Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 31

Faraday’s Law - all with Video Answers

Educators

CB

GA

Chapter Questions

Problem 1

A 50 -turn rectangular coil of dimensions $5.00 \mathrm{~cm} \times$ $10.0 \mathrm{~cm}$ is allowed to fall from a position where $B=0$ to a new position where $B=0.500 \mathrm{~T}$ and is directed perpendicular to the plane of the coil. Calculate the magnitude of the average emf induced in the coil if the displacement occurs in $0.250 \mathrm{~s}$.

Vishal Gupta

Vishal Gupta

Numerade Educator

Problem 2

A flat loop of wire consisting of a single turn of crosssectional area $8.00 \mathrm{~cm}^{2}$ is perpendicular to a magnetic field that increases uniformly in magnitude from $0.500 \mathrm{~T}$ to $2.50 \mathrm{~T}$ in $1.00 \mathrm{~s}$. What is the resulting induced current if the loop has a resistance of $2.00 \Omega ?$

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 3

A 25-turn circular coil of wire has a diameter of $1.00 \mathrm{~m}$. It is placed with its axis along the direction of the Earth's magnetic field of $50.0 \mu \mathrm{T}$, and then in $0.200 \mathrm{~s}$ it is flipped $180^{\circ}$. An average emf of what magnitude is generated in the coil?

Vishal Gupta

Numerade Educator

Problem 4

A rectangular loop of area $A$ is placed in a region where the magnetic field is perpendicular to the plane of the loop. The magnitude of the field is allowed to vary in time according to the expression $B=B_{\max } e^{-t / \tau}$, where $B_{\max }$ and $\tau$ are constants. The field has the constant value $B_{\max }$ for $t<0$. (a) Use Faraday's law to show that the emf induced in the loop is given by
$$\boldsymbol{\varepsilon}=\left(A B_{\max } / \tau\right) e^{-t / \tau}$$
(b) Obtain a numerical value for $\mathcal{E}$ at $t=4.00 \mathrm{~s}$ when
$A=0.160 \mathrm{~m}^{2}, B_{\max }=0.350 \mathrm{~T}$, and $\tau=2.00 \mathrm{~s}$. (c) For
the values of $A, B_{\max }$, and $\tau$ given in part (b), what is the maximum value of $\mathcal{E}$ ?

Vishal Gupta

Numerade Educator

Problem 5

A strong electromagnet produces a uniform field of $1.60 \mathrm{~T}$ over a cross-sectional area of $0.200 \mathrm{~m}^{2}$. A coil having 200 turns and a total resistance of $20.0 \Omega$ is placed around the electromagnet. The current in the electromagnet is then smoothly decreased until it reaches zero in $20.0 \mathrm{~ms}$. What is the current induced in the coil?

Vishal Gupta

Numerade Educator

Problem 6

A magnetic field of $0.200$ T exists within a solenoid of 500 turns and a diameter of $10.0 \mathrm{~cm}$. How rapidly (that is, within what period of time) must the field be reduced to zero if the average induced emf within the coil during this time interval is to be $10.0 \mathrm{kV} ?$

Vishal Gupta

Numerade Educator

Problem 7

An aluminum ring with a radius of $5.00 \mathrm{~cm}$ and a resistance of $3.00 \times 10^{-4} \Omega$ is placed on top of a long aircore solenoid with 1000 turns per meter and a radius of $3.00 \mathrm{~cm}$, as shown in Figure $\mathrm{P} 31.7$. Assume that the axial component of the field produced by the solenoid over the area of the end of the solenoid is one-half as strong as at the center of the solenoid. Assume that the solenoid produces negligible field outside its crosssectional area. (a) If the current in the solenoid is increasing at a rate of $270 \mathrm{~A} / \mathrm{s}$, what is the induced current in the ring? (b) At the center of the ring, what is the magnetic field produced by the induced current in the ring? (c) What is the direction of this field?

Vishal Gupta

Numerade Educator

Problem 8

An aluminum ring of radius $r_{1}$ and resistance $R$ is placed on top of a long air-core solenoid with $n$ turns per meter and smaller radius $r_{2}$, as shown in Figure P31.7. Assume that the axial component of the field produced by the solenoid over the area of the end of the solenoid is one-half as strong as at the center of the solenoid. Assume that the solenoid produces negligible field outside its cross-sectional area. (a) If the current in the solenoid is increasing at a rate of $\Delta I / \Delta t$, what is the induced current in the ring? (b) At the center of the ring, what is the magnetic field produced by the induced current in the ring? (c) What is the direction of this field?

Vishal Gupta

Numerade Educator

Problem 9

A loop of wire in the shape of a rectangle of width $w$ and length $L$ and a long, straight wire carrying a current $I$ lie on a tabletop as shown in Figure $\mathrm{P} 31.9 .$
(a) Determine the magnetic flux through the loop due to the current $I$. (b) Suppose that the current is changing with time according to $I=a+b t$, where $a$ and $b$ are constants. Determine the induced emf in the loop if $b=10.0 \mathrm{~A} / \mathrm{s}, h=1.00 \mathrm{~cm}, w=10.0 \mathrm{~cm}$, and $L=$ $100 \mathrm{~cm}$. What is the direction of the induced current in the rectangle?

