Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 31

Inductance - all with Video Answers

Educators

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Chapter Questions

Problem 1

A $2.00-\mathrm{H}$ inductor carries a steady current of $0.500 \mathrm{~A}$. When the switch in the circuit is opened, the current is effectively zero after $10.0 \mathrm{~ms}$. What is the average induced emf in the inductor during this time interval?

Keshav Singh

Keshav Singh

Numerade Educator

Problem 2

A coiled telephone cord forms a spiral with 70.0 turns, a diameter of $1.30 \mathrm{~cm},$ and an unstretched length of $60.0 \mathrm{~cm} .$ Determine the inductance of one conductor in the unstretched cord.

Keshav Singh

Numerade Educator

Problem 3

An emf of $24.0 \mathrm{mV}$ is induced in a 500 -turn coil when the current is changing at the rate of $10.0 \mathrm{~A} / \mathrm{s}$. What is the magnetic flux through each turn of the coil at an instant when the current is $4.00 \mathrm{~A}$ ?

Zachary Warner

Numerade Educator

Problem 4

A 40.0 -mA current is carried by a uniformly wound air-core solenoid with 450 turns, a $15.0-\mathrm{mm}$ diameter, and $12.0-\mathrm{cm}$ length. Compute (a) the magnetic field inside the solenoid, (b) the magnetic flux through each turn, and (c) the inductance of the solenoid. (d) What If? If the current were different, which of these quantities would change?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 5

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

Eduard Sanchez

Numerade Educator

Problem 6

A toroid has a major radius $R$ and a minor radius rand is tightly wound with $N$ turns of wire on a hollow cardboard torus. Figure $\mathrm{P} 31.6$ shows half of this toroid, allowing us to see its cross section. If $R \gg r,$ 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 $$L \approx \frac{1}{2} \mu_{0} N^{2} \frac{r^{2}}{R}$$

Eduard Sanchez

Numerade Educator

Problem 7

A $10.0-\mathrm{mH}$ 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-in-
duced emf as a function of time?

Eduard Sanchez

Numerade Educator

Problem 8

The current in a $4.00 \mathrm{mH}$ -inductor varies in time as shown
in Figure $\mathrm{P} 31.8 .$ Construct a graph of the self-induced emf across the inductor over the time interval $t=0$ to $t=12.0 \mathrm{~ms}$

Eduard Sanchez

Numerade Educator

Problem 9

You are working as an electrical technician. One day, out in the field, you need an inductor but cannot find one. Looking in your wire supply cabinet, you find a cardboard tube with single-conductor wire wrapped uniformly around it to form a solenoid. You carefully count the turns of wire and find that there are 580 turns. The diameter of the tube is $8.00 \mathrm{~cm},$ and the length of the wire-wrapped portion is $36.0 \mathrm{~cm}$. You pull out your calculator to determine (a) the inductance of the coil and
(b) the emf generated in it if the current in the wire increases at the rate of $4.00 \mathrm{~A} / \mathrm{s}$

Eduard Sanchez

Numerade Educator

Problem 10

A 510 -turn solenoid has a radius of $8.00 \mathrm{~mm}$ and an overall 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?

Eduard Sanchez

Numerade Educator

Problem 11

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 $R ?$ (b) What is the time constant?

Keshav Singh

Numerade Educator

Problem 12

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 $\tau=L / R$

Keshav Singh

Numerade Educator

Problem 13

A circuit consists of a coil, a switch, and a battery, all in series. The internal resistance of the battery is negligible compared with that of the coil. The switch is originally open. It is thrown closed, and after a time interval $\Delta t,$ the current in the circuit reaches $80.0 \%$ of its final value. The switch then remains closed for a time interval much longer than $\Delta t$. The wires connected to the terminals of the bat-
tery are then short-circuited with another wire and removed from the battery, so that the current is uninterrupted. (a) At an instant that is a time interval $\Delta t$ after the short circuit, the current is what percentage of its maximum value? (b) At the moment $2 \Delta t$ after the coil is short-circuited, the current in the coil is what percentage of its maximum value?

