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Understanding Physics

Karen Cummings, Priscilla W. Laws, Edward F. Redish

Chapter 31

Induction and Maxwell’s Equations - all with Video Answers

Educators

Chapter Questions

02:39

Problem 1

Antenna A UHF television loop antenna has a diameter of $11 \mathrm{~cm}$. The magnetic field of a TV signal is normal to the plane of the loop and, at one instant of time, its magnitude is changing at the rate $0.16 \mathrm{~T} / \mathrm{s}$. The magnetic field is uniform. What emf is induced in the antenna?

Vishal Gupta
Vishal Gupta
Numerade Educator
04:22

Problem 2

A small loop of area $A$ is inside of, and has its axis in the same direction as, a long solenoid of $n$ turns per unit length and current $i$. If $i=I^{\max } \sin \omega t$, find the magnitude of the $\mathrm{cmf}$ induced in the loop.

Vishal Gupta
Vishal Gupta
Numerade Educator
06:59

Problem 3

Magnetic Flux The magnetic flux encircled by the loop shown in Fig. $31-29$ increases according to the relation $\Phi^{\text {mal }}=\left(6.0 \mathrm{mWb} / \mathrm{s}^{2}\right) t^{2}+(3.7 \mathrm{mWb} / \mathrm{s}) t .(\mathrm{a})$
What is the magnitude of the emf induced in the loop when $t=2.0 \mathrm{~s}$ ? (b) What is the direction of the current through $R ?$

Vishal Gupta
Vishal Gupta
Numerade Educator
05:47

Problem 4

Calculate emf The magnitude of the magnetic field encircled by a single loop of wire. $12 \mathrm{~cm}$ in radius and of $8.5 \Omega$ resistance, changes with time as shown in Fig. 31-30. Calculate the magnitude of the emf in the loop as a function of time. Con- FIGURE $31-30=$ Problem 4 sider the time intervals (a) $t_{1}=$ $0.0 \mathrm{~s}$ to $t_{2}=2.0 \mathrm{~s},(\mathrm{~b}) t_{2}=2.0 \mathrm{~s}$ to $t_{3}=4.0 \mathrm{~s},(\mathrm{c}) t_{3}=4.0 \mathrm{~s}$ to $t_{4}=6.0 \mathrm{~s}$
The (uniform) magnetic field is perpendicular to the plane of the loop.

Vishal Gupta
Vishal Gupta
Numerade Educator
04:51

Problem 5

Uniform Magnetic Field A uniform magnetic field is normal to the plane of a circular loop $10 \mathrm{~cm}$ in diameter and made of copper wire (of diameter $2.5 \mathrm{~mm}$ ). (a) Calculate the resistance of the wire. (See Table 26-2.) (b) At what rate must the magnetic field change with time if an induced current of $10 \mathrm{~A}$ is to appear in the loop?

Vishal Gupta
Vishal Gupta
Numerade Educator
04:04

Problem 6

The current in the solenoid of Touchstone Example $31-1$ changes, not as stated there, but according to $i=$ $(3.0 \mathrm{~A} / \mathrm{s}) t+\left(1.0 \mathrm{~A} / \mathrm{s}^{2}\right) t^{2}$. (a) Plot the
induced emf in the coil from $t_{1}=0.0 \mathrm{~s}$ to $t_{2}=4.0 \mathrm{~s}$. (b) The resis-
tance of the coil is $0.15 \Omega$. What is the current in the coil at $t=2.0 \mathrm{~s}$ ?

Amit Srivastava
Amit Srivastava
Numerade Educator
03:01

Problem 7

In Fig. 31 - $\begin{array}{lll}31 & \text { a } 120 \text { -turn } & \text { coil } & \text { of } \text { radius }\end{array}$ $1.8 \mathrm{~cm}$ and resistance $5.3 \Omega$ is placed outside a solcnoid like that of Touchstone Example $31-1 .$ If the current in the solenoid is changed as in that sample problem, what current appears in the coil while the solenoid current is being changed?

Amit Srivastava
Amit Srivastava
Numerade Educator
03:27

Problem 8

Elastic Conducting Material An elastic conducting material is stretched into a circular loop of $12.0 \mathrm{~cm}$ radius. It is placed with its plane perpendicular to a uniform $0.800$ T magnetic field. When released, the radius of the loop starts to shrink at an instantaneous rate of $75.0 \mathrm{~cm} / \mathrm{s}$. What magnitude of emf is induced in the loop at that instant?

Vishal Gupta
Vishal Gupta
Numerade Educator
03:58

Problem 9

Square Loop A square loop of wire is held in a uniform, magnetic field $0.24$ T directed perpendicularly to the plane of the loop. The length of each side of the square is decreasing at a constant rate of $5.0 \mathrm{~cm} / \mathrm{s}$. What emf is induced in the loop when the length is $12 \mathrm{~cm} ?$

Vishal Gupta
Vishal Gupta
Numerade Educator
01:18

Problem 10

Rectangular Loop A rectangular loop (area $=0.15 \mathrm{~m}^{2}$ ) turns in a uniform magnetic field, $B=0.20 \mathrm{~T}$. When the angle between the field and the normal to the plane of the loop is $\pi / 2$ rad and increasing at $0.60 \mathrm{rad} / \mathrm{s}$, what emf is induced in the loop?

Averell Hause
Averell Hause
Carnegie Mellon University
09:41

Problem 11

Two Parallel Loops Though not to scale, Fig. $31-32$ shows two parallel loops of wire with a common axis. The smaller loop (radius $r$ ) is above the larger loop (radius $R$ ) by a distance $x \geqslant R$. Consequently, the magnetic field due to the current $i$ in the larger loop is nearly constant throughout the smaller loop. Suppose that $x$ is increasing at the constant FIGURE $31-32=$ rate of $d x / d t=v .$ (a) Determine the Problem 11 . magnetic flux at the area bounded by the smaller loop as a function of $x$. (Hint: See Eq. $30-29 .$ ) In the smaller loop, find (b) the induced emf and (c) the direction of the induced current.

Vishal Gupta
Vishal Gupta
Numerade Educator
04:05

Problem 12

Circular Loop In Fig. 31-33, a circular loop of wire $10 \mathrm{~cm}$ in diameter (seen edge-on) is placed with its normal at an angle $\theta=30^{\circ}$ with the direction of a uniform magnetic field $\vec{B}$ of magnitude $0.50 \mathrm{~T}$. The loop is then rotated such that the normal rotates in a cone about the field direction at the constant rate of $100 \mathrm{rev} / \mathrm{min}$ :
the angle $\theta$ remains unchanged during the process. What is the emf induced in the loop?

