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Fundamentals of Physics, Volume 2

David Halliday & Robert Resnick & Jearl Walker

Chapter 30

Induction and Inductance - all with Video Answers

Educators

Chapter Questions

04:05

Problem 1

Fig. 30.13, a circular loop of wire $10 \mathrm{~cm}$ in diameter (seen edge-on) is placed with its normal $\vec{N}$ 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 $\vec{N}$ rotates in a cone about the field direction at the rate $100 \mathrm{rev} / \mathrm{min}$; angle $\theta$ remains unchanged during the process. What is the emf induced in the loop?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:54

Problem 2

A certain 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 $0.800 \mathrm{~T}$ magnetic field. When released, the radius of the loop starts to shrink at an instantaneous rate of $75.0 \mathrm{~cm} / \mathrm{s}$. What emf is induced in the loop at that instant?

Vishal Gupta
Vishal Gupta
Numerade Educator
05:50

Problem 3

In Fig. 30.14, a $120-$ turn coil of radius $1.8 \mathrm{~cm}$ and resistance $5.3 \Omega$ is coaxial with a solenoid of 220 turns $/ \mathrm{cm}$ and diameter $3.2 \mathrm{~cm}$. The solenoid current drops from $1.5 \mathrm{~A}$ to zero in time interval $\Delta t=25 \mathrm{~ms}$. What current is induced in the coil during $\Delta r$ ?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:43

Problem 4

A wire loop of radius $12 \mathrm{~cm}$ and resistance $8.5 \Omega$ is located in a uniform magnetic field $\vec{B}$ that changes in magnitude as given in Fig. 30.15. The vertical axis scale is set by $B_s=0.50 \mathrm{~T}$, and the horizontal axis scale is set by $t_s=6.00 \mathrm{~s}$. The loop's plane is perpendicular to $\vec{B}$. What emf is induced in the loop during time intervals (a) 0 to $2.0 \mathrm{~s}$, (b) $2.0 \mathrm{~s}$ to $4.0 \mathrm{~s}$, and (c) $4.0 \mathrm{~s}$ to $6.0 \mathrm{~s}$ ?

Averell Hause
Averell Hause
Carnegie Mellon University
02:59

Problem 5

In Fig. 30.16, a wire forms a closed circular loop, of 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 $5.0 \mathrm{~A}$ rightward. Thereafter, the current changes according to $i=5.0 \mathrm{~A}-\left(2.0 \mathrm{~A} / \mathrm{s}^2\right) t^2$. (The straight wire is insulated; so there is no electrical contact between it and the wire of the loop.) What is the magnitude of the current induced in the loop at times $t>0$ ?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
02:48

Problem 6

Figure 30.17a shows a circuit consisting of an ideal battery with emf $\mathscr{E}=6.00 \mu \mathrm{V}$, a resistance $R$, and a small wire loop of area $5.0 \mathrm{~cm}^2$. For the time interval $t=10 \mathrm{~s}$ to $t=20 \mathrm{~s}$, an external magnetic field is set up throughout the loop. The field is uniform, its direction is into the page in Fig. 30.17a, and the field magnitude is given by $B=a t$, where $B$ is in teslas, $a$ is a constant, and $t$ is in seconds. Figure 30.17b gives the current $i$ in the circuit before, during, and after the external field is set up. The vertical axis scale is set by $i_s=2.0 \mathrm{~mA}$. Find the constant $a$ in the equation for the field magnitude.

Keshav Singh
Keshav Singh
Numerade Educator
04:58

Problem 7

In Fig. 30.18, the magnetic flux through the loop increases according to the relation $\Phi_B=6.0 t^2+7.0 t$, where $\Phi_B$ is in milliwebers and $t$ is in seconds. (a) What is the magnitude of the emf induced in the loop when $t=2.0 \mathrm{~s} ?$ (b) Is the direction of the current through $R$ to the right or left?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
05:44

Problem 8

A uniform magnetic field $\vec{B}$ is perpendicular to the plane of a circular loop of diameter $10 \mathrm{~cm}$ formed from wire of diameter $2.5 \mathrm{~mm}$ and resistivity $1.69 \times 10^{-8} \Omega \cdot \mathrm{m}$. At what rate must the magnitude of $\vec{B}$ change to induce a $10 \mathrm{~A}$ current in the loop?

Vishal Gupta
Vishal Gupta
Numerade Educator
04:43

Problem 9

A small loop of area $6.8 \mathrm{~mm}^2$ is placed inside a long solenoid that has 854 turns/cm and carries a sinusoidally varying current $i$ of amplitude $1.28 \mathrm{~A}$ and angular frequency $212 \mathrm{rad} / \mathrm{s}$. The central axes of the loop and solenoid coincide. What is the amplitude of the emf induced in the loop?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
03:04

Problem 10

Figure 30.19 shows a closed loop of wire that consists of a pair of equal semicircles, of radius $3.7 \mathrm{~cm}$, lying in mutually perpendicular planes. The loop was formed by folding a flat circular loop along a diameter until the two halves became perpendicular to each other. A uniform magnetic field $\vec{B}$ of magnitude $76 \mathrm{mT}$ is directed perpendicular to the fold diameter and makes equal angles (of $45^{\circ}$ ) with the planes of the semicircles. The magnetic field is reduced to zero at a uniform rate during a time interval of $4.5 \mathrm{~ms}$. During this interval, what are the (a) magnitude and (b) direction (clockwise or counterclockwise when viewed along the direction of $\vec{B}$ ) of the emf induced in the loop?