Vishal Gupta

Numerade Educator

Problem 10

A coil of 15 turns and radius $10.0 \mathrm{~cm}$ surrounds a long solenoid of radius $2.00 \mathrm{~cm}$ and $1.00 \times 10^{3}$ turns per meter (Fig. $\mathrm{P} 31.10$ ). If the current in the solenoid changes as $I=(5.00 \mathrm{~A}) \sin (120 t)$, find the induced emf in the 15 -turn coil as a function of time.

Vishal Gupta

Numerade Educator

Problem 11

Find the current through section $P Q$ of length $a=$ $65.0 \mathrm{~cm}$ shown in Figure $\mathrm{P} 31.11$. The circuit is located in a magnetic field whose magnitude varies with time according to the expression $B=\left(1.00 \times 10^{-3} \mathrm{~T} / \mathrm{s}\right) t$. Assume that the resistance per length of the wire is $0.100 \Omega / \mathrm{m}$.

CB

Casey Bogh

Numerade Educator

Problem 12

A 30-turn circular coil of radius $4.00 \mathrm{~cm}$ and resistance $1.00 \Omega$ is placed in a magnetic field directed perpendicular to the plane of the coil. The magnitude of the magnetic field varies in time according to the expression $B=0.0100 t+0.0400 t^{2}$, where $t$ is in seconds and $B$ is in tesla. Calculate the induced emf in the coil at $t=5.00 \mathrm{~s} .$

Vishal Gupta

Numerade Educator

Problem 13

A long solenoid has 400 turns per meter and carries a current $I=(30.0 \mathrm{~A})\left(1-e^{-1.60 t}\right) .$ Inside the solenoid and coaxial with it is a coil that has a radius of $6.00 \mathrm{~cm}$ and consists of a total of 250 turns of fine wire (Fig. P31.13). What emf is induced in the coil by the changing current?

Vishal Gupta

Numerade Educator

Problem 14

A long solenoid has $n$ turns per meter and carries a current $I=I_{\max }\left(1-e^{-\alpha t}\right)$. Inside the solenoid and coaxial with it is a coil that has a radius $R$ and consists of a total of $N$ turns of fine wire (see Fig. P31.13). What emf is induced in the coil by the changing current?

Vishal Gupta

Numerade Educator

Problem 15

A coil formed by wrapping 50 turns of wire in the shape $\quad$ 19. $A$ of a square is positioned in a magnetic field so that the normal to the plane of the coil makes an angle of $30.0^{\circ}$ with the direction of the field. When the magnetic field is increased uniformly from $200 \mu \mathrm{T}$ to $600 \mu \mathrm{T}$ in $0.400 \mathrm{~s}$, an emf of magnitude $80.0 \mathrm{mV}$ is induced in the coil. What is the total length of the wire?

Vishal Gupta

Numerade Educator

Problem 16

A closed loop of wire is given the shape of a circle with a radius of $0.500 \mathrm{~m} .$ It lies in a plane perpendicular to a uniform magnetic field of magnitude $0.400 \mathrm{~T}$. If in $0.100 \mathrm{~s}$ the wire loop is reshaped into a square but remains in the same plane, what is the magnitude of the average induced emf in the wire during this time?

Vishal Gupta

Numerade Educator

Problem 17

A toroid having a rectangular cross-section $(a=$ $2.00 \mathrm{~cm}$ by $b=3.00 \mathrm{~cm})$ and inner radius $R=4.00 \mathrm{~cm}$
consists of 500 turns of wire that carries a current $I=I_{\max } \sin \omega t$, with $I_{\max }=50.0 \mathrm{~A}$ and a frequency
$f=\omega / 2 \pi=60.0 \mathrm{~Hz}$. A coil that consists of 20 turns of wire links with the toroid, as shown in Figure P31.17. Determine the emf induced in the coil as a function of time.

Vishal Gupta

Numerade Educator

Problem 18

A single-turn, circular loop of radius $R$ is coaxial with a long solenoid of radius $r$ and length $\ell$ and having $N$ turns (Fig. P31.18). The variable resistor is changed so that the solenoid current decreases linearly from $I_{1}$ to $I_{2}$ in an interval $\Delta t$. Find the induced emf in the loop.

Vishal Gupta

Numerade Educator

Problem 19

A circular coil enclosing an area of $100 \mathrm{~cm}^{2}$ is made of 200 turns of copper wire, as shown in Figure P31.19. Initially, a 1.10-T uniform magnetic field points in a perpendicular direction upward through the plane of the coil. The direction of the field then reverses. During the time the field is changing its direction, how much charge flows through the coil if $R=5.00 \Omega ?$

Vishal Gupta

Numerade Educator

Problem 20

Consider the arrangement shown in Figure P31.20. Assume that $R=6.00 \Omega, \ell=1.20 \mathrm{~m}$, and a uniform 2.50-T magnetic field is directed into the page. At what speed should the bar be moved to produce a current of $0.500 \mathrm{~A}$ in the resistor?

