problem 60

In the circuit diagrammed in Figure P32.21, assume the switch has been closed for a long time interval and is opened at $t=0 .$ Also assume $R=4.00 \Omega, L=1.00 \mathrm{H},$ and $\mathcal{E}=10.0 \mathrm{V}$ . (a) Before the switch is opened, does the inductor behave as an open circuit, a short circuit, a resistor of some particular resistance, or none of those choices? (b) What current does the inductor carry? (c) How much energy is stored in the inductor for $t < 0$ ? (d) After the switch is opened, what happens to the energy previously stored in the inductor? (e) Sketch a graph of the current in the inductor for $t \geq 0$ . Label the initial and final values and the time constant.

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Problem 61

(a) A flat, circular coil does not actually produce a uniform magnetic field in the area it encloses. Nevertheless, estimate the inductance of a flat, compact, circular coil with radius $R$ and $N$ turns by assuming the field at its center is uniform over its area. (b) A circuit on a laboratory table consists of a 1.50 -volt battery, a $270-\Omega$ resistor, a switch, and three 30.0 -cm-long patch connecting them. Suppose the circuit is arranged to be circular. Think of it as a flat coil with one turn. Compute the order of magnitude of its inductance and $(\mathrm{c})$ of the time constant describing how fast the current increases when you close the switch.

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Problem 62

At the moment $t=0,$ a $24.0-\mathrm{V}$ battery is connected to a $5.00-\mathrm{mH}$ coil and a $6.00-\Omega$ resistor. (a) Immediately thereafter, how does the potential difference across the resistor compare to the emf across the coil? (b) Answer the same question about the circuit several seconds later. (c) Is there an instant at which these two voltages are equal in magnitude? If so, when? Is there more than one such instant? (d) After a $4.00-A$ current is established in the resistor and coil, the battery is suddenly replaced by a short circuit. Answer parts (a), (b), and (c) again with reference to this new circuit.

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Problem 63

A time-varying current $I$ is sent through a $50.0-\mathrm{mH}$ inductor from a source as shown in Figure $\mathrm{P} 32.63 \mathrm{a}$ . The current is constant at $I=-1.00 \mathrm{mA}$ until $t=0$ and then varies with time afterward as shown in Figure $\mathrm{P} 32.63 \mathrm{b}$ . Make a graph of the emf across the inductor as a function of time.

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Problem 64

Why is the following situation impossible? You are working on an experiment involving a series circuit consisting of a charged $500-\mu \mathrm{F}$ capacitor, a $32.0-\mathrm{mH}$ inductor, and a resistor $R$ . You discharge the capacitor through the inductor and resistor and observe the decaying oscillations of the current that are perfect for your needs.

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Problem 65

A wire of nonmagnetic material, with radius R, carries current uniformly distributed over its cross section. The total current carried by the wire is I. Show that the magnetic energy per unit length inside the wire is $\mu_{0} I^{2} / 16 \pi$

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Problem 66

At $t=0$ , the open switch in Figure $\mathrm{P} 32.66$ is thrown closed. We wish to find a symbolic expression for the current in the inductor for time $t>0 .$ Let this current be called $I$ and choose it to be downward in the inductor in Figure $\mathrm{P} 32.66$ . Identify $I_{1}$ as the current to the right through $R_{1}$ and $I_{2}$ as the current downward through $R_{2}$ . (a) Use Kirchhoff's junction rule to find a relation among the three currents. (b) Use Kirchhoff's loop rule around the left loop to find another relationship. (c) Use Kirchhoff's loop rule around the outer loop to find a third relationship. (d) Eliminate $I_{\text { and }} I_{2}$ among the three equations to find an equation involving only the current $I$ . (e) Compare the equation in part (d) with Equation 32.6 in the text. Use this comparison to rewrite Equation 32.7 in the text for the situation in this problem and show that

$$

\begin{array}{c}{I(t)=\frac{\varepsilon}{R_{1}}\left[1-e^{-(R / I) t ]}\right.} \\ {\text { where } R^{\prime}=R_{1} R_{2} /\left(R_{1}+R_{2}\right)}\end{array}

$$

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Problem 67

The toroid in Figure $\mathrm{P} 32.67$ consists of $N$ turns and has a rectangular cross section. Its inner and outer radii are a and $b$ , respectively. The figure shows half of the toroid to allow us to see its cross-section. Compute the inductance of a 500 -turn toroid for which $a=10.0 \mathrm{cm}, b=12.0 \mathrm{cm},$ and $h=1.00 \mathrm{cm} .$

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Problem 68

The toroid in Figure P32.67 consists of N turns and has a rectangular cross section. Its inner and outer radii are a and b, respectively. Find the inductance of the toroid.