Eduard Sanchez

Numerade Educator

Problem 14

You are working as a demonstration assistant for a physics professor. He shows you the circuit in Figure $\mathrm{P} 31.14$, which he wants you to build for an upcoming class. The lightbulb is a household incandescent bulb that receives energy at the rate of $40.0 \mathrm{~W}$ when operating at $120 \mathrm{~V}$. It has a resistance $R_{1},$ which, for simplicity, we will assume is constant at all operating voltages. The battery in the circuit has an emf of $12.0 \mathrm{~V}$. When the switch has been closed for a long time, the bulb glows dimly, since it is powered by only $12.0 \mathrm{~V}$. When the switch is opened, however, the bulb flashes brightly and then gradually dims to darkness. Your professor want you to determine two values: (a) the resistance $R_{2}$ that is necessary for the bulb to initially flash, when the switch is opened, at the same brightness it would have if plugged into a $120-\mathrm{V}$ socket; (b) the inductance $L$ necessary to keep the current in the lightbulb above $50.0 \%$ of its value when the switch is opened, for a time interval of $2.00 \mathrm{~s}$ after it is opened. Assume a resistance-free inductor and that the resistance of the lightbulb does not vary with temperature.

Victor Salazar

Numerade Educator

Problem 15

The switch in Figure $\mathrm{P} 31.15$ is open for $t<0$ and is then thrown closed at time $t=0 .$ Assume $R=4.00 \Omega, L=1.00 \mathrm{H}$ and $\mathcal{E}=10.0 \mathrm{~V}$. Find (a) the current in the inductor and (b) the current in the switch as functions of time thereafter.

Eduard Sanchez

Numerade Educator

Problem 16

The switch in Figure $\mathrm{P} 31.15$ 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.

Vishal Gupta

Numerade Educator

Problem 17

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
(b) at $t=1.50 \mathrm{~s}$ ?

Keshav Singh

Numerade Educator

Problem 18

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 $L_{\mathrm{eq}}=L_{1}+I_{2}$. (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 $L_{e q}=L_{1}+I_{2}$ and $R_{\mathrm{eq}}=R_{1}+R_{\mathrm{q}^{*}}(\mathrm{~d})$ 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 $1 / R_{\mathrm{eq}}=1 / R_{1}+1 / R_{2}^{2}$ Explain your answer.

Victor Salazar

Numerade Educator

Problem 19

Consider the current pulse $i(t)$ shown in Figure $\mathrm{P} 31.19 \mathrm{a}$. The current begins at zero, becomes 10.0 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 $\mathrm{P} 31.19 \mathrm{~b}$. Determine the current in the inductor as a function of time.

Victor Salazar

Numerade Educator

Problem 20

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

Keshav Singh

Numerade Educator

Problem 21

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

Keshav Singh

Numerade Educator

Problem 22

Complete the calculation in Example 31.3 by proving that
$$\int_{0}^{\infty} e^{-2 R t / L} d t=\frac{L}{2 R}$$

Keshav Singh

Numerade Educator

Problem 23

A $24.0-\mathrm{V}$ battery is connected in series with a resistor and an inductor, with $R=8.00 \Omega$ and $L=4.00 \mathrm{H},$ respectively. Find the energy stored in the inductor (a) when the current reaches its maximum value and (b) at an instant that is a time interval of one time constant after the switch is closed.

Penny Riley

Numerade Educator

Problem 24

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 $3.00 \mathrm{~A}$. (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.

Victor Salazar

Numerade Educator

Problem 25

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?

Keshav Singh

Numerade Educator

Problem 26

Two solenoids $\mathrm{A}$ and $\mathrm{B},$ spaced close to each other and sharing the same cylindrical axis, have 400 and 700 turns, respectively. A current of $3.50 \mathrm{~A}$ in solenoid A produces an average flux of $300 \mu \mathrm{Wb}$ through each turn of $\mathrm{A}$ and a flux of $90.0 \mu \mathrm{Wb}$ through each turn of B. (a) Calculate the mutual inductance of the two solenoids. (b) What is the inductance of A? (c) What emf is induced in B when the current in A changes at the rate of $0.500 \mathrm{~A} / \mathrm{s}$ ?