Vishal Gupta

Numerade Educator

Problem 21

A conducting rod of length $\ell$ moves on two horizontal, frictionless rails, as shown in Figure P31.20. If a constant force of $1.00 \mathrm{~N}$ moves the bar at $2.00 \mathrm{~m} / \mathrm{s}$ through a magnetic field $\mathbf{B}$ that is directed into the page, (a) what is the current through an $8.00-\Omega$ resistor $R ?(\mathrm{~b})$ What is the rate at which energy is delivered to the resistor? (c) What is the mechanical power delivered by the force $\mathbf{F}_{\text {app }} ?$

GA

Gabriel A

Numerade Educator

Problem 22

A conducting rod of length $\ell$ moves on two horizontal, frictionless rails, as shown in Figure $\mathrm{P} 31.20 .$ If a constant force of $1.00 \mathrm{~N}$ moves the bar at $2.00 \mathrm{~m} / \mathrm{s}$ through a magnetic field $\mathbf{B}$ that is directed into the page, (a) what is the current through an $8.00-\Omega$ resistor $R ?(\mathrm{~b})$ What is the rate at which energy is delivered to the resistor? (c) What is the mechanical power delivered by the force $\mathbf{F}_{\text {app }} ?$

CB

Casey Bogh

Numerade Educator

Problem 23

A Boeing- 747 jet with a wing span of $60.0 \mathrm{~m}$ is flying horizontally at a speed of $300 \mathrm{~m} / \mathrm{s}$ over Phoenix, Arizona, at a location where the Earth's magnetic field is $50.0 \mu \mathrm{T}$ at $58.0^{\circ}$ below the horizontal. What voltage is generated between the wingtips?

Vishal Gupta

Numerade Educator

Problem 24

The square loop in Figure $\mathrm{P} 31.24$ is made of wires with total series resistance $10.0 \Omega$. It is placed in a uniform $0.100$ -T magnetic field directed perpendicular into the plane of the paper. The loop, which is hinged at each corner, is pulled as shown until the separation between points $A$ and $B$ is $3.00 \mathrm{~m}$. If this process takes $0.100 \mathrm{~s}$, what is the average current generated in the loop? What is the direction of the current?

Vishal Gupta

Numerade Educator

Problem 25

A helicopter has blades with a length of $3.00 \mathrm{~m}$ extending outward from a central hub and rotating at $2.00 \mathrm{rev} / \mathrm{s}$. If the vertical component of the Earth's magnetic field is $50.0 \mu \mathrm{T}$, what is the emf induced between the blade tip and the center hub?

Vishal Gupta

Numerade Educator

Problem 26

Use Lenz's law to answer the following questions concerning the direction of induced currents: (a) What is the direction of the induced current in resistor $R$ shown in Figure $\mathrm{P} 31.26 \mathrm{a}$ when the bar magnet is moved to the left? (b) What is the direction of the current induced in the resistor $R$ right after the switch $\mathrm{S}$ in Figure $\mathrm{P} 31.26 \mathrm{~b}$ is closed? (c) What is the direction of the induced current in $R$ when the current $I$ in Figure $\mathrm{P} 31.26 \mathrm{c}$ decreases rapidly to zero? (d) A copper bar is moved to the right while its axis is maintained in a direction perpendicular to a magnetic field, as shown in Figure $\mathrm{P} 31.26 \mathrm{~d}$. If the top of the bar becomes positive relative to the bottom, what is the direction of the magnetic field?

Vishal Gupta

Numerade Educator

Problem 27

A rectangular coil with resistance $R$ has $N$ turns, each of length $\ell$ and width $w$ as shown in Figure P31.27. The coil moves into a uniform magnetic field $\mathbf{B}$ with a velocity $\mathbf{v}$. What are the magnitude and direction of the resultant force on the coil (a) as it enters the magnetic field, (b) as it moves within the field, and (c) as it leaves the field?

Vishal Gupta

Numerade Educator

Problem 28

In 1832 Faraday proposed that the apparatus shown in Figure P31.28 could be used to generate electric current from the water flowing in the Thames River. ${ }^{4}$ Two conducting plates of lengths $a$ and widths $b$ are placed facing each other on opposite sides of the river, a distance $w$ apart, and are immersed entirely. The flow velocity of the river is $\mathbf{v}$ and the vertical component of the Earth's magnetic field is $B$. (a) Show that the current in the load resistor $R$ is
$$I=\frac{a b v B}{\rho+a b R / w}$$
where $\rho$ is the electrical resistivity of the water. (b) Calculate the short-circuit current $(R=0)$ if $a=100 \mathrm{~m}$, $b=5.00 \mathrm{~m}, v=3.00 \mathrm{~m} / \mathrm{s}, \bar{B}=50.0 \mu \mathrm{T}$, and $\rho=$
$100 \Omega \cdot \mathrm{m}$

Mahnoor Amin

Numerade Educator

Problem 29

In Figure $\mathrm{P} 31.29$, the bar magnet is moved toward the loop. Is $V_{a}-V_{b}$ positive, negative, or zero? Explain.

Vishal Gupta

Numerade Educator

Problem 30

A metal bar spins at a constant rate in the magnetic field of the Earth as in Figure $31.10$. The rotation occurs in a region where the component of the Earth's magnetic field perpendicular to the plane of rotation is $3.30 \times 10^{-5} \mathrm{~T}$. If the bar is $1.00 \mathrm{~m}$ in length and its angular speed is $5.00 \pi \mathrm{rad} / \mathrm{s}$, what potential difference is developed between its ends?