Vishal Gupta
Vishal Gupta
Numerade Educator
05:38

Problem 13

In Fig. $31-29$ let the flux encircled by the loop be $\Phi^{\text {male }}(0)$ at time $t_{1}=0 .$ Then let the magnetic field $\vec{B}$ vary in a continuous but unspecified way, in both magnitude and direction, so that at time $t_{2}$ the flux is represented by $\Phi^{\operatorname{mag}}\left(t_{2}\right)$. (a) Show that the net charge $q\left(t_{2}\right)$ that has passed through resistor $R$ in time $t_{2}$ is
$$
q\left(t_{2}\right)=\frac{1}{R}\left[\Phi^{\operatorname{mag}}(0)-\Phi^{\operatorname{mag}}\left(t_{2}\right)\right]
$$
and is independent of the way $\vec{B}$ has changed. (b) If $\Phi^{\operatorname{mag}}\left(t_{2}\right)=$ $\Phi^{\text {mag }}(0)$ in a particular case, we have $q\left(t_{2}\right)=0 .$ Is the induced current necessarily zero throughout the interval from 0 to $t_{2}$ ?

Vishal Gupta
Vishal Gupta
Numerade Educator
07:51

Problem 14

Big Loop, Little Loop A small circular loop of area $2.00 \mathrm{~cm}^{2}$ is placed in the plane of, and concentric with, a large circular loop of radius $1.00 \mathrm{~m}$. The current in the large loop is changed uniformly from 200 A to $-200 \mathrm{~A}$ (a change in direction) in a time of $1.00 \mathrm{~s}$, beginning at $t_{1}=0 .$ (a) What is the magnitude of the magnetic field at the center of the small circular loop due to the current in the large loop at $t_{1}=0 \mathrm{~s}, t_{2}=0.500 \mathrm{~s}$, and $t_{3}=1.00 \mathrm{~s}$ ? (b) What is the magnitude of the emf induced in the small loop at $t_{2}=0.500 \mathrm{~s}$ ? (Since the inner loop is small, assume the ficld $\vec{B}$ due to the outer loop is uniform over the area of the smaller loop.)

Vishal Gupta
Vishal Gupta
Numerade Educator
04:34

Problem 15

Copper Wire on Wooden Core One hundred turns of insulated copper wire are wrapped around a wooden cylindrical core of cross-sectional area $1.20 \times 10^{-3} \mathrm{~m}^{2}$. The two ends of the wire are connected to a resistor. The total resistance in the circuit is $13.0 \Omega$. If an externally applied uniform longitudinal magnetic field in the core changes from $1.60 \mathrm{~T}$ in one direction to $1.60 \mathrm{~T}$ in the opposite direction, how much charge flows through the circuit? (Hint: See Problem 13.)

Vishal Gupta
Vishal Gupta
Numerade Educator
06:54

Problem 16

Earth's Field At a certain place, Earth's magnetic field has magnitude $|\vec{B}|=0.590$ gauss and is inclined downward at an angle of $70.0^{\circ}$ to the horizontal. A flat horizontal circular coil of wire with a radius of $10.0 \mathrm{~cm}$ has 1000 turns and a total resistance of $85.0 \Omega$. It is connected to a meter with $140 \Omega$ resistance. The coil is flipped through a half-revolution about a diameter, so that it is again horizontal. How much charge flows through the meter during the flip? (Hint: See Problem 13.)

Vishal Gupta
Vishal Gupta
Numerade Educator
08:04

Problem 17

Square Loop A square wire loop with $2.00 \mathrm{~m}$ sides is perpendicular to a uniform magnetic field. with half the area of the loop in the field as shown in Fig. $31-34$. The loop contains a $20.0 \mathrm{~V}$ battery with negligible internal resistance. If the magnitude of the field varies with time according to $B=\left(\begin{array}{ccc}0.0420 & \mathrm{~T})-\end{array}\right.$
$(0.870 \mathrm{~T} / \mathrm{s}) t$, what are (a) the magnitude of the net emf in the circuit and
(b) the direction of the current through the battery?

Vishal Gupta
Vishal Gupta
Numerade Educator
08:05

Problem 18

Three Circular Segments $\mathrm{A}$ wire is bent into three circular segments, each of radius $r=10 \mathrm{~cm}$, as shown in Fig. 31-35. Each segment is a quadrant of a circle, $a b$ lying in the $x y$ plane, $b c$ lying in the $y z$ plane, and $c a$ lying in the $z x$ plane. (a) If a uniform magnetic field $\vec{B}$ points in the positive $x$ direction, what is the magnitude of the emf developed in the wire when $\vec{B}$ increases at the rate of $3.0 \mathrm{mT} / \mathrm{s}$ in the $x$ direction? (b) What is the direction of the current in segment bc?

Vishal Gupta
Vishal Gupta
Numerade Educator
07:38

Problem 19

Rectangular Coil A rectangular coil of $N$ turns and of length $a$ and width $b$ is rotated at frequency $f$ in a uniform magnetic field $\vec{B}$, as indicated in Fig. $31-36$. The coil is connected to co-rotating cylinders, against which metal brushes slide to make contact. If we arbitrarily define $\mathrm{emf}$ as being positive during the first quarter-turn,
(a) show that the emf induced in the coil is given (as a function of time $t$ ) by
$$
\mathscr{E}=2 \pi f N a b B \sin (2 \pi f t)=\mathscr{E}_{0} \sin (2 \pi f t)
$$
This is the principle of the commercial alternating-current generator. (b) Design a loop that will produce an emf with $8_{0}=150 \mathrm{~V}$ when rotated at $60.0 \mathrm{rcv} / \mathrm{s}$ in a uniform magnetic field of $0.500 \mathrm{~T}$.

Vishal Gupta
Vishal Gupta
Numerade Educator
08:25

Problem 20

Semicircle $A$ stiff wire bent into a semicircle of radius $a$ is rotated with frequency $\bar{f}$ in a uniform magnetic ficld, as suggested in Fig. 31-37. What are (a) the frequency and (b) the amplitude of the varying emf induced in the loop?

Vishal Gupta
Vishal Gupta
Numerade Educator
03:44

Problem 21

Electric Generator An clectric generator consists of FIGURE $31-37=$ Problem 20 . 100 turns of wire formed into a rectangular loop $50.0 \mathrm{~cm}$ by $30.0 \mathrm{~cm}$, placed cntirely in a uniform magnetic field with magnitude $B=3.50 \mathrm{~T}$. What is the maximum value of the emf produced when the loop is spun at 1000 rev/min about an axis perpendicular to $\vec{B}$ ?