Averell Hause
Averell Hause
Carnegie Mellon University
03:26

Problem 11

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. 30.20. The coil is connected to co-rotating cylinders, against which metal brushes slide to make contact. (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) What value of Nab gives an emf with $\mathscr{E}_0=150 \mathrm{~V}$ when the loop is rotated at $60.0 \mathrm{rev} / \mathrm{s}$ in a uniform magnetic field of $0.500 \mathrm{~T}$ ?

Keshav Singh
Keshav Singh
Numerade Educator
14:28

Problem 12

In Fig. 30.21, a wire loop of lengths $L=40.0 \mathrm{~cm}$ and $W=25.0 \mathrm{~cm}$ lies in a magnetic field $\vec{B}$. What are the (a) magnitude $\mathscr{E}$ and (b) direction (clockwise or counterclockwise-or "none" if $\mathscr{E}$ $=0$ ) of the emf induced in the loop if $\vec{B}=$ $\left(4.00 \times 10^{-2} \mathrm{~T} / \mathrm{m}\right) y \hat{\mathrm{k}}$ ? What are (c) $\mathscr{E}$ and (d) direction if $\vec{B}=\left(6.00 \times 10^{-2} \mathrm{~T} / \mathrm{s}\right) t \hat{\mathrm{k}}$ ? What are (e) $\mathscr{E}$ and (f) direction if $\vec{B}=\left(8.00 \times 10^{-2} \mathrm{~T} / \mathrm{m} \cdot \mathrm{s}\right) y t \mathrm{k}$ ? What are (g) $\mathscr{E}$ and (h) direction if $\left.\vec{B}=\left(3.00 \times 10^{-2} \mathrm{~T} / \mathrm{m} \cdot \mathrm{s}\right) x i\right)^{\prime}$ ? What are (i) $\mathscr{E}$ and (j) direction if $\vec{B}=\left(5.00 \times 10^{-2} \mathrm{~T} / \mathrm{m} \cdot \mathrm{s}\right) y t i$ ?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:59

Problem 13

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 a point in the circuit during the change?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:26

Problem 14

In Fig. $30.22 a$, a uniform magnetic field $\vec{B}$ increases in magnitude with time $t$ as given by Fig. $30.22 b$, where the vertical axis scale is set by $B_s=9.0 \mathrm{mT}$ and the horizontal scale is set by $t_s=3.0 \mathrm{~s}$. 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 $q$ passing point $A$ on the loop is given in Fig. $30.22 c$ as a function of $t$, with the vertical axis scale set by $q_s=6.0 \mathrm{mC}$ and the horizontal axis scale again set by $t_s=3.0 \mathrm{~s}$. What is the loop's resistance?

Averell Hause
Averell Hause
Carnegie Mellon University
08:04

Problem 15

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. 30.23. The loop contains an ideal battery with emf $\mathscr{E}=20.0 \mathrm{~V}$. If the magnitude of the field varies with time according to $B=0.0420-0.870 t$, with $B$ in teslas and $t$ in seconds, what are (a) the net emf in the circuit and (b) the direction of the (net) current around the loop?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:49

Problem 16

Figure $30.24 a$ shows a wire that forms a rectangle $(W=20 \mathrm{~cm}, H=30 \mathrm{~cm})$ and has a resistance of $5.0 \mathrm{~m} \Omega$. Its interior is split into three equal areas, with magnetic fields $\vec{B}_1, \vec{B}_2$, and $\vec{B}_3$. The fields are uniform within each region and directly out of or into the page as indicated. Figure $30.24 \mathrm{~b}$ gives the change in the $z$ components $B_z$ of the three fields with time $t$; the vertical axis scale is set by $B_s=4.0 \mu \mathrm{T}$ and $B_b=-2.5 B_s$, and the horizontal axis scale is set by $t_s=2.0 \mathrm{~s}$. What are the (a) magnitude and (b) direction of the current induced in the wire?