Vishal Gupta

Numerade Educator

Problem 31

Two parallel rails with negligible resistance are $10.0 \mathrm{~cm}$ apart and are connected by a $5.00-\Omega$ resistor. The circuit also contains two metal rods having resistances of $10.0 \Omega$ and $15.0 \Omega$ sliding along the rails (Fig. P31.31). The rods are pulled away from the resistor at constant speeds $4.00 \mathrm{~m} / \mathrm{s}$ and $2.00 \mathrm{~m} / \mathrm{s}$, respectively. A uniform magnetic field of magnitude $0.0100 \mathrm{~T}$ is applied perpendicular to the plane of the rails. Determine the current in the $5.00-\Omega$ resistor.

Vishal Gupta

Numerade Educator

Problem 32

For the situation described in Figure P31.32, the magnetic field changes with time according to the expression $B=\left(2.00 t^{3}-4.00 t^{2}+0.800\right) \mathrm{T}$, and $r_{2}=2 R=$ $5.00 \mathrm{~cm}$
(a) Calculate the magnitude and direction of the force exerted on an electron located at point $P_{2}$ when $t=2.00 \mathrm{~s}$. (b) At what time is this force equal to zero?

Vishal Gupta

Numerade Educator

Problem 33

A magnetic field directed into the page changes with time according to $B=\left(0.0300 t^{2}+1.40\right) \mathrm{T}$, where $t$ is in seconds. The field has a circular cross-section of radius $R=2.50 \mathrm{~cm}$ (see Fig. P31.32). What are the magnitude and direction of the electric field at point $P_{1}$ when $t=3.00 \mathrm{~s}$ and $r_{1}=0.0200 \mathrm{~m}$ ?

Vishal Gupta

Numerade Educator

Problem 34

A solenoid has a radius of $2.00 \mathrm{~cm}$ and 1000 turns per meter. Over a certain time interval the current varies with time according to the expression $I=3 e^{0.2 t}$, where $I$ is in amperes and $t$ is in seconds. Calculate the electric field $5.00 \mathrm{~cm}$ from the axis of the solenoid at $t=10.0 \mathrm{~s}$.

Vishal Gupta

Numerade Educator

Problem 35

A long solenoid with 1000 turns per meter and radius $2.00 \mathrm{~cm}$ carries an oscillating current $I=$ $(5.00 \mathrm{~A}) \sin (100 \pi t) .$ (a) What is the electric field induced at a radius $r=1.00 \mathrm{~cm}$ from the axis of the solenoid? (b) What is the direction of this electric field when the current is increasing counterclockwise in the coil?

CB

Casey Bogh

Numerade Educator

Problem 36

In a 250-turn automobile alternator, the magnetic flux in each turn is $\Phi_{B}=\left(2.50 \times 10^{-4} \mathrm{~T} \cdot \mathrm{m}^{2}\right) \cos (\omega t)$, where $\omega$ is the angular speed of the alternator. The alternator is geared to rotate three times for each engine revolution. When the engine is running at an angular speed of 1000 rev/min, determine (a) the induced emf in the alternator as a function of time and (b) the maximum emf in the alternator.

Vishal Gupta

Numerade Educator

Problem 37

A coil of area $0.100 \mathrm{~m}^{2}$ is rotating at $60.0$ rev $/ \mathrm{s}$ with the axis of rotation perpendicular to a $0.200$ -T magnetic field. (a) If there are 1000 turns on the coil, what is the maximum voltage induced in it? (b) What is the orientation of the coil with respect to the magnetic field when the maximum induced voltage occurs?

Vishal Gupta

Numerade Educator

Problem 38

A square coil $(20.0 \mathrm{~cm} \times 20.0 \mathrm{~cm})$ that consists of 100 turns of wire rotates about a vertical axis at 1 500 rev/min, as indicated in Figure P31.38. The horizontal component of the Earth's magnetic field at the location of the coil is $2.00 \times 10^{-5} \mathrm{~T}$. Calculate the maximum emf induced in the coil by this field.

Vishal Gupta

Numerade Educator

Problem 39

A long solenoid, with its axis along the $x$ axis, consists of 200 turns per meter of wire that carries a steady current of $15.0 \mathrm{~A}$. A coil is formed by wrapping 30 turns of thin wire around a circular frame that has a radius of $8.00 \mathrm{~cm} .$ The coil is placed inside the solenoid and mounted on an axis that is a diameter of the coil and coincides with the $y$ axis. The coil is then rotated with an angular speed of $4.00 \pi \mathrm{rad} / \mathrm{s}$. (The plane of the coil is in the $y z$ plane at $t=0 .$ ) Determine the emf developed in the coil as a function of time.

Vishal Gupta

Numerade Educator

Problem 40

A bar magnet is spun at constant angular speed $\omega$ around an axis, as shown in Figure P31.40. A flat rectangular conducting loop surrounds the magnet, and at $t=0$, the magnet is oriented as shown. Make a qualitative graph of the induced current in the loop as a function of time, plotting counterclockwise currents as positive and clockwise currents as negative.

Vishal Gupta

Numerade Educator

Problem 41

(a) What is the maximum torque delivered by an electric motor if it has 80 turns of wire wrapped on a rectangular coil of dimensions $2.50 \mathrm{~cm}$ by $4.00 \mathrm{~cm}$ ? Assume that the motor uses $10.0 \mathrm{~A}$ of current and that a uniform $0.800$ -T magnetic field exists within the motor.
(b) If the motor rotates at $3600 \mathrm{rev} / \mathrm{min}$, what is the peak power produced by the motor?