Vishal Gupta
Vishal Gupta
Numerade Educator
07:47

Problem 22

Closed Circular Loop In Fig. $31-38$, a wire forms a closed circular loop, with radius $R=2.0 \mathrm{~m}$ and resistance $4.0 \Omega$. The circle is centered on a long straight wire; at time $t=0$, the current in the long straight wire is $5.0$ A rightward. Thereafter, the current changes according to $i=5.0 \mathrm{~A}-\left(2.0 \mathrm{~A} / \mathrm{s}^{2}\right) t^{2} . \quad$ Problem $22 .$
(The straight wire is insulated, $\underline{\text { so }}$ there is no electrical contact between it and the wire of the loop.) What are the magnitude and direction of the current induced in the loop at times $t>0 ?$

Vishal Gupta
Vishal Gupta
Numerade Educator
07:40

Problem 23

Square Loop Two In Fig. $31-39$. the square loop of wire has sides of length $2.0 \mathrm{~cm}$. A magnetic field is directed out of the page; its magnitude is given by $B=\left(4.0 \mathrm{~T} / \mathrm{m} \cdot \mathrm{s}^{2}\right) t^{2} y$, where
$B$ is in teslas, $t$ is in seconds, and $y$ is in meters. Determine the emf around the square at $t=2.5 \mathrm{~s}$ and indicate whether its direction is clockwise or counterclockwise.

Vishal Gupta
Vishal Gupta
Numerade Educator
09:59

Problem 24

Square Loop Three For the situation shown in Fig. $31-40, a=12.0$ $\mathrm{cm}$ and $b=16.0 \mathrm{~cm}$. The current in the long straight wire is given by $\quad i=\left(4.50 \mathrm{~A} / \mathrm{s}^{2}\right) t^{2}-(10.0 \mathrm{~A} / \mathrm{s}) t$,
where $i$ is in amperes and $t$ is in seconds. (a) Find the magnitude of the emf in the square loop at $t=3.00 \mathrm{~s}$.
(b) Indicate whether the direction of the induced current in the loop is clockwise or counterclockwise at $t=$ $3.00 \mathrm{~s}$

Vishal Gupta
Vishal Gupta
Numerade Educator
05:40

Problem 25

Parallel $\quad$ Copper $\quad$ Wires Tiwo long, parallel copper wires of diameter $2.5 \mathrm{~mm}$ carry currents of 10 A in opposite directions. (a) Assuming that their central axes are 20 $\mathrm{mm}$ apart, calculate the magnetic flux per meter of wire that exists in the space between those axes. (b) What fraction of this flux lies inside the wires? (c) Repeat part (a) for parallel currents.

Amit Srivastava
Amit Srivastava
Numerade Educator
09:57

Problem 26

Rectangular Wire Loop A rectangular loop of wire with length $a$, width $b$, and resistance $R$ is placed near an infinitely long wire carrying current $i$, as shown in Fig. $31-41 .$ The distance from the long wire to the center of the loop is $r$. Find (a) the magnitude of the magnetic flux encircled by the loop and
(b) the amount of induced current in the loop $\left|i^{\text {ind }}\right|$ as it moves away from the long wire with velocity $\vec{v}$. (c) Indicate whether the induced current FIGURE $31-41$ w. is clockwise or counterclockwise. $\quad$ Problem $26 .$

Vishal Gupta
Vishal Gupta
Numerade Educator
06:23

Problem 27

Internal Energy If $50.0 \mathrm{~cm}$ of copper wire (diameter $=1.00 \mathrm{~mm}$ ) is formed into a circular loop and placed perpendicular to a uniform magnetic field that is increasing at the constant rate of $10.0$ $\mathrm{mT} / \mathrm{s}$, at what rate does internal energy increase in the loop?

Vishal Gupta
Vishal Gupta
Numerade Educator
03:15

Problem 28

Loop Antenna A loop antenna of area $A$ and resistance $R$ is perpendicular to a uniform magnetic field $\vec{B}$. The field drops lincarly to zero in a time interval $\Delta t .$ Find an cxpression for the total internal energy added to the loop.

Vishal Gupta
Vishal Gupta
Numerade Educator
05:23

Problem 29

Rod on Rails A metal rod is forced to move with constant velocity $\vec{v}$ along two parallel metal rails, connected with a strip of metal at one end, as shown in Fig. $31-42 . \mathrm{A}$ magnetic field of magnitude $|\vec{B}|=$ $0.350 \mathrm{~T}$ points out of the page. (a) If the rails are separated by $25.0 \mathrm{~cm}$ and the speed of the rod is $55.0$ $\mathrm{cm} / \mathrm{s}$, what emf is generated? (b) If the rod has a resistance of $18.0$ $\Omega$ and the rails and connector have negligible resistance, what is the current in the rod? (c) At what rate is mechanical energy being transformed to thermal energy?

Vishal Gupta
Vishal Gupta
Numerade Educator
05:32

Problem 30

Find Terminal Speed In Fig. 31 -
43, a long rectangular conducting loop, of width $L$, resistance $R$, and mass $m$, is hung in a horizontal, uniform magnetic field $\vec{B}$ that is directed into the page and that exists only above line $a a$. The loop is then dropped; during its fall, it accelerates until it reaches a certain terminal speed $v_{i} .$ Ignoring air drag, find that terminal speed.

Vishal Gupta
Vishal Gupta
Numerade Educator
08:10

Problem 31

The conducting rod shown in Fig. $31-42$ has FIGURE $31-43=$ length $L$ and is being pulled along $\quad$ Problem $30 .$ horizontal, frictionless conducting rails at a constant velocity $\vec{v} .$ The rails are connected at one end with a metal strip. A uniform magnetic field $\vec{B}$, directed out of the pagc, fills the region in which the rod moves. Assume that $L=10$ $\mathrm{cm}, v=5.0 \mathrm{~m} / \mathrm{s}$, and $B=1.2 \mathrm{~T}$. (a) What is the magnitude of the emf induced in the rod? (b) What is the magnitude and direction (clockwise or counterclockwise) of the current in the conducting loop? Assume that the resistance of the rod is $0.40 \Omega$ and that the resistance of the rails and metal strip is negligibly small. (c) At what rate is thermal energy added to the rod? (d) What magnitude of force must be applied to the rod by an external agent to maintain its motion? (c) At what rate does this external agent do work on the rod? Compare this answer with the answer to (c).