Keshav Singh
Keshav Singh
Numerade Educator
05:11

Problem 17

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 at a constant rate from $200 \mathrm{~A}$ to $-200 \mathrm{~A}$ (a change in direction) in a time of $1.00 \mathrm{~s}$, starting at $t=0$. What is the magnitude of the magnetic field $\vec{B}$ at the center of the small loop due to the current in the large loop at (a) $t=0$, (b) $t=0.500 \mathrm{~s}$, and (c) $t=1.00 \mathrm{~s}$ ? (d) From $t=0$ to $t=1.00 \mathrm{~s}$, is $\vec{B}$ reversed? Because the inner loop is small, assume $\vec{B}$ is uniform over its area. (e) What emf is induced in the small loop at $t=0.500 \mathrm{~s}$ ?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
05:16

Problem 18

In Fig. 30.25, two straight conducting rails form a right angle. A conducting bar in contact with the rails starts at the vertex at time $t=0$ and moves with a constant velocity of $5.20 \mathrm{~m} / \mathrm{s}$ along them. A magnetic field with $B=0.350 \mathrm{~T}$ is directed out of the page. Calculate (a) the flux through the triangle formed by the rails and bar at $t=3.00 \mathrm{~s}$ and (b) the emf around the triangle at that time. (c) If the emf is $\mathscr{E}=a r^n$, where $a$ and $n$ are constants, what is the value of $n$ ?

Km Neeraj
Km Neeraj
Numerade Educator
03:37

Problem 19

An electric generator contains a coil of 100 turns of wire, each forming a rectangular loop $50.0 \mathrm{~cm}$ by $30.0 \mathrm{~cm}$. The coil is placed entirely in a uniform magnetic field with magnitude $B=3.50 \mathrm{~T}$ and with $\vec{B}$ initially perpendicular to the coil's plane. What is the maximum value of the emf produced when the coil is spun at $1000 \mathrm{rev} / \mathrm{min}$ about an axis perpendicular to $\vec{B}$ ?

Vishal Gupta
Vishal Gupta
Numerade Educator
05:54

Problem 20

At a certain place, Earth's magnetic field has magnitude $B=0.590$ 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 in series 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?

Vishal Gupta
Vishal Gupta
Numerade Educator
04:51

Problem 21

In Fig. 30.26, a stiff wire bent into a semicircle of radius $a=2.0 \mathrm{~cm}$ is rotated at constant angular speed $40 \mathrm{rev} / \mathrm{s}$ in a uniform $20 \mathrm{mT}$ magnetic field. What are the (a) frequency and (b) amplitude of the emf induced in the loop?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:18

Problem 22

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 $\pi / 2 \mathrm{rad}$ 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 23

Figure 30.27 shows two parallel loops of wire having a common axis. The smaller loop (radius $r$ ) is above the larger loop (radius $R$ ) by a distance $x \& R$. Consequently, the magnetic field due to the counterclockwise current $i$ in the larger loop is nearly uniform throughout the smaller loop. Suppose that $x$ is increasing at the constant rate $d x / d t=\nu$. (a) Find an expression for the magnetic flux through the area of the smaller loop as a function of $x$.

Vishal Gupta
Vishal Gupta
Numerade Educator
02:45

Problem 24

A wire is bent into three circular segments, each of radius $r=10 \mathrm{~cm}$, as shown in Fig. 30.28. 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 ca 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 $B$ increases at the rate of $3.0 \mathrm{mT} / \mathrm{s}$ ?
(b) What is the direction of the current in segment $b c$ ?

Keshav Singh
Keshav Singh
Numerade Educator
06:39

Problem 25

Two long, parallel copper wires of diameter $2.5 \mathrm{~mm}$ carry currents of $10 \mathrm{~A}$ 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 percentage of this flux lies inside the wires? (c) Repeat part (a) for parallel currents.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
09:59

Problem 26

For the wire arrangement in Fig. 30.29, $a=12.0 \mathrm{~cm}$ and $b=16.0 \mathrm{~cm}$. The current in the long straight wire is $i=4.50 t^2-10.0 t$, where $i$ is in amperes and $t$ is in seconds. (a) Find the emf in the square loop at $t=3.00 \mathrm{~s}$. (b) What is the direction of the induced current in the loop?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:25

Problem 27

As seen in Fig 30.30 , a square loop of wire ha sides of length $2.0 \mathrm{~cm}$. A mag netic field is directed out of th page; its magnitude is given b $B=4.0 t^2 y$, where $B$ is in teslas $t$ is in seconds, and $y$ is in meter At $t=2.5 \mathrm{~s}$, what are the (a) mag nitude and (b) direction of the em induced in the loop?

Keshav Singh
Keshav Singh
Numerade Educator
05:52

Problem 28

In Fig. 30.31, a rectangular loop of wire with length $a=2.2 \mathrm{~cm}$, width $b=0.80 \mathrm{~cm}$, and resistance $R=0.40 \mathrm{~m} \Omega$ is placed near an infinitely long wire carrying current $i=4.7 \mathrm{~A}$. The loop is then moved away from the wire at constant speed $v=3.2 \mathrm{~mm} / \mathrm{s}$. When the center of the loop is at distance $r=1.5 b$, what are (a) the magnitude of the magnetic flux through the loop and (b) the current induced in the loop?