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Problem 69

Problems 69 through 72 apply ideas from this and earlier chapters to some properties of superconductors, which were introduced in Section 27.5.

Review. A novel method of storing energy has been proposed. A huge underground superconducting coil, 1.00 $\mathrm{km}$ in diameter, would be fabricated. It would carry a maximum current of 50.0 $\mathrm{kA}$ through each winding of a $150-$ turn $\mathrm{Nb}_{3} \mathrm{Sn}$ solenoid. (a) If the inductance of this huge coil were $50.0 \mathrm{H},$ what would be the total energy stored? (b) What would be the compressive force per unit length acting between two adjacent windings 0.250 $\mathrm{m}$ apart?

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Problem 70

Problems 69 through 72 apply ideas from this and earlier chapters to some properties of superconductors, which were introduced in Section 27.5.

Review. In an experiment carried out by S. C. Collins between 1955 and 1958 , a current was maintained in a superconducting lead ring for 2.50 yr with no observed loss, even though there was no energy input. If the inductance of the ring were $3.14 \times 10^{-8} \mathrm{H}$ and the sensitivity of the experiment were 1 part in $10^{9},$ what was the maximum resistance of the ring? Suggestion: Treat the ring as an $R L$ circuit carrying decaying current and recall that the approximation $e^{-x} \approx 1-x$ is valid for small $x$ .

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Problem 71

Problems 69 through 72 apply ideas from this and earlier chapters to some properties of superconductors, which were introduced in Section 27.5.

Review. The use of superconductors has been proposed for power transmission lines. A single coaxial cable (Fig. P32.71) coakial cable (Fig. P32.71) could carry a power of $1.00 \times 10^{3} \mathrm{MW}$ (the output of a large power a distance of $1.00 \times 10^{3} \mathrm{km}$ without loss. An inner wire of radius $a=$ $2.00 \mathrm{cm},$ made from the superconductor $\mathrm{Nb}_{3} \mathrm{Sn}$ , carries the current $I$ in one direction. A surrounding superconducting cylinder of radius $b=5.00 \mathrm{cm}$ would carry the return current $I .$ In such a system, what is the magnetic field (a) at the surface of the inner conductor and (b) at the inner surface of the outer conductor? (c) How much energy would be stored in the magnetic field in the space between the conductors in a $1.00 \times 10^{3} \mathrm{km}$ superconducting line? (d) What is the pressure exerted on the outer conductor due to the current in the inner conductor?

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Problem 72

Problems 69 through 72 apply ideas from this and earlier chapters to some properties of superconductors, which were introduced in Section 27.5.

Review. A fundamental property of a type I superconducting material is perfect diamagnetism, or demonstration of the Meissner effect, illustrated in Figure 30.27 in Section 30.6 and described as follows. If a sample of superconducting material is placed into an externally produced magnetic field or is cooled to become superconducting while it in a magnetic field, electric currents appear on the surface of the sample. The currents have precisely the strength and orientation required to make the total magnetic field be

zero throughout the interior of the sample. This problem will help you understand the magnetic force that can then act on the sample. Compare this problem with Problem 63 in Chapter 26 , pertaining to the force attracting a perfect dielectric into a strong electric field. A vertical solenoid with a length of 120 $\mathrm{cm}$ and a diameter of 2.50 $\mathrm{cm}$ consists of 1400 turns of copper wire carrying a counterclockwise current (when viewed from above) of 2.00 $\mathrm{A}$ as shown in Figure $\mathrm{P} 32.72 \mathrm{a}$ . (a) Find the magnetic field in the vacuum inside the solenoid. (b) Find the