Vishal Gupta

Numerade Educator

Problem 42

A semicircular conductor of radius $R=0.250 \mathrm{~m}$ is rotated about the axis $A C$ at a constant rate of 120 rev/min (Fig. P31.42). A uniform magnetic field in all of the lower half of the figure is directed out of the plane of rotation and has a magnitude of $1.30 \mathrm{~T}$. (a) Calculate the maximum value of the emf induced in the conductor. (b) What is the value of the average induced emf for each complete rotation? (c) How would the answers to parts (a) and (b) change if $\mathbf{B}$ were allowed to extend a distance $R$ above the axis of rotation? Sketch the emf versus time (d) when the field is as drawn in Figure $\mathrm{P} 31.42$ and $(\mathrm{e})$ when the field is extended as described in part (c).

Vishal Gupta

Numerade Educator

Problem 43

A $0.150-\mathrm{kg}$ wire in the shape of a closed rectangle $1.00 \mathrm{~m}$ wide and $1.50 \mathrm{~m}$ long has a total resistance of $0.750 \Omega .$ The rectangle is allowed to fall through a magnetic field directed perpendicular to the direction of motion of the rectangle (Fig. P31.44). The rectangle accelerates downward as it approaches a terminal speed of $2.00 \mathrm{~m} / \mathrm{s}$, with its top not yet in the region of the field. Calculate the magnitude of $\mathbf{B}$.

CB

Casey Bogh

Numerade Educator

Problem 44

A $0.150-\mathrm{kg}$ wire in the shape of a closed rectangle $1.00 \mathrm{~m}$ wide and $1.50 \mathrm{~m}$ long has a total resistance of $0.750 \Omega$. The rectangle is allowed to fall through a magnetic field directed perpendicular to the direction of motion of the rectangle (Fig. P31.44). The rectangle accelerates downward as it approaches a terminal speed of $2.00 \mathrm{~m} / \mathrm{s}$, with its top not yet in the region of the field. Calculate the magnitude of $\mathbf{B}$.

CB

Casey Bogh

Numerade Educator

Problem 45

A conducting rectangular loop of mass $M$, resistance $R$, and dimensions $w$ by $\ell$ falls from rest into a magnetic field $\mathbf{B}$ as in Figure $\mathrm{P} 31.44$. The loop approaches terminal speed $v_{t} .$ (a) Show that $$v_{t}=\frac{M g R}{B^{2} w^{2}}$$
(b) Why is $v_{t}$ proportional to $R$ ? (c) Why is it inversely proportional to $B^{2}$ ?

CB

Casey Bogh

Numerade Educator

Problem 46

Figure P31.46 represents an electromagnetic brake that utilizes eddy currents. An electromagnet hangs from a railroad car near one rail. To stop the car, a large steady current is sent through the coils of the electromagnet. The moving electromagnet induces eddy currents in the rails, whose fields oppose the change in the field of the electromagnet. The magnetic fields of the eddy currents exert force on the current in the electromagnet, thereby slowing the car. The direction of the car's motion and the direction of the current in the electromagnet are shown correctly in the picture. Determine which of the eddy currents shown on the rails is correct. Explain your answer.

Vishal Gupta

Numerade Educator

Problem 47

A proton moves through a uniform electric field $\mathbf{E}=50.0 \mathbf{j} \mathrm{V} / \mathrm{m}$ and a uniform magnetic field $\mathbf{B}=$ $(0.200 \mathbf{i}+0.300 \mathbf{j}+0.400 \mathbf{k}) \mathrm{T}$. Determine the acceleration of the proton when it has a velocity $\mathbf{v}=200 \mathbf{i} \mathrm{m} / \mathrm{s}$.

Vishal Gupta

Numerade Educator

Problem 48

An electron moves through a uniform electric field $\mathbf{E}=$ $(2.50 \mathbf{i}+5.00 \mathbf{j}) \mathrm{V} / \mathrm{m}$ and a uniform magnetic field $\mathbf{B}=$ $0.400 \mathbf{k}$ T. Determine the acceleration of the electron when it has a velocity $\mathbf{v}=10.0 \mathbf{i} \mathrm{m} / \mathrm{s}$.

Vishal Gupta

Numerade Educator

Problem 49

A steel guitar string vibrates (see Fig. $31.5)$. The component of the magnetic field perpendicular to the area of a pickup coil nearby is given by $$B=50.0 \mathrm{mT}+(3.20 \mathrm{mT}) \sin (2 \pi 523 t / \mathrm{s}).$$ The circular pickup coil has 30 turns and radius $2.70 \mathrm{~mm}$. Find the emf induced in the coil as a function of time.

Vishal Gupta

Numerade Educator

Problem 50

Figure $\mathrm{P} 31.50$ is a graph of the induced emf versus time for a coil of $N$ turns rotating with angular velocity $\omega$ in a uniform magnetic field directed perpendicular to the axis of rotation of the coil. Copy this graph (on a larger scale), and on the same set of axes show the graph of emf versus $t$ (a) if the number of turns in the coil is doubled, (b) if instead the angular velocity is doubled, and (c) if the angular velocity is doubled while the number of turns in the coil is halved.