Vishal Gupta
Vishal Gupta
Numerade Educator
04:32

Problem 32

Two straight conducting rails form a right angle where their cnds are joined. $\mathrm{A}$ conducting bar in contact with the rails starts at the vertex at time $t=0$ and moves with a constant velocity of magnitude $5.20 \mathrm{~m} / \mathrm{s}$ along them, as shown in Fig. 31-44. A magnetic field FIGURE $31-44=$ of magnitude $B=0.350 \mathrm{~T}$ is di- Problem 32 . rected out of the page. Calculate (a) the flux through the triangle formed by the rails and bar at $t=3.00$ s and (b) the magnitude of emf around the triangle at that time. (c) If we write the $\mathrm{emf}$ as $\mathscr{B}=a t^{n}$, where $a$ and $n$ are constants, what is the value of $n ?$

Amit Srivastava
Amit Srivastava
Numerade Educator
06:16

Problem 33

Rod on Conducting Rails Two Figure $31-45$ shows a rod of length $L$ caused to move at constant speed $v$ along horizontal conducting rails. The magnetic field in which the rod moves is not uniform but is provided by a current $i$ in a long wire parallel to the rails. Assume that $v=5.00$ $\mathrm{m} / \mathrm{s}, a=10.0 \mathrm{~mm}, L=10.0 \mathrm{~cm}$, and
$i=100$ A. (a) Calculate the magnitude of the emf induced in the rod.
(b) What is the magnitude of the current in the conducting loop? Assume that the resistance of the rod is $0.400 \Omega$ and that the resistance of the rails and the strip that connects them at the right is negligible. (c) At what rate is internal energy added to the rod? (d) What magnitude of force must be applied to the rod by an external agent to maintain its motion? (e) At what rate does this external agent do work on the rod? Compare this answer to that for (c).

Amit Srivastava
Amit Srivastava
Numerade Educator
05:31

Problem 34

Two Circular Regions Figure $31-46$ shows two circular regions $R_{1}$ and $R_{2}$ with radii $r_{1}=20.0 \mathrm{~cm}$ and $r_{2}=30.0 \mathrm{~cm} .$ In $R_{1}$ there is a uniform magnetic field of magnitude $B_{1}=50.0 \mathrm{~m} \mathrm{~T}$ into the page, and in $R_{2}$ there is a uniform magnetic field of magnitude $B_{2}=75.0 \mathrm{mT}$ out of the page (ignore any fringing of these fields). Both fields are decreasing at the rate of $8.50 \mathrm{mT} / \mathrm{s}$. Calculate the integral $\oint \vec{E} \cdot d \vec{s}^{*}$ for each of the three dashed paths.

Vishal Gupta
Vishal Gupta
Numerade Educator
06:38

Problem 35

Long Solenoid A long solenoid has a diameter of $12.0 \mathrm{~cm}$. When a current $i$ exists in its windings, a uniform magnetic field of magnitude $B=30.0 \mathrm{~m} \mathrm{~T}$ is produced in its interior. By decreasing $i .$ the field is caused to decrease at the rate of $6.50 \mathrm{mT} / \mathrm{s}$. Calculate the magnitude of the induced electric field (a) $2.20 \mathrm{~cm}$ and (b) $8.20 \mathrm{~cm}$ from the axis of the solenoid.

Vishal Gupta
Vishal Gupta
Numerade Educator
03:53

Problem 36

Magnet Lab Early in 1981 the Francis Bitter National Magnet Laboratory at M.I.T. commenced operation of a 3.3-cm-diameter cylindrical magnet that produces a $30 \mathrm{~T}$ field, then the world's largest steady-state field. The field magnitude can be varied sinusoidally between the limits of $29.6$ and $30.9 \mathrm{~T}$ at a frequency of $15 \mathrm{~Hz}$. When this is donc, what is the maximum value of the magnitude of the induced electric field at a radial distance of $1.6 \mathrm{~cm}$ from the axis? (Hint: See Touchstone Example $31-3 .$ )

Amit Srivastava
Amit Srivastava
Numerade Educator
02:41

Problem 37

Drop to Zero Prove that the electric field $\vec{E}$ in a charged parallel-plate capacitor cannot drop abruptly to zero (as is suggested at point $a$ in Fig. $31-47$ ), as one moves perpendicular to the ficld, say, along the horizontal arrow in the figurc. Fringing of the ficld lines always occurs in actual capacitors, which means that $\vec{E}$ approaches zero in a continuous and gradual way (see Problem 35 in Chapter 30 ). (Hint: Apply Faraday's law to the rectangular path shown by the FIGURE $31-47$ w Problem dashed lines). $37 .$

Amit Srivastava
Amit Srivastava
Numerade Educator
03:09

Problem 38

Charging Capacitor Touchstone Example $31-4$ describes the charging of a parallel-plate capacitor with circular plates of radius $55.0 \mathrm{~mm}$. At what two radii $r$ from the central axis of the capacitor is the magnitude of the induced magnetic field equal to $50 \%$ of its maximum value?

Amit Srivastava
Amit Srivastava
Numerade Educator
01:04

Problem 39

The induced magnetic field $6.0 \mathrm{~mm}$ from the central axis of a circular parallel-plate capacitor and between the plates has magnitude of $2.0 \times 10^{-7} \mathrm{~T}$. The plates have radius $3.0 \mathrm{~mm}$. At what rate $|d \vec{E} / d t|$ is the electric field magnitude between the plates changing?

Raj Bala
Raj Bala
Numerade Educator
08:43

Problem 40

Parallel-Plate Capacitor Suppose that a parallel-plate capacitor has circular plates with radius $R=30 \mathrm{~mm}$ and a plate separation of $5.0 \mathrm{~mm}$. Suppose also that a sinusoidal potential difference with a maximum value of $150 \mathrm{~V}$ and a frequency of $60 \mathrm{~Hz}$ is applied across the plates. That is,
$$
\Delta V=(150 \mathrm{~V}) \sin [2 \pi(60 \mathrm{~Hz}) t]
$$
(a) Find $B^{\max }(R)$, the maximum value of the magnitude of the induced magnetic field that occurs at $r=R .$ (b) Plot $B^{\max }(r)$ for $0<$ $r<10 \mathrm{~cm}$

Amit Srivastava
Amit Srivastava
Numerade Educator
03:17

Problem 41

Uniform Electric Flux Figure $31-48$ shows a circular region of radius $R=3.00 \mathrm{~cm}$ in which a uniform electric flux is directed out of the page. The total electric flux enclosed by the region is given by $\Phi^{\text {elec }}=(3.00 \mathrm{~m} \mathrm{~V} \cdot \mathrm{m} / \mathrm{s}) t$, where $t$ is time. What is the magnitude of the magnetic field that is induced at radial distances (a) $2.00$ $\mathrm{cm}$ and $(\mathrm{b}) 5.00 \mathrm{~cm} ?$

Amit Srivastava
Amit Srivastava
Numerade Educator
03:01

Problem 42

Nonuniform Electric Flux Figure 31-48 through 44 , and shows a circular region of radius $R=3.00 \mathrm{~cm} \quad 57,59$, and 60 . in which an electric flux is directed out of the page. The flux encircled by a concentric circle of radius $r$ is given by $\Phi^{\text {elee }}=(0.600 \mathrm{~V} \cdot \mathrm{m} / \mathrm{s})(r / R) t$, where $r \leq R$ and $t$ is time. What is the magnitude of the induced magnetic field at radial distances
(a) $2.00 \mathrm{~cm}$ and
(b) $5.00 \mathrm{~cm}$ ?