Averell Hause
Averell Hause
Carnegie Mellon University
01:59

Problem 29

In Fig. 30.32, 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. A magnetic field of magnitude $B=0.350 \mathrm{~T}$ points out of the page. (a) If the rails are separated by $L=25.0 \mathrm{~cm}$ 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 energy being transferred to thermal energy?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
03:45

Problem 30

In Fig. 30.33a, a circular loop of wire is concentric with a solenoid and lies in a plane perpendicular to the solenoid's central axis. The loop has radius $6.00 \mathrm{~cm}$. The solenoid has radius $2.00 \mathrm{~cm}$, consists of 8000 turns $/ \mathrm{m}$, and has a current $i_{\text {sol }}$ varying with time $t$ as given in Fig. 30.33b, where the vertical axis scale is set by $i_s=1.00 \mathrm{~A}$ and the horizontal axis scale is set by $t_s=2.0$ s. Figure $30.33 c$ shows, as a function of time, the energy $E_{\mathrm{th}}$ that is transferred to thermal energy of the loop; the vertical axis scale is set by $E_s=100.0 \mathrm{~nJ}$. What is the loop's resistance?

Keshav Singh
Keshav Singh
Numerade Educator
04:57

Problem 31

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 is thermal energy generated in the loop?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:32

Problem 32

A loop antenna of area $2.00 \mathrm{~cm}^2$ and resistance $5.21 \mu \Omega$ is perpendicular to a uniform magnetic field of magnitude $17.0 \mu \mathrm{T}$. The field magnitude drops to zero in $2.96 \mathrm{~ms}$. How much thermal energy is produced in the loop by the change in field?

Averell Hause
Averell Hause
Carnegie Mellon University
12:14

Problem 33

Figure 30.34 shows a rod of length $L=10.0 \mathrm{~cm}$ that is forced to move at constant speed $v=5.00 \mathrm{~m} / \mathrm{s}$ along horizontal rails. The rod, rails, and connecting strip at the right form a conducting loop. The rod has resistance $0.400 \Omega$; the rest of the loop has negligible resistance. A current $i=100 \mathrm{~A}$ through the long straight wire at distance $a=10.0 \mathrm{~mm}$ from the loop sets up a (nonuniform) magnetic field through the loop. Find the (a) emf and (b) current induced in the loop. (c) At what rate is thermal energy generated in the rod? (d) What is the magnitude of the force that must be applied to the rod to make it move at constant speed? (e) At what rate does this force do work on the rod?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
05:32

Problem 34

In Fig. 30.35, 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 $v_t$. Ignoring air drag, find an expression for $v_i$.

Vishal Gupta
Vishal Gupta
Numerade Educator
04:30

Problem 35

The conducting rod shown in Fig. 30.32 has length $L$ and is being pulled along 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 page, fills the region in which the rod moves. Assume that $L=10 \mathrm{~cm}, v=5.0 \mathrm{~m} / \mathrm{s}$, and $B=1.2 \mathrm{~T}$. What are the (a) magnitude and (b) direction (up or down the page) of the emf induced in the rod? What are the (c) size and (d) direction 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. (e) At what rate is thermal energy being generated in the rod? (f) What external force on the rod is needed to maintain $\vec{v}$ ? (g) At what rate does this force do work on the rod?

Keshav Singh
Keshav Singh
Numerade Educator
03:17

Problem 36

Figure 30.36 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 $B_1=50.0 \mathrm{mT}$ directed into the page, and in $R_2$ there is a uniform magnetic field of magnitude $B_2=75.0 \mathrm{mT}$ directed out of the page (ignore fringing). Both fields are decreasing at the rate of $8.50 \mathrm{mT} / \mathrm{s}$. Calculate $\phi \vec{E} \cdot d \vec{s}$ for (a) path 1 , (b) path 2, and (c) path 3.

Averell Hause
Averell Hause
Carnegie Mellon University
06:38

Problem 37

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 $B=30.0 \mathrm{mT}$ 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
02:01

Problem 38

A circular region in an $x y$ plane is penetrated by a uniform magnetic field in the positive direction of the $z$ axis. The field's magnitude $B$ (in teslas) increases with time $t$ (in seconds) according to $B=a t$, where $a$ is a constant. The magnitude $E$ of the electric field set up by that increase in the magnetic field is given by Fig. 30.37 versus radial distance $r$, the vertical axis scale is set by $E_s=300 \mu \mathrm{N} / \mathrm{C}$, and the horizontal axis scale is set by $r_s=4.00 \mathrm{~cm}$. Find $a$.

Keshav Singh
Keshav Singh
Numerade Educator
04:53

Problem 39

The magnetic field of a cylindrical magnet that has a pole-face diameter of $3.3 \mathrm{~cm}$ can be varied sinusoidally between $29.6 \mathrm{~T}$ and $30.0 \mathrm{~T}$ at a frequency of $15 \mathrm{~Hz}$. (The current in a wire wrapped around a permanent magnet is varied to give this variation in the net field.) At a radial distance of $1.6 \mathrm{~cm}$, what is the amplitude of the electric field induced by the variation?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:21

Problem 40

The inductance of a closely packed coil of 400 turns is $8.0 \mathrm{mH}$. Calculate the magnetic flux through the coil when the current is $5.0 \mathrm{~mA}$.