Vishal Gupta

Numerade Educator

Problem 51

A technician wearing a brass bracelet enclosing an area of $0.00500 \mathrm{~m}^{2}$ places her hand in a solenoid whose magnetic field is $5.00 \mathrm{~T}$ directed perpendicular to the plane of the bracelet. The electrical resistance around the circumference of the bracelet is $0.0200 \Omega$. An unexpected power failure causes the field to drop to $1.50 \mathrm{~T}$ in a time of $20.0 \mathrm{~ms}$. Find (a) the current induced in the bracelet and (b) the power delivered to the resistance of the bracelet. (Note: As this problem implies, you should not wear any metallic objects when working in regions of strong magnetic fields.)

Vishal Gupta

Numerade Educator

Problem 52

Two infinitely long solenoids (seen in cross-section) thread a circuit as shown in Figure P31.52. The magnitude of $\mathbf{B}$ inside each is the same and is increasing at the rate of $100 \mathrm{~T} / \mathrm{s}$. What is the current in each resistor?

Vishal Gupta

Numerade Educator

Problem 53

A conducting rod of length $\ell=35.0 \mathrm{~cm}$ is free to slide on two parallel conducting bars, as shown in Figure P31.53. Two resistors $R_{1}=2.00 \Omega$ and $R_{2}=5.00 \Omega$ are connected across the ends of the bars to form a loop. A constant magnetic field $B=2.50 \mathrm{~T}$ is directed perpendicular into the page. An external agent pulls the rod to the left with a constant speed of $v=8.00 \mathrm{~m} / \mathrm{s}$. Find (a) the currents in both resistors, (b) the total power delivered to the resistance of the circuit, and (c) the magnitude of the applied force that is needed to move the rod with this constant velocity.

Vishal Gupta

Numerade Educator

Problem 54

Suppose you wrap wire onto the core from a roll of cellophane tape to make a coil. Describe how you can use a bar magnet to produce an induced voltage in the coil. What is the order of magnitude of the emf you generate? State the quantities you take as data and their values.

GA

Gabriel A

Numerade Educator

Problem 55

A bar of mass $m$, length $d$, and resistance $R$ slides without friction on parallel rails, as shown in Figure P31.55. A battery that maintains a constant $\operatorname{emf} \boldsymbol{\varepsilon}$ is connected between the rails, and a constant magnetic field $\mathbf{B}$ is directed perpendicular to the plane of the page. If the bar starts from rest, show that at time $t$ it moves with a speed
$$v=\frac{\boldsymbol{\varepsilon}}{B d}\left(1-e^{-B^{2} d^{2} t / m R}\right).$$

Dominador Tan

Numerade Educator

Problem 56

An automobile has a vertical radio antenna $1.20 \mathrm{~m}$ long. The automobile travels at $65.0 \mathrm{~km} / \mathrm{h}$ on a horizontal road where the Earth's magnetic field is $50.0 \mu \mathrm{T}$ directed toward the north and downward at an angle of $65.0^{\circ}$ below the horizontal. (a) Specify the direction that the automobile should move to generate the maximum motional emf in the antenna, with the top of the antenna positive relative to the bottom. (b) Calculate the magnitude of this induced emf.

Vishal Gupta

Numerade Educator

Problem 57

The plane of a square loop of wire with edge length $a=0.200 \mathrm{~m}$ is perpendicular to the Earth's magnetic field at a point where $B=15.0 \mu \mathrm{T}$, as shown in Figure P31.57. The total resistance of the loop and the wires connecting it to the galvanometer is $0.500 \Omega$. If the loop is suddenly collapsed by horizontal forces as shown, what total charge passes through the galvanometer?

Prabhu Ramji

Numerade Educator

Problem 58

Magnetic field values are often determined by using a device known as a search coil. This technique depends on the measurement of the total charge passing through a coil in a time interval during which the magnetic flux linking the windings changes either because of the motion of the coil or because of a change in the value of $B$. (a) Show that as the flux through the coil changes from $\Phi_{1}$ to $\Phi_{2}$, the charge transferred through the coil will be given by $Q=N\left(\Phi_{2}-\Phi_{1}\right) / R$, where $R$ is the resistance of the coil and associated circuitry (galvanometer) and $N$ is the number of turns. (b) As a specific example, calculate $B$ when a 100 -turn coil of resistance $200 \Omega$ and cross-sectional area $40.0 \mathrm{~cm}^{2}$ produces the following results. A total charge of $5.00 \times 10^{-4}$ C passes through the coil when it is rotated in a uniform field from a position where the plane of the coil is perpendicular to the field to a position where the coil's plane is parallel to the field.

Vishal Gupta

Numerade Educator

Problem 59

In Figure P31.59, the rolling axle, $1.50 \mathrm{~m}$ long, is pushed along horizontal rails at a constant speed $v=3.00 \mathrm{~m} / \mathrm{s}$. A resistor $R=0.400 \Omega$ is connected to the rails at points $a$ and $b$, directly opposite each other. (The wheels make good electrical contact with the rails, and so the axle, rails, and $R$ form a closed-loop circuit. The only significant resistance in the circuit is $R$.) There is a uniform magnetic field $B=0.0800 \mathrm{~T}$ vertically downward. (a) Find the induced current $I$ in the resistor. (b) What horizontal force $F$ is required to keep the axle rolling at constant speed? (c) Which end of the resistor, $a$ or $b$, is at the higher electric potential? (d) After the axle rolls past the resistor, does the current in $R$ reverse direction? Explain your answer.