Amit Srivastava
Amit Srivastava
Numerade Educator
02:12

Problem 43

Uniform Electric Field In Fig. $31-48$, a uniform electric field is directed out of the page within a circular region of radius $R=$ $3.00 \mathrm{~cm} .$ The magnitude of the electric field is given by $E=(4.5 \times$ $\left.10^{-3} \mathrm{~V} / \mathrm{m} \cdot \mathrm{s}\right) t$, where $t$ is time. What is the magnitude of the induced magnetic field at radial distances (a) $2.00 \mathrm{~cm}$ and (b) $5.00 \mathrm{~cm}$ ?

Amit Srivastava
Amit Srivastava
Numerade Educator
02:34

Problem 44

Nonuniform Electric Field In Fig. $31-48$, an clectric ficld is directed out of the page within a circular region of radius $R=$ $3.00 \mathrm{~cm}$. The magnitude of the electric field is given by $E=$ $(0.500 \mathrm{~V} / \mathrm{m}+\mathrm{s})(1-r / R) t$, where $t$ is
the time and $r$ is the radial distance $(r \leq R)$. What is the magnitude of the induced magnetic field at radial $L$ $H_{0------1}{w}$ distances (a) $2.00 \mathrm{~cm}$ and (b) $5.00$ $\mathrm{cm} ?$

Amit Srivastava
Amit Srivastava
Numerade Educator
01:48

Problem 45

Discharging Capacitor A capacitor with square plates of edge length $L$ is being discharged by a current of $0.75$ A. Figure $31-49$ is a Fic head-on view of one of the plates Pro from inside the capacitor. A dashed rectangular path is shown. If $L=12 \mathrm{~cm}$, $W=4.0 \mathrm{~cm}$, and $H=2.0 \mathrm{~cm}$, what is the value of $\{\vec{B} \cdot d \vec{s}$ around the dashed path?

Amit Srivastava
Amit Srivastava
Numerade Educator
03:35

Problem 46

Charging Capacitor The circuit in Fig. $31-50$ consists of switch $\mathrm{S}$, a $12.0 \mathrm{~V}$ ideal battery, a $20.0 \mathrm{M} \Omega$ resistor, and an air-filled capacitor. The capacitor has parallel circular plates of radius $5.00 \mathrm{~cm}$, separated by $3.00 \mathrm{~mm}$. At time $t=0 \mathrm{~s}$, switch $\mathrm{S}$ is closed to begin charging the capacitor. The electric field between the plates is uniform. At $t=250 \mu \mathrm{s}$, what is the magnitude of the magnetic field within the capacitor, at radial distance $3.00 \mathrm{~cm}$ ?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
01:56

Problem 47

Prove That Displacement Prove that the displacement current in a parallel-plate capacitor of capacitance $C$ can be written as $i^{\text {dis }}=$ $C(d \Delta V / d t)$, where $\Delta V$ is the potential difference between the plates.

Amit Srivastava
Amit Srivastava
Numerade Educator
02:15

Problem 48

At What Rate At what rate must the potential difference between the plates of a parallel-plate capacitor with a $2.0 \mu \mathrm{F}$ capacitance be changed to produce a displacement current of $1.5 \mathrm{~A}$ ?

Carlos Henrique De Lima
Carlos Henrique De Lima
Numerade Educator
01:31

Problem 49

Current Density For the situation of Touchstone Example $31-4$, show that the magnitude of the current density of the displacement current is $J^{\text {dis }}=\varepsilon_{0}(d E / d t)$ for $r \leq R$.

Amit Srivastava
Amit Srivastava
Numerade Educator
01:50

Problem 50

Being Discharged A parallel-plate capacitor with circular plates of radius $0.10 \mathrm{~m}$ is being discharged. A circular loop of radius $0.20 \mathrm{~m}$ is concentric with the capacitor and halfway between the plates. The displacement current through the loop is $2.0 \mathrm{~A}$. At what rate is the magnitude of the electric field between the plates changing?

Amit Srivastava
Amit Srivastava
Numerade Educator
03:33

Problem 51

Displacement Current As a parallel-plate capacitor with circular plates $20 \mathrm{~cm}$ in diameter is being charged, the current density of the displacement current in the region between the plates is uniform and has a magnitude of $20 \mathrm{~A} / \mathrm{m}^{2}$. (a) Calculate the magnitude $B$ of the magnetic field at a distance $r=50 \mathrm{~mm}$ from the axis of symmetry of this region. (b) Calculate $d E / d t$ in this region.

Prabhu Ramji
Prabhu Ramji
Numerade Educator
03:30

Problem 52

Electric Field The magnitude of the electric field between the two circular parallel plates in Fig. $31-51$ is $E=(4.0 \times$ $\left.10^{5} \mathrm{~V} \cdot \mathrm{m}\right)-\left(6.0 \times 10^{4} \mathrm{~V} \cdot \mathrm{m} / \mathrm{s}\right) t$, with $E$ in
volts per meter and $t$ in seconds. At $t=$ $0 \mathrm{~s}$, the field is upward as shown. The plate area is $4.0 \times 10^{-2} \mathrm{~m}^{2}$. For $t \geq 0 \mathrm{~s}$, (a) what FiGURE $31-51$ =
are the magnitude and direction of the Problem 52 . displacement current between the plates and (b) is the direction of the induced magnetic field clockwise or counterclockwise around the plates?

Amit Srivastava
Amit Srivastava
Numerade Educator
04:58

Problem 53

Magnitude of Electric Field The magnitude of a uniform electric field collapses to zero from an initial strength of $6.0 \times 10^{5} \mathrm{~N} / \mathrm{C}$ in a time of $15 \mu \mathrm{s}$ in the manner shown in Fig. $31-52 .$ Calculate the amount of displacement current, $|i|$, through a $1.6 \mathrm{~m}^{2}$ area perpendicular to the field, during each of the time intervals, $a, b$, and $c$ shown on the graph. (Ignore the behavior at the ends of the intervals.)