Khaled Yasein

Numerade Educator

Problem 27

Solenoid $\mathrm{S}_{1}$ 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 $S_{1} .$ It lies inside $S_{1},$ 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 $S_{1}$ is proportional. (c) State how the results of parts (a) and
(b) compare with each other.

Keshav Singh

Numerade Educator

Problem 28

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 $R=20.0 \mathrm{~cm}$

Keshav Singh

Numerade Educator

Problem 29

The battery $\mathrm{emf}$ 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?

Narayan Hari

Numerade Educator

Problem 30

The $L C$ circuit shown in Figure $\mathrm{P} 31.30$ 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 $L$. coscillations. At periodic instants, the energies stored by the capacitor and the inductor are equal, with each of the two components storing $250 \mu \mathrm{J}$

Sanat Mukherjee

Sanat Mukherjee

Numerade Educator

Problem 31

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 in the circuit is $0.100 \mathrm{~A},$ what is the greatest potential difference across the capacitor?

Keshav Singh

Numerade Educator

Problem 32

An $L C$ circuit like that in Figure $\mathrm{P} 31.30$ 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 $t=2.00 \mathrm{~ms}$
(a) the energy stored in the capacitor, (b) the energy stored in the inductor, and (c) the total energy in the circuit.

Keshav Singh

Numerade Educator

Problem 33

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?

Keshav Singh

Numerade Educator

Problem 34

Show that Equation 31.24 in the text is Kirchhoff's loop rule as applied to the circuit in Figure $31.15 \mathrm{~b}$.

Victor Salazar

Numerade Educator

Problem 35

Electrical oscillations are initiated in a series circuit con-
taining 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?

Penny Riley

Numerade Educator

Problem 36

Consider a capacitor with vacuum between its large, closely spaced, oppositely charged parallel plates. (a) 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.

Victor Salazar

Numerade Educator

Problem 37

A capacitor in a series $L C$ 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 $C$

Narayan Hari

Numerade Educator

Problem 38

In the circuit diagrammed in Figure $\mathrm{P} 31.15$, 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.

Sam Stansfield

Numerade Educator

Problem 39

(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 $30.0-\mathrm{cm}$ -long patch cords 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
(c) of the time constant describing how fast the current increases when you close the switch.

Victor Salazar

Numerade Educator

Problem 40

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.

Keshav Singh

Numerade Educator

Problem 41

You are working on an $L C$ circuit for an experiment you are performing in your basement. You have an appropriate capacitor, but you need to build your own inductor. You wish to cut a wooden ring with a rectangular cross section, as shown in Figure $\mathrm{P} 31.41$, from wood with thickness $h=1.00 \mathrm{~cm}$. You want to wrap 500 turns of wire around it to form a toroidal inductor. For your experiment, you need to have $1.82 \times 10^{-4} \mathrm{~J}$ of energy stored in the inductor when it carries a current of 2.00 A. In order to cut the appropriate wooden ring, you need to determine the ratio $b / a .$ Ignore any effect of the wood core on the magnetic field.

Linda Winkler

Numerade Educator

Problem 42

You are working on an $L C$ circuit for an experiment you are performing in your basement. You have an appropriate capacitor, but you need to build your own inductor. You wish to cut a wooden ring with a rectangular cross section, as shown in Figure $\mathrm{P} 31.41$, from wood with thickness $h$. You want to wrap $N$ turns of wire around it to form a toroidal inductor. For your experiment, you need to have energy $U_{B}$ stored in the inductor when it carries a current i. In order to cut the appropriate wooden ring, you need to determine the ratio $b / a$. Ignore any effect of the wood core on the magnetic field.