Vishal Gupta

Numerade Educator

Problem 60

A conducting rod moves with a constant velocity $\mathbf{v}$ perpendicular to a long, straight wire carrying a current $I$ as shown in Figure $\mathrm{P} 31.60$. Show that the magnitude of the emf generated between the ends of the rod is
$$|\boldsymbol{\varepsilon}|=\frac{\mu_{0} v I}{2 \pi r} \ell$$
In this case, note that the emf decreases with increasing $r$, as you might expect.

Dading Chen

Numerade Educator

Problem 61

A circular loop of wire of radius $r$ is in a uniform magnetic field, with the plane of the loop perpendicular to the direction of the field (Fig. P31.61). The magnetic field varies with time according to $B(t)=a+b t$, where $a$ and $b$ are constants. (a) Calculate the magnetic flux through the loop at $t=0 .$ (b) Calculate the emf induced in the loop. (c) If the resistance of the loop is $R$, what is the induced current? (d) At what rate is electrical energy being delivered to the resistance of the loop?

Vishal Gupta

Numerade Educator

Problem 62

In Figure $\mathrm{P} 31.62$, a uniform magnetic field decreases at a constant rate $d B / d t=-K$, where $K$ is a positive constant. A circular loop of wire of radius $a$ containing a resistance $R$ and a capacitance $C$ is placed with its plane normal to the field. (a) Find the charge $Q$ on the capacitor when it is fully charged. (b) Which plate is at the higher potential? (c) Discuss the force that causes the separation of charges.

Vishal Gupta

Numerade Educator

Problem 63

A rectangular coil of 60 turns, dimensions $0.100 \mathrm{~m}$ by $0.200 \mathrm{~m}$ and total resistance $10.0 \Omega$, rotates with angular speed $30.0 \mathrm{rad} / \mathrm{s}$ about the $y$ axis in a region where a $1.00-\mathrm{T}$ magnetic field is directed along the $x$ axis. The rotation is initiated so that the plane of the coil is perpendicular to the direction of $\mathbf{B}$ at $t=0 .$ Calculate (a) the maximum induced emf in the coil, (b) the maximum rate of change of magnetic flux through the coil, (c) the induced emf at $t=0.0500 \mathrm{~s}$, and $(\mathrm{d})$ the torque exerted on the coil by the magnetic field at the instant when the emf is a maximum.

GA

Gabriel A

Numerade Educator

Problem 64

A small circular washer of radius $0.500 \mathrm{~cm}$ is held directly below a long, straight wire carrying a current of $10.0 \mathrm{~A}$. The washer is located $0.500 \mathrm{~m}$ above the top of the table (Fig. P31.64). (a) If the washer is dropped from rest, what is the magnitude of the average induced emf in the washer from the time it is released to the moment it hits the tabletop? Assume that the magnetic field is nearly constant over the area of the washer and equal to the magnetic field at the center of the washer. (b) What is the direction of the induced current in the washer?

Vishal Gupta

Numerade Educator

Problem 65

To monitor the breathing of a hospital patient, a thin belt is wrapped around the patient's chest. The belt is a 200-turn coil. When the patient inhales, the area encircled by the coil increases by $39.0 \mathrm{~cm}^{2}$. The magnitude of the Earth's magnetic field is $50.0 \mu \mathrm{T}$ and makes an angle of $28.0^{\circ}$ with the plane of the coil. If a patient takes $1.80 \mathrm{~s}$ to inhale, find the average induced emf in the coil during this time.

Vishal Gupta

Numerade Educator

Problem 66

A conducting rod of length $\ell$ moves with velocity $\mathbf{v}$ parallel to a long wire carrying a steady current $I$. The axis of the rod is maintained perpendicular to the wire with the near end a distance $r$ away, as shown in Figure P31.66. Show that the magnitude of the emf induced in the rod is $$|\boldsymbol{\varepsilon}|=\frac{\mu_{0} I}{2 \pi} v \ln \left(1+\frac{\ell}{r}\right).$$

Vishal Gupta

Numerade Educator

Problem 67

A rectangular loop of dimensions $\ell$ and $w$ moves with a constant velocity $\mathbf{v}$ away from a long wire that carries a current $I$ in the plane of the loop (Fig. $\mathrm{P} 31.67)$. The total resistance of the loop is $R .$ Derive an expression that gives the current in the loop at the instant the near side is a distance $r$ from the wire.

Vishal Gupta

Numerade Educator

Problem 68

A horizontal wire is free to slide on the vertical rails of a conducting frame, as shown in Figure $\mathrm{P} 31.68$. The wire has mass $m$ and length $\ell$, and the resistance of the circuit is $R$. If a uniform magnetic field is directed perpendicular to the frame, what is the terminal speed of the wire as it falls under the force of gravity?

Aniket Bajaj

Numerade Educator

Problem 69

The magnetic flux threading a metal ring varies with time $t$ according to $\Phi_{B}=3\left(a t^{3}-b t^{2}\right) \mathrm{T} \cdot \mathrm{m}^{2}$, with $a=2.00 \mathrm{~s}^{-3}$ and $b=6.00 \mathrm{~s}^{-2}$. The resistance of the ring is $3.00 \Omega$. Determine the maximum current induced in the ring during the interval from $t=0$ to $t=2.00 \mathrm{~s}$.