energy density of the magnetic field. Now a superconducting bar 2.20 $\mathrm{cm}$ in diameter is inserted partway into the solenoid. Its upper end is far outside the solenoid, where the magnetic field is negligible. The lower end of the bar is deep inside the solenoid. (c) Explain how you identify the direction required for the current on the curved surface of the bar so that the total magnetic field is zero within the bar. The field created by the supercurrents is sketched in Figure $\mathrm{P} 32.72 \mathrm{b}$ , and the total field is sketched in Figure $\mathrm{P} 32.72 \mathrm{c} .$ (d) The field of the solenoid exerts a force on the current in the superconductor. Explain how you determine the direction of the force on the bar. (e) Noting that the units $J / \mathrm{m}^{3}$ of energy density are the same as the units $\mathrm{N} / \mathrm{m}^{2}$ of pressure, calculate the magnitude of the force by multiplying the energy density of the solenoid field times the area of the bottom end of the superconducting bar.

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Problem 73

Assume the magnitude of the magnetic field outside a sphere of radius $R$ is $B=B_{0}(R / r)^{2},$ where $B_{0}$ is a constant. (a) Determine the total energy stored in the magnetic field outside the sphere. (b) Evaluate your result from part (a) for $B_{0}=5.00 \times 10^{-5} \mathrm{T}$ and $R=6.00 \times 10^{6} \mathrm{m},$ values appropriate for the Earth's magnetic field.

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Problem 74

In earlier times when many households received nondigital television signals from an antenna, the lead-in wires from the antenna were often constructed in the form of two parallel wires (Fig. P32.74). The two wires carry currents of equal magnitude in opposite directions. The center-to-center separation of the wires is w, and a is their radius. Assume w is large enough compared with a that the wires carry the current uniformly distributed over their surfaces and negligible magnetic field exists inside the wires. (a) Why does this configuration of conductors have an inductance? (b) What constitutes the flux loop for this

configuration? (c) Show that the inductance of a length x of this type of lead-in is

$$

L=\frac{\mu_{0} x}{\pi} \ln \left(\frac{w-a}{a}\right)

$$

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Problem 75

Two inductors having inductances $L_{1}$ and $L_{2}$ are connected in parallel as shown in Figure $\mathrm{P} 32.75 \mathrm{a}$ . The mutual inductance between the two inductors is $M .$ Determine the inductance between the two inductors is $M .$ Determine the equivalent inductance $L_{\mathrm{cq}}$ for the system (Fig. P32.75b).

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Problem 76

In Figure $\mathrm{P} 32.76,$ the battery has emf $\mathcal{E}=18.0 \mathrm{V}$ and the other circuit elements have values $L=0.400 \mathrm{H}, R_{1}=$ $2.00 \mathrm{k} \Omega,$ and $R_{2}=6.00 \mathrm{k} \Omega .$ The switch is closed for $t<0,$ and steady-state conditions are established. The switch is then opened at $t=0 .$ (a) Find the emf across $L$ immediately after $t=0 .$ (b) Which end of the coil, $a$ or $b,$ is at the higher potential? (c) Make graphs of the currents in $R_{1}$ and in $R_{2}$ as a function of time, treating the steady-state directions as positive. Show values before and after $t=0$ .

(d) At what moment after $t=0$ does the current in $R_{2}$ have the value 2.00 $\mathrm{mA}$ ?

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Problem 77

To prevent damage from arcing in an electric motor, a discharge resistor is sometimes placed in parallel with the armature. If the motor is suddenly unplugged while running, this resistor limits the voltage that appears across the armature coils. Consider a $12.0-\mathrm{V}$ DC motor with an armature that has a resistance of 7.50$\Omega$ and an inductance of 450 $\mathrm{mH}$ . Assume the magnitude of the self-induced emf in the armature coils is 10.0 $\mathrm{V}$ when the motor is running at normal speed. (The equivalent circuit for the armature is shown in Fig. P32.77.) Calculate the maximum resistance $R$

that limits the voltage across the armature to 80.0 $\mathrm{V}$ when the motor is unplugged.

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