Averell Hause
Averell Hause
Carnegie Mellon University
04:43

Problem 41

A circular coil has a $10.0 \mathrm{~cm}$ radius and consists of 30.0 closely wound turns of wire. An externally produced magnetic field of magnitude $2.60 \mathrm{mT}$ is perpendicular to the coil. (a) If no current is in the coil, what magnetic flux links its turns? (b) When the current in the coil is $3.80 \mathrm{~A}$ in a certain direction, the net flux through the coil is found to vanish. What is the inductance of the coil?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
03:00

Problem 42

Figure 30.38 shows a copper strip of width $W=16.0 \mathrm{~cm}$ that has been bent to form a shape that consists of a tube of radius $R=1.8 \mathrm{~cm}$ plus two parallel flat extensions. Current $i=35 \mathrm{~mA}$ is distributed uniformly across the width so that the tube is effectively a one-turn solenoid. Assume that the magnetic field outside the tube is negligible and the field inside the tube is uniform. What are (a) the magnetic field magnitude inside the tube and (b) the inductance of the tube (excluding the flat extensions)?

Averell Hause
Averell Hause
Carnegie Mellon University
06:04

Problem 43

Two identical long wires of radius $a=1.53 \mathrm{~mm}$ are parallel and carry identical currents in opposite directions. Their center-to-center separation is $d=14.2 \mathrm{~cm}$. Neglect the flux within the wires but consider the flux in the region between the wires. What is the inductance per unit length of the wires?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:08

Problem 44

A $12 \mathrm{H}$ inductor carries a current of $2.0 \mathrm{~A}$. At what rate must the current be changed to produce a $60 \mathrm{~V}$ emf in the inductor?

Averell Hause
Averell Hause
Carnegie Mellon University
01:45

Problem 45

At a given instant the current and self-induced emf in an inductor are directed as indicated in Fig. 30.39. (a) Is the current increasing or decreasing? (b) The induced emf is $17 \mathrm{~V}$, and the rate of change of the current is $25 \mathrm{kA} / \mathrm{s}$; find the inductance.

Km Neeraj
Km Neeraj
Numerade Educator
03:13

Problem 46

The current $i$ through a $4.6 \mathrm{H}$ inductor varies with time $t$ as shown by the graph of Fig. 30.40, where the vertical axis scale is set by $i_s=8.0 \mathrm{~A}$ and the horizontal axis scale is set by $t_s=6.0 \mathrm{~ms}$. The inductor has a resistance of $12 \Omega$. Find the magnitude of the induced emf $\mathscr{E}$ during time intervals (a) 0 to $2 \mathrm{~ms}$, (b) $2 \mathrm{~ms}$ to $5 \mathrm{~ms}$, and (c) $5 \mathrm{~ms}$ to $6 \mathrm{~ms}$. (Ignore the behavior at the ends of the intervals.)

Averell Hause
Averell Hause
Carnegie Mellon University
02:38

Problem 47

Inductors in series. Two inductors $L_1$ and $L_2$ are connected in series and are separated by a large distance so that the magnetic field of one cannot affect the other. (a) Show that the equivalent inductance is given by
$$
L_{\mathrm{eq}}=L_1+L_2 .
$$

Km Neeraj
Km Neeraj
Numerade Educator
04:09

Problem 48

Inductors in parallel. Two inductors $L_1$ and $L_2$ are connected in parallel and separated by a large distance so that the magnetic field of one cannot affect the other. (a) Show that the equivalent inductance is given by
$$
\frac{1}{L_{\mathrm{eq}}}=\frac{1}{L_1}+\frac{1}{L_2} .
$$

Vishal Gupta
Vishal Gupta
Numerade Educator
02:47

Problem 49

The inductor arrangement of Fig. 30.41 , with $L_1=30.0 \mathrm{mH}$, $L_2=50.0 \mathrm{mH}, L_3=20.0 \mathrm{mH}$, and $L_4=15.0 \mathrm{mH}$, is to be connected to a varying current source. What is the equivalent inductance of the arrangement? (First see Problems 47 and 48.)

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:55

Problem 50

The current in an $R L$ circuit builds up to one-third of its steady-state value in $5.00 \mathrm{~s}$. Find the inductive time constant.

Averell Hause
Averell Hause
Carnegie Mellon University
03:55

Problem 51

The current in an $R L$ circuit drops from $1.0 \mathrm{~A}$ to $10 \mathrm{~mA}$ in the first second following removal of the battery from the circuit. If $L$ is $10 \mathrm{H}$, find the resistance $R$ in the circuit.

Vishal Gupta
Vishal Gupta
Numerade Educator
03:23

Problem 52

The switch in Fig. 30.6.1 is closed on $a$ at time $t=0$. What is the ratio $\mathscr{E}_L / \mathscr{E}$ of the inductor's self-induced emf to the battery's emf (a) just after $t=0$ and (b) at $t=2.00 \tau_L$ ? (c) At what multiple of $\tau_L$ will $\mathscr{E}_L / \mathscr{E}=0.500$ ?