Vishal Gupta

Numerade Educator

Problem 69

The magnetic flux threading a metal ring varies with time $t$ according to $\Phi_{B}=3\left(a t^{3}-b t^{2}\right) \mathrm{T} \cdot \mathrm{m}^{2}$, with $a=2.00 \mathrm{~s}^{-3}$ and $b=6.00 \mathrm{~s}^{-2}$. The resistance of the ring is $3.00 \Omega$. Determine the maximum current induced in the ring during the interval from $t=0$ to $t=2.00 \mathrm{~s}$.

Vishal Gupta

Numerade Educator

Problem 70

The bar of mass $m$ shown in Figure P31.70 is pulled horizontally across parallel rails by a massless string that passes over an ideal pulley and is attached to a suspended mass $M$. The uniform magnetic field has a magnitude $B$, and the distance between the rails is $\ell$. The rails are connected at one end by a load resistor $R$. Derive an expression that gives the horizontal speed of the bar as a function of time, assuming that the suspended mass is released with the bar at rest at $t=0$. Assume no friction between rails and bar.

Vishal Gupta

Numerade Educator

Problem 70

The bar of mass $m$ shown in Figure P31.70 is pulled horizontally across parallel rails by a massless string that passes over an ideal pulley and is attached to a suspended mass $M$. The uniform magnetic field has a magnitude $B$, and the distance between the rails is $\ell$. The rails are connected at one end by a load resistor $R$. Derive an expression that gives the horizontal speed of the bar as a function of time, assuming that the suspended mass is released with the bar at rest at $t=0$. Assume no friction between rails and bar.

Vishal Gupta

Numerade Educator

Problem 71

A solenoid wound with 2000 turns $/ \mathrm{m}$ is supplied with current that varies in time according to $I=$ $4 \sin (120 \pi t)$, where $I$ is in $\mathrm{A}$ and $t$ is in s. A small coaxial circular coil of 40 turns and radius $r=5.00 \mathrm{~cm}$ is located inside the solenoid near its center. (a) Derive an expression that describes the manner in which the emf in the small coil varies in time. (b) At what average rate is energy transformed into internal energy in the small coil if the windings have a total resistance of $8.00 \Omega$ ?

Vishal Gupta

Numerade Educator

Problem 71

A solenoid wound with 2000 turns $/ \mathrm{m}$ is supplied with current that varies in time according to $I=$ $4 \sin (120 \pi t)$, where $I$ is in $\mathrm{A}$ and $t$ is in s. A small coaxial circular coil of 40 turns and radius $r=5.00 \mathrm{~cm}$ is located inside the solenoid near its center. (a) Derive an expression that describes the manner in which the emf in the small coil varies in time. (b) At what average rate is energy transformed into internal energy in the small coil if the windings have a total resistance of $8.00 \Omega$ ?

Vishal Gupta

Numerade Educator

Problem 72

A wire $30.0 \mathrm{~cm}$ long is held parallel to and $80.0 \mathrm{~cm}$ above a long wire carrying $200 \mathrm{~A}$ and resting on the floor (Fig. P31.72). The $30.0$ -cm wire is released and falls, remaining parallel with the current-carrying wire as it falls. Assume that the falling wire accelerates at $9.80 \mathrm{~m} / \mathrm{s}^{2}$ and derive an equation for the emf induced in it. Express your result as a function of the time $t$ after the wire is dropped. What is the induced emf $0.300 \mathrm{~s}$ after the wire is released?

Vishal Gupta

Numerade Educator

Problem 73

A long, straight wire carries a current $I=I_{\max } \sin (\omega t+$ $\phi$ ) and lies in the plane of a rectangular coil of $N$ turns of wire, as shown in Figure $\mathrm{P} 31.9$. The quantities $I_{\max }$, $\omega$, and $\phi$ are all constants. Determine the emf induced in the coil by the magnetic field created by the current in the straight wire. Assume $I_{\max }=50.0 \mathrm{~A}, \omega=$ $200 \pi \mathrm{s}^{-1}, N=100, h=w=5.00 \mathrm{~cm}$, and $L=20.0 \mathrm{~cm} .$

GA

Gabriel A

Numerade Educator

Problem 74

A dime is suspended from a thread and hung between the poles of a strong horseshoe magnet as shown in Figure P31.74. The dime rotates at constant angular speed $\omega$ about a vertical axis. Letting $\theta$ represent the angle between the direction of $\mathbf{B}$ and the normal to the face of the dime, sketch a graph of the torque due to induced currents as a function of $\theta$ for $0<\theta<2 \pi$.

Dading Chen

Numerade Educator

Problem 75

The wire shown in Figure $\mathrm{P} 31.75$ is bent in the shape of a tent, with $\theta=60.0^{\circ}$ and $L=1.50 \mathrm{~m}$, and is placed in a uniform magnetic field of magnitude $0.300 \mathrm{~T}$ perpendicular to the tabletop. The wire is rigid but hinged at points $a$ and $b$. If the "tent" is flattened out on the table in $0.100 \mathrm{~s}$, what is the average induced emf in the wire during this time?

Vishal Gupta

Numerade Educator