Victor Salazar

Numerade Educator

Problem 43

You are trying out to represent your campus in the Physics Olympics. You have just been given a problem involving the circuit shown in Figure $\mathrm{P} 31.43 .$ The values of the circuit elements are $\mathcal{E}=12.0 \mathrm{~V}, R=10.0 \Omega, C=5.00 \mu \mathrm{F},$ and
$L=2.00 \mathrm{mH} .$ The inductor is resistance-free and the capacitor begins with zero charge. Switch $\mathrm{S}$ has been set to position $a$ for a long time. At $t=0,$ switch $\mathrm{S}$ is thrown to position
b. What is the charge on the capacitor at $t=1.00 \mathrm{~s}$ ? To qualify for the team, you must be the first contestant to determine the answer! Ready? Go!

Sam Stansfield

Numerade Educator

Problem 44

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 in the circuit. When the resistance $R$ is $8.00 \Omega,$ the decay in the oscillations is too slow for your experimental design. To make the decay faster, you double the resistance. As a result, you generate decaying oscillations of the current that are perfect for your needs.

Penny Riley

Numerade Educator

Problem 45

A time-varying current $i$ is sent through a $50.0-\mathrm{mH}$ inductor from a source as shown in Figure $\mathrm{P} 31.45 \mathrm{a}$. The current is constant at $i=-1.00 \mathrm{~mA}$ until $t=0$ and then varies with time afterward as shown in Figure $\mathrm{P} 31.45 \mathrm{~b}$. Make a graph of the emf across the inductor as a function of time.

Victor Salazar

Numerade Educator

Problem 46

At $t=0,$ the open switch in Figure $\operatorname{P} 31.46$ 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 $\mathrm{P} 31.46$. 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 $i_{1}$ 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 31.6 in the text. Use this comparison to rewrite Equation 31.7 in the text for the situation in this problem and show that $$\begin{array}{c}i(t)=\frac{\varepsilon}{R_{1}}\left[1-e^{-\left(R^{\prime} / L\right) t}\right] \\\text { where } R^{\prime}=R_{1} R_{2} /\left(R_{1}+R_{2}\right)\end{array}$$

Victor Salazar

Numerade Educator

Problem 47

The use of superconductors has been proposed for power transmission lines. A single coaxial cable (Fig. $\mathrm{P} 31.47$ ) could carry a power of $1.00 \times 10^{3} \mathrm{MW}$ (the output of a large power plant) at $200 \mathrm{kV}, \mathrm{DC},$ over 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 \begin{tabular}{l} direction. A surrounding superconducting cylinder of \\ \hline \end{tabular} 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?

Victor Salazar

Numerade Educator

Problem 48

A fundamental property of a type I superconducting material is perfect diamagnetism, or demonstration of the Meissner effect, illustrated in Figure 29.27 in Section 29.6 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 is 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 39 in Chapter 25 , 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 A as shown in Figure $\mathrm{P} 31.48$ 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 $\mathrm{P} 31.48 \mathrm{~b},$ and the total field is sketched in Figure $\mathrm{P} 31.48 \mathrm{c} .$ (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 $\mathrm{J} / \mathrm{m}^{3}$ of energy density are the 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.

Victor Salazar

Numerade Educator

Problem 49

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 $\mu_{0} I^{2} / 16 \pi$

Keshav Singh

Numerade Educator

Problem 50

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. $\mathrm{P} 31.50$ ). 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
$$L=\frac{\mu_{0} x}{\pi} \ln \left(\frac{w-a}{a}\right)$$

Victor Salazar

Numerade Educator

Problem 51

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 appropri-
ate for the Earth's magnetic field.

Penny Riley

Numerade Educator

Problem 52

The battery has emf $\varepsilon=18.0 \mathrm{~V}$ 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 $2.00 \mathrm{~mA} ?$

Victor Salazar

Numerade Educator

Problem 53

Two inductors having inductances $L_{1}$ and $L_{2}$ are connected in parallel as shown in Figure $\mathrm{P} 31.53 \mathrm{a}$. The mutual inductance between the two inductors is $M .$ Determine the equivalent inductance $L_{e q}$ for the system.

Victor Salazar

Numerade Educator