Vishal Gupta
Vishal Gupta
Numerade Educator
01:12

Problem 54

Displacement Current Two A parallel-plate capacitor with circular plates is being charged. Consider a circular loop centered on the central axis between the plates. The loop radius is $0.20 \mathrm{~m}$, the plate radius is $0.10 \mathrm{~m}$, and the displacement current through the loop is $2.0 \mathrm{~A}$. What is the rate at which the magnitude of the electric field between the plates is changing?

Amit Srivastava
Amit Srivastava
Numerade Educator
02:31

Problem 55

Square Plates A parallel$\begin{array}{llll}\text { plate capacitor } & \text { has } & \text { square }\end{array}$ plates $1.0 \mathrm{~m}$ on a side as shown in Fig. $31-53$. A current of $2.0 \mathrm{~A}$ charges the capacitor, producing a uniform electric field $\vec{E}$ \begin{tabular}{l} between the plates, with $\vec{E}$ per- \\ \hline \end{tabular} Edge view $\quad$ Top view pendicular to the plates. (a) What is the displacement curFiGURE $31-53=$ Problem 55 . rent $i^{\text {dis }}$ through the region between the plates? (b) What is $d E / d t$ in this region? (c) What is the displacement current through the square dashed path between the plates? (d) What is $\oint \vec{B} \cdot d s^{\text {' }}$ around this square dashed path?

Amit Srivastava
Amit Srivastava
Numerade Educator
02:59

Problem 56

Consider a Loop A capacitor with parallel circular plates of radius $R$ is discharging via a current of $12.0$ A. Consider a loop of radius $R / 3$ that is centered on the central axis between the plates.
(a) How much displacement current is encircled by the loop? The maximum induced magnetic ficld has a magnitude of $12.0 \mathrm{mT}$.
(b) At what radial distance from the central axis of the plate is the magnitude of the induced magnetic field $3.00 \mathrm{mT}$ ?

Amit Srivastava
Amit Srivastava
Numerade Educator
02:30

Problem 57

Uniform Displacement-Current Density. Figure $31-48$ shows a circular region of radius $R=3.00 \mathrm{~cm}$ in which a displacement current is directed out of the page. The magnitude of the displacement current has a uniform density $J^{\text {dis }}=6.00 \mathrm{~A} / \mathrm{m}^{2} .$ What is the magnitude of the magnetic field due to the displacement current at radial distances (a) $2.00 \mathrm{~cm}$ and (b) $5.00 \mathrm{~cm}$ ?

Amit Srivastava
Amit Srivastava
Numerade Educator
10:08

Problem 58

Actual and Displacement $\quad P_{\bullet}$ Figure $31-54 a$ shows current $i$ that is produced in a wire of resistivity $1.62 \times 10^{-8} \Omega \cdot \mathrm{m}$ in the direction indicated. The magni- $\quad(a)$ tude of the current versus time $t$ is shown in Fig. $31-54 b$. Point $P$ is at radius $9.00 \mathrm{~mm}$ from the wire's center. Determine the magnitude of the magnetic field at point $P$ due to the real current $i$ in the wire at (a) $t_{1}=20 \mathrm{~ms}$, (b) $t_{2}=40 \mathrm{~ms}$, (c) $t_{3}=60 \mathrm{~ms}$, and
(d) $t_{4}=70 \mathrm{~ms}$. Next, assume that
the electric field driving the cur- rent is confined to the wire. FIGURE $31-54$ a Problem 58 . Then determine the magnitude of the magnetic field at point $P$ due to the displacement current $i_{-1 s}$ in the wire at (e) $t_{1}=20 \mathrm{~ms}$, (f) $t_{2}=40 \mathrm{~ms},(\mathrm{~g}) t_{3}=60 \mathrm{~ms}$, and (h)
$t_{4}=70 \mathrm{~ms}$. (i) When both magnetic fields are present at point $P .$ what are their directions in Fig. $31-54 a ?$

Amit Srivastava
Amit Srivastava
Numerade Educator
03:18

Problem 59

Nonuniform Displacement-Current Density. Figure $31-48$ shows a circular region of radius $R=3.00 \mathrm{~cm}$ in which a displacement current is directed out of the page. The displacement current has a density of magnitude $J^{\text {dis }}=\left(4.00 \mathrm{~A} / \mathrm{m}^{2}\right)(1-r / R)$, where $r$ is the radial distance $r \leq R$. What is the magnitude of the magnetic field due to the displacement current at radial distances
(a) $2.00 \mathrm{~cm}$ and
(b) $5.00 \mathrm{~cm} ?$

Amit Srivastava
Amit Srivastava
Numerade Educator
01:30

Problem 60

Uniform Displacement Current. Figure $35-48$ shows a circular region of radius $R=3.00 \mathrm{~cm}$ in which a uniform displacement current $i^{\text {dis }}=0.500 \mathrm{~A}$ is directed out of the page. What is the magnitude of the magnetic field due to the displacement current at radial distances (a) $2.00 \mathrm{~cm}$ and $(\mathrm{b}) 5.00 \mathrm{~cm} ?$

Amit Srivastava
Amit Srivastava
Numerade Educator
03:03

Problem 61

Rolling a Sheet of Paper Imagine rolling a sheet of paper into a cylinder and placing a bar magnet near its end as shown in Fig. $31-55 .$ (a) FIGURE $31-55=$ Problem 61 . Sketch the magnetic field lines that pass through the surface of the cylinder. (b) What can you say about the sign of $\vec{B} \cdot d \vec{A}$ for cvery area $d \vec{A}$ on the surface? (c) Does this result contradict Gauss' law for magnetism? Explain.

Amit Srivastava
Amit Srivastava
Numerade Educator
02:04

Problem 62

Die Suppose the magnetic flux at each of five faces of a die (singular of "dice") is given by \Phi'mag $=\pm N \mathrm{~Wb}$, where $N(=1$ to 5$)$ is the number of spots on the face. The flux is positive (outward) for $N$ even and negative (inward) for $N$ odd. What is the flux at the sixth face of the die? Is it directed in or out?

Amit Srivastava
Amit Srivastava
Numerade Educator
02:06

Problem 63

Right Circular Cylinder A Gaussian surface in the shape of a right circular cylinder with end caps has a radius of $12.0 \mathrm{~cm}$ and a length of $80.0 \mathrm{~cm} .$ One end encircles an inward magnetic flux of $25.0 \mu \mathrm{Wb}$. At the other end there is a uniform magnetic field of $1.60 \mathrm{mT}$, normal to the surface and directed outward. What is the net magnetic flux at the curved surface?