Km Neeraj
Km Neeraj
Numerade Educator
04:48

Problem 53

A solenoid having an inductance of $6.30 \mu \mathrm{H}$ is connected in series with a $1.20 \mathrm{k} \Omega$ resistor. (a) If a $14.0 \mathrm{~V}$ battery is connected across the pair, how long will it take for the current through the resistor to reach $80.0 \%$ of its final value? (b) What is the current through the resistor at time $t=1.0 \tau_L$ ?
d in region $2 ?$

Vishal Gupta
Vishal Gupta
Numerade Educator
07:38

Problem 54

In Fig. $30.42, \mathscr{E}=100 \mathrm{~V}, R_1=$ $10.0 \Omega, R_2=20.0 \Omega, R_3=30.0 \Omega$, and $L=2.00 \mathrm{H}$. Immediately after switch $S$ is closed, what are (a) $i_1$ and (b) $i_2$ ? (Let currents in the indicated directions have positive values and currents in the opposite directions have negative values.) A long time later, what are (c) $i_1$ and (d) $i_2$ ? The switch is then reopened. Just then, what are (e) $i_1$ and (f) $i_2$ ? A long time later, what are (g) $i_1$ and (h) $i_2$ ?

Km Neeraj
Km Neeraj
Numerade Educator
02:36

Problem 55

A battery is connected to a series $R L$ circuit at time $t=0$. At what multiple of $\tau_L$ will the current be $0.100 \%$ less than its equilibrium value?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:07

Problem 56

In Fig. 30.43, the inductor has 25 turns and the ideal battery has an emf of $16 \mathrm{~V}$. Figure 30.44 gives the magnetic flux $\Phi$ through each turn versus the current $i$ through the inductor. The vertical axis scale is set by $\Phi_s=4.0 \times 10^{-4} \mathrm{~T} \cdot \mathrm{m}^2$, and the horizontal axis scale is set by $i_s=2.00 \mathrm{~A}$. If switch $\mathrm{S}$ is closed at time $t=0$, at what rate $d i / d t$ will the current be changing at $t=1.5 \tau_L$ ?

Keshav Singh
Keshav Singh
Numerade Educator
02:16

Problem 57

In Fig. 30.45, $R=15 \Omega, L=5.0 \mathrm{H}$, the ideal battery has $\mathscr{E}=10 \mathrm{~V}$, and the fuse in the upper branch is an ideal 3.0 A fuse. It has zero resistance as long as the current through it remains less than $3.0 \mathrm{~A}$. If the current reaches $3.0 \mathrm{~A}$, the fuse "blows" and thereafter has infinite resistance. Switch $\mathrm{S}$ is closed at time $t=0$. (a) When does the fuse blow? (b) Sketch a graph of the current $i$ through the inductor as a function of time. Mark the time at which the fuse blows.

Keshav Singh
Keshav Singh
Numerade Educator
02:06

Problem 58

Suppose the emf of the battery in the circuit shown in Fig. 30.6.2 varies with time $t$ so that the current is given by $i(t)=3.0+5.0 t$, where $i$ is in amperes and $t$ is in seconds. Take $R=4.0 \Omega$ and $L=6.0 \mathrm{H}$, and find an expression for the battery emf as a function of $t$.

Km Neeraj
Km Neeraj
Numerade Educator
05:53

Problem 59

In Fig. 30.46, after switch $\mathrm{S}$ is closed at time $t=0$, the emf of the source is automatically adjusted to maintain a constant current $i$ through $\mathrm{S}$. (a) Find the current through the inductor as a function of time. (b) At what time is the current through the resistor equal to the current through the inductor?

Km Neeraj
Km Neeraj
Numerade Educator
05:33

Problem 60

A wooden toroidal core with a square cross section has an inner radius of $10 \mathrm{~cm}$ and an outer radius of $12 \mathrm{~cm}$. It is wound with one layer of wire (of diameter $1.0 \mathrm{~mm}$ and resistance per meter $0.020 \Omega / \mathrm{m}$ ). What are (a) the inductance and (b) the inductive time constant of the resulting toroid? Ignore the thickness of the insulation on the wire.

Averell Hause
Averell Hause
Carnegie Mellon University
04:17

Problem 61

A coil is connected in series with a $10.0 \mathrm{k} \Omega$ resistor. An ideal $50.0 \mathrm{~V}$ battery is applied across the two devices, and the current reaches a value of $2.00 \mathrm{~mA}$ after $5.00 \mathrm{~ms}$. (a) Find the inductance of the coil. (b) How much energy is stored in the coil at this same moment?

Vishal Gupta
Vishal Gupta
Numerade Educator
View

Problem 62

A coil with an inductance of $2.0 \mathrm{H}$ and a resistance of $10 \Omega$ is suddenly connected to an ideal battery with $\mathscr{E}=100 \mathrm{~V}$. At $0.10 \mathrm{~s}$ after the connection is made, what is the rate at which (a) energy is being stored in the magnetic field, (b) thermal energy is appearing in the resistance, and (c) energy is being delivered by the battery?