Amit Srivastava
Amit Srivastava
Numerade Educator
02:21

Problem 64

Weird Shape Figure $31-56$ shows a closed surface. Along the flat top face, which has a radius of $2.0 \mathrm{~cm}$, a magnetic field $\vec{B}$ of magnitude $0.30 \mathrm{~T}$ is directed outward. Along the flat bottom face, a magnetic flux of $0.70 \mathrm{mWb}$ is directed outward. What are (a) the magnitude and (b) the net magnetic flux at the curved part of the surface?

Amit Srivastava
Amit Srivastava
Numerade Educator
04:12

Problem 65

Power from a Tether A few years ago, the space shuttle $\mathrm{Co}$ lumbia tricd an cxperiment with a tethered satellite. The satellite was released from the shuttle and slowly reeled out on a long conducting cable as shown in Fig. $31-57$ (not to scale). For this problem we will make the following approximations:

The shuttle is moving at a constant velocity. The Farth's magnetic ficld is constant and uniform.

The line of the tether, the velocity of the system, and the magnetic field are all perpendicular to each other.
The Earth's field produces an FiGURE 31-57 = Problem 65 . emf from one end of the cable to the other. The idea is to use a system like this to generate electric power in space more efficiently than with solar panels.
(a) Explain why a voltage difference is produced.
(b) If the Earth's magnetic field is given by a magnitude $\vec{B}$, the shuttle- satellite system is moving with a velocity $\vec{v}$, and the tether has a length $L$, calculate the magnitude of the emf $\mathscr{E}$ from one end of the tether to the other.
(c) At the shuttle's altitude, the Earth's field is about $0.3$ gauss and the shuttle's speed is about $7.5 \mathrm{~km} / \mathrm{s}$. The tether is $20 \mathrm{~km}$ long (!). What is the expected potential difference in volts?
(d) At the altitude of the shuttle, the thin atmosphere is lightly ionized, allowing a current of about $0.5$ amps to flow from the satellite hack to the shuttle through the thin air. What is the resistance of the $20 \mathrm{~km}$ of ionized air?

Amit Srivastava
Amit Srivastava
Numerade Educator
08:46

Problem 66

Building a Generator The apparatus shown in Fig. $31-58$ can be used to build a motor. This device can also be used to build a generator that will produce a voltagc. (a) Explain the setup that one would use to make a motor and explain how it works. Do the
same for the generator. (b) Estimate the maximum voltage that would be produced if you cranked the generator by hand. (Hint:
As a comparison for estimating the strength of the bar magnet, the Earth's magnetic field FIGURE $31-58=$ Problem $66 .$ at our location is about $0.4$ gauss.)

Linda Winkler
Linda Winkler
Numerade Educator
02:12

Problem 67

Faraday's Law Faraday's law describes the emf produced by magnetic fields in a variety of circumstances. State and discuss Faraday's law, being careful to include a discussion of different physical situations that may be described by the statement of the law.

Amit Srivastava
Amit Srivastava
Numerade Educator
03:06

Problem 68

Magnetic Field, Force, and Torque Figure $31-59$ shows two long. current-carrying wires and a bar magnet. At the right is shown a compass specifying set of direction labels. For each of the vectors
(a) - (e) below, select the direction label that best gives the direction of the item. If the magnitude of the item is zero, write 0 . If none of the directions are correct, write $\mathrm{N}$.
(a) The magnetic field due to the lower wire at the center of the upper wire
(b) The force on the lower wire due to the magnetic field from the upper wire
(c) The net torque acting on the upper wire
(d) The magnetic field due to the currents at the center of the magnet
(e) The net force acting on the lower wire due to the bar magnet

Amit Srivastava
Amit Srivastava
Numerade Educator
03:56

Problem 69

$B$ Increases in Time In Fig. $31-60 a$, a uniform magnetic field $\vec{B}$ increases in magnitude with time $t$ as given by Fig. $31-60 b .$ A circular conducting loop of area $8.0 \times 10^{-4} \mathrm{~m}^{2}$ lies in the field, in the plane of the page. The amount of charge $q$ that has passed point $A$ on the loop is given in Fig. $31-60 c$ as a function of $t .$ What is the loop's resistance?

Amit Srivastava
Amit Srivastava
Numerade Educator
04:17

Problem 70

Circular Loop Around a Solenoid In Fig. $31-61 a$, a circular loop of wire is concentric with a solenoid and lies in a plane that is perpendicular to the solcnoid's central axis. The loop has radius $6.00 \mathrm{~cm}$. The solenoid has radius $2.00 \mathrm{~cm}$, consists of 8000 turns per meter, and has a current $i_{\text {sol }}$ that varies with time $t$ as given in Fig. $31-61 b$. Figure $31-61 c$ shows, as a function of time, the energy $E^{\text {thermal }}$ that is transformed to thermal energy in the loop. What is the loop's resistance?

Amit Srivastava
Amit Srivastava
Numerade Educator
05:00

Problem 71

Magnitudes and Direction Figure $31-62 a$ shows a wire that forms a rectangle and has a resistance of $5.0 \mathrm{~m} \Omega$. Its interior is split into three equal areas with different magnetic fields $\vec{B}_{1}, \vec{B}_{2}$, and $\vec{B}_{3}$ that are either directly out of or into the page, as indicated. The fields are uniform within each region. Figure $31-62 b$ gives the change in the $z$ components $B_{z}$ of the three fields with time $t$. What are the magnitude and direction of the current induced in the wire?

Amit Srivastava
Amit Srivastava
Numerade Educator
04:27

Problem 72

Two Concentric Regions Figure $31-63 a$ shows two concentric circular regions in which uniform magnetic fields can change. Region 1, with radius $r_{1}=1.0 \mathrm{~cm}$, has an outward magnetic field $\vec{B}_{1}$ that is increasing in magnitude. Region 2, with radius $r_{2}=2.0 \mathrm{~cm}$, has an outward magnetic field $\vec{B}_{2}$ that may also be changing. Imagine that a conducting ring of radius $R$ is centered on the regions and then the emf $\mathscr{E}$ around the ring is determined. Figure $31-63 b$ gives emf 8 as a function of the square of the ring's radius, $R^{2}$, to the outer edge of region 2. What are the rates of $B$ -field magnitude change (a) $d B_{1} / d t$ and (b) $d B \sqrt{d d t ?}$
(c) Is the magnitude of $\vec{B}_{2}$ increasing, decreasing, or remaining constant?