Averell Hause
Averell Hause
Carnegie Mellon University
03:47

Problem 63

At $t=0$, a battery is connected to a series arrangement of a resistor and an inductor. If the inductive time constant is $37.0 \mathrm{~ms}$, at what time is the rate at which energy is dissipated in the resistor equal to the rate at which energy is stored in the inductor's magnetic field?

Keshav Singh
Keshav Singh
Numerade Educator
02:09

Problem 64

At $t=0$, a battery is connected to a series arrangement of a resistor and an inductor. At what multiple of the inductive time constant will the energy stored in the inductor's magnetic field be 0.500 its steady-state value?

Averell Hause
Averell Hause
Carnegie Mellon University
04:41

Problem 65

For the circuit of Fig. 30.6.2, assume that $\mathscr{E}=10.0 \mathrm{~V}, R=6.70 \Omega$, and $L=5.50 \mathrm{H}$. The ideal battery is connected at time $t=0$. (a) How much energy is delivered by the battery during the first $2.00 \mathrm{~s}$ ? (b) How much of this energy is stored in the magnetic field of the inductor? (c) How much of this energy is dissipated in the resistor?

Km Neeraj
Km Neeraj
Numerade Educator
02:01

Problem 66

A circular loop of wire $50 \mathrm{~mm}$ in radius carries a current of $100 \mathrm{~A}$. Find the (a) magnetic field strength and (b) energy density at the center of the loop.

Averell Hause
Averell Hause
Carnegie Mellon University
04:54

Problem 67

A solenoid that is $85.0 \mathrm{~cm}$ long has a cross-sectional area of $17.0 \mathrm{~cm}^2$. There are 950 turns of wire carrying a current of $6.60 \mathrm{~A}$. (a) Calculate the energy density of the magnetic field inside the solenoid. (b) Find the total energy stored in the magnetic field there (neglect end effects).

Vishal Gupta
Vishal Gupta
Numerade Educator
02:55

Problem 68

A toroidal inductor with an inductance of $90.0 \mathrm{mH}$ encloses a volume of $0.0200 \mathrm{~m}^3$. If the average energy density in the toroid is $70.0 \mathrm{~J} / \mathrm{m}^3$, what is the current through the inductor?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:24

Problem 69

What must be the magnitude of a uniform electric field if it is to have the same energy density as that possessed by a $0.50 \mathrm{~T}$ magnetic field?

Vishal Gupta
Vishal Gupta
Numerade Educator
05:29

Problem 70

Figure $30.47 a$ shows, in cross section, two wires that are straight, parallel, and very long. The ratio $i_1 / i_2$ of the current carried by wire 1 to that carried by wire 2 is $1 / 3$. Wire 1 is fixed in place. Wire 2 can be moved along the positive side of the $x$ axis so as to change the magnetic energy density $u_B$ set up by the two currents at the origin. Figure $30.47 b$ gives $u_B$ as a function of the position $x$ of wire 2 . The curve has an asymptote of $u_B=1.96 \mathrm{~nJ} / \mathrm{m}^3$ as $x \rightarrow \infty$, and the horizontal axis scale is set by $x_s=60.0 \mathrm{~cm}$. What is the value of (a) $i_1$ and (b) $i_2$ ?

Averell Hause
Averell Hause
Carnegie Mellon University
04:31

Problem 71

A length of copper wire carries a current of $10 \mathrm{~A}$ uniformly distributed through its cross section. Calculate the energy density of (a) the magnetic field and (b) the electric field at the surface of the wire. The wire diameter is $2.5 \mathrm{~mm}$, and its resistance per unit length is $3.3 \Omega / \mathrm{km}$.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
02:41

Problem 72

Coil 1 has $L_1=25 \mathrm{mH}$ and $N_1=100$ turns. Coil 2 has $L_2=40 \mathrm{mH}$ and $N_2=200$ turns. The coils are fixed in place; their mutual inductance $M$ is $3.0 \mathrm{mH}$. A $6.0 \mathrm{~mA}$ current in coil 1 is changing at the rate of $4.0 \mathrm{~A} / \mathrm{s}$. (a) What magnetic flux $\Phi_{12}$ links coil 1, and (b) what self-induced emf appears in that coil? (c) What magnetic flux $\Phi_{21}$ links coil 2, and (d) what mutually induced emf appears in that coil?

Averell Hause
Averell Hause
Carnegie Mellon University
04:20

Problem 73

Two coils are at fixed locations. When coil 1 has no current and the current in coil 2 increases at the rate $15.0 \mathrm{~A} / \mathrm{s}$, the emf in coil 1 is $25.0 \mathrm{mV}$. (a) What is their mutual inductance? (b) When coil 2 has no current and coil 1 has a current of $3.60 \mathrm{~A}$, what is the flux linkage in coil 2 ?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:37

Problem 74

Two solenoids are part of the spark coil of an automobile. When the current in one solenoid falls from $6.0 \mathrm{~A}$ to zero in $2.5 \mathrm{~ms}$, an emf of $30 \mathrm{kV}$ is induced in the other solenoid. What is the mutual inductance $M$ of the solenoids?