Amit Srivastava
Amit Srivastava
Numerade Educator
03:51

Problem 73

Pulled at Constant Speed Figure $31-64 a$ shows a rectangular conducting loop of resistance $R=0.020 \Omega$, height $H=1.5 \mathrm{~cm}$, and length $D=2.5 \mathrm{~cm}$ being pulled at constant speed $v=40 \mathrm{~cm} / \mathrm{s}$ through two regions of uniform magnetic ficld. Figure $31-64 b$ gives the current $i$ induced in the loop as a function of the position $x$ of the right side of the loop. For example, a current of $3.0 \mu \mathrm{A}$ is induced clockwise as the loop enters region $1 .$ What are the magnitudes and directions of the magnetic field in (a) region 1 and (b) region $2 ?$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:54

Problem 74

Plane Loop A plane loop of wire consisting of a single turn of area $8.0 \mathrm{~cm}^{2}$ is perpendicular to a magnetic field that increases uniformly in magnitude from $0.50 \mathrm{~T}$ to $2.5 \mathrm{~T}$ in a time of $1.0 \mathrm{~s}$. What is the resulting induced current if the coil has a total resistance of $2.0 \Omega$ ?

Vishal Gupta
Vishal Gupta
Numerade Educator
03:02

Problem 75

At What Rate Must $B$ Change The plane of a rectangular coil of dimensions $5.0 \mathrm{~cm}$ by $8.0 \mathrm{~cm}$ is perpendicular to the direction of magnetic field $B$. If the coil has 75 turns and a total resistance of $8.0 \Omega$, at what rate must the magnitude of $B$ change in order to induce a current of $0.10 \mathrm{~A}$ in the windings of the coil?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:48

Problem 76

Rod on Rails 3 In the arrangement shown in Fig. $31-65$, a conducting rod rolls to the right along parallel conducting rails connected on one end by a $6.0 \Omega$ resistor. A $2.5 \mathrm{~T}$ magnetic field is directed into the paper. Let $L=1.2 \mathrm{~m}$. Neglect the mass of the bar and friction. (a) Calculate the applied force required to move the bar to the right at a constant speed of $2.0 \mathrm{~m} / \mathrm{s}$. (b) At what rate is energy dissipated in the resistor?

Amit Srivastava
Amit Srivastava
Numerade Educator
06:58

Problem 77

An Engineer An engineer has designed a setup with a small pickup coil placed in the center of a large ficld coil as shown in Fig. $31-66 .$ Both coils have many turns of conducting wire. The field coil produces a magnetic field that is proportional in magnitude to the amount of current flowing through its wires. The pickup coil is smaller and its many turns can sense or "pick up" the changing magnetic field in the field coil. The pickup coil produces an emf that is proportional in magnitude to the rate of change of the magnetic ficld and the angle $\phi .$ Here $\phi$ is the angle between the normal to the ficld coil and the normal to the pickup coil. You have been hired as a consultant to check on the reliability of the engineer's work. You figure out how to use Faraday's law along with proportional reasoning to check on the validity of the results that have been reported without doing any formal calculations or measurements. Sketches from the engineer's notebook are shown in Fig. $31-66 b .$ (a) Look at the graph pair in Fig. $31-66 b$. Sketch the measured emf induced in the pickup coil if the engineer has adjusted the scope so the maximum emf is the first positive grid line and the minimum emf is on the first negative grid line. Assume that the normal to each of the coils is pointing in the same direction. (b) According to the engineer's notebook, she fed exactly the same pattern of current to the field coil but she turned the pickup coil so its normal makes an angle of $+45^{\circ}$ with respect to the normal to the plane of the field coil. Carefully sketch the pattern of emf observed in the pickup coil. What is the maximum and minimum amplitude of the emf in "grid" units? (c) What happens when she flips the pickup coil over around so its normal is $180^{\circ}$ from the normal to the ficld coil? Sketch the emf and use the correct signs for the values of the induced emf for this situation. Explain the reasons for the shape and magnitude of your sketch in each case.

Amit Srivastava
Amit Srivastava
Numerade Educator
06:58

Problem 78

Engineer Task 2 You are still double-checking the work of the engincer from Problem 77. Consider the graph shown in Fig. $31-67 .$
(a) What should our honest and competent engineer have reported for the pattern of emf values as a function of time? Assume that. once again the normal to the pickup coil is in the same direction as the normal to the field coil. Please take care to sketch not only the shape of the emf graph but also its proper magnitude using the same gain setting on the oscilloscope as you did in Problem $77 .$ Use a solid line for your sketch. (b) Suppose the engineer reduced the number of turns in the pickup coil by a factor of 2 and redid the measurements. Sketch a new graph showing the shape and proper magnitudes for the expected pickup coil emf using a dashed line. Explain the reasons for the shape and magnitude of your sketch in each case.

Amit Srivastava
Amit Srivastava
Numerade Educator
06:58

Problem 79

Engineer Task 3 You are still double-checking the work of the engineer from Problem $31-77$. Assume that the number of turns in both the field and pickup coils is the same as in that problem, as is the oscilloscope setting. Consider the graph shown in Fig. $31-68 .(\mathrm{a})$ What should our honest and competent engineer have reported for the pattern of current fed into the field coil as a function of time? Assume that once again the normal to the pickup coil is in the same direction as the normal to the field coil. Please take care to sketch not only the shape of the emf graph but also its proper magnitude using the same gain setting on the oscilloscope as you did in Problem $31-77$. Use a solid line for your sketch. (b) Suppose the engineer reduced the number of turns in the pickup coil by a factor of 2 and redid the measurements. Sketch a new graph showing the shape and proper magnitudes for emf in the field coil using a dashed line. Explain the reasons for the shape and magnitude of your sketch in each case.

Amit Srivastava
Amit Srivastava
Numerade Educator
06:37

Problem 80

Engineer Task 4 You are still double-checking the work of the engineer from Problem 31-77. Assume that the number of turns in both the field and pickup coils is the same as in that problem. Consider the graph shown in Fig. $31-69 .$ What should our honcst and competent engineer have reported for the pattern of emf induced in the pickup coil if the oscilloscope gain is adjusted to give a maximum value of emf of $+2$ oscilloscope grid units and a minimum value of $-2$ oscilloscope units? (Hint: What is the derivative of the sine function?) Explain the reason for the shape and magnitude of your sketch in each case.

Vishal Gupta
Vishal Gupta
Numerade Educator
04:16

Problem 81

Ring of Copper Figure $31-70$ shows a ring of copper with its plane perpendicular to the axis of the nearby rod-shaped magnet. In which of the following situations will a current be induced in the ring? Choose all correct an-
swers.
(a) The magnet is moved FIGURE $31-70=$ Problem 81 . horizontally toward the left.
(b) The ring is moved away from the magnet.
(c) The ring is rotated around any of its diameters.
(d) The magnet is moved up or down.
(e) The ring is rotated around its center in the plane in which it lies.

Amit Srivastava
Amit Srivastava
Numerade Educator