Vishal Gupta
Vishal Gupta
Numerade Educator
06:50

Problem 75

A rectangular loop of $N$ closely packed turns is positioned near a long straight wire as shown in Fig. 30.48. What is the mutual inductance $M$ for the loop-wire combination if $N=100, a=1.0 \mathrm{~cm}, b=8.0 \mathrm{~cm}$, and $l=30 \mathrm{~cm}$ ?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:15

Problem 76

A coil $\mathrm{C}$ of $N$ turns is placed around a long solenoid $\mathrm{S}$ of radius $R$ and $n$ turns per unit length, as in Fig. 30.49. (a) Show that the mutual inductance for the coil-solenoid combination is given by $M=\mu_0 \pi R^2 n N$. (b) Explain why $M$ does not depend on the shape, size, or possible lack of close packing of the coil.

Km Neeraj
Km Neeraj
Numerade Educator
05:05

Problem 77

Two coils connected as shown in Fig. 30.50 separately have inductances $L_1$ and $L_2$. Their mutual inductance is $M$.
(a) Show that this combination can be replaced by a single coil of equivalent inductance given by
$$
L_{e q}=L_1+L_2+2 M
$$
(b) How could the coils in Fig. 30.50 be reconnected to yield an equivalent inductance of
$$
L_{e q}=L_1+L_2-2 M ?
$$
(This problem is an extension of Problem 47, but the requirement that the coils be far apart has been removed.)

Km Neeraj
Km Neeraj
Numerade Educator
03:26

Problem 78

At time $t=0$, a $12.0 \mathrm{~V}$ potential difference is suddenly applied to the leads of a coil of inductance $23.0 \mathrm{mH}$ and a certain resistance $R$. At time $t=0.150 \mathrm{~ms}$, the current through the inductor is changing at the rate of $280 \mathrm{~A} / \mathrm{s}$. Evaluate $R$.

Averell Hause
Averell Hause
Carnegie Mellon University
05:51

Problem 79

In Fig. 30.51, the battery is ideal and $\mathscr{E}=10 \mathrm{~V}, R_1=5.0 \Omega$, $R_2=10 \Omega$, and $L=5.0 \mathrm{H}$. Switch $\mathrm{S}$ is closed at time $t=0$. Just afterwards, what are (a) $i_1$, (b) $i_2$, (c) the current $i_{\mathrm{S}}$ through the switch, (d) the potential difference $V_2$ across resistor 2 , (e) the potential difference $V_L$ across the inductor, and (f) the rate of change $d i v / d r$ ? A long time later. what are (g) $i_1$, (h) $i_2$, (i) $i_{\mathrm{S}}$, (j) $V_2,(\mathrm{k}) V_L$, and (1) $d i_2 / d r$ ?

Keshav Singh
Keshav Singh
Numerade Educator
03:50

Problem 80

In Fig. $30.43, R=4.0 \mathrm{k} \Omega, L=8.0 \mu \mathrm{H}$, and the ideal battery has $\mathscr{E}=20 \mathrm{~V}$. How long after switch $\mathrm{S}$ is closed is the current $2.0 \mathrm{~mA}$ ?

Vishal Gupta
Vishal Gupta
Numerade Educator
03:58

Problem 81

Figure $30.52 a$ 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 field. Figure $30.52 b$ gives the current $i$ induced in the loop as a function of the position $x$ of the right side of the loop. The vertical axis scale is set by $i_s=3.0 \mu \mathrm{A}$. For example, a current equal to $i_s$ is induced clockwise as the loop enters region 1 . What are the (a) magnitude and (b) direction (into or out of the page) of the magnetic field in region 1? What are the (c) magnitude and (d) direction of the magnetic fiel

Narayan Hari
Narayan Hari
Numerade Educator
02:45

Problem 82

A uniform magnetic field $\vec{B}$ is perpendicular to the plane of a circular wire loop of radius $r$. The magnitude of the field varies with time according to $B=B_0 e^{-t / \tau}$, where $B_0$ and $\tau$ are constants. Find an expression for the emf in the loop as a function of time.

Vishal Gupta
Vishal Gupta
Numerade Educator
03:47

Problem 83

Switch $\mathrm{S}$ in Fig. 30.43 is closed at time $t=0$, initiating the buildup of current in the $15.0 \mathrm{mH}$ inductor and the $20.0 \Omega$ resistor. At what time is the emf across the inductor equal to the potential difference across the resistor?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
02:04

Problem 84

Figure $30.53 a$ 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 two regions and then the emf $\mathscr{E}$ around the ring is determined. Figure $30.53 \mathrm{~b}$ gives emf $\mathscr{E}$ as a function of the square $R^2$ of the ring's radius, to the outer edge of region 2 . The vertical axis scale is set by $\mathscr{E}_s=20.0 \mathrm{nV}$. What are the rates (a) $d B_1 / d t$ and (b) $d B_2 / d r$ ? (c) Is the magnitude of $\vec{B}_2$ increasing, decreasing, or remaining constant?

Narayan Hari
Narayan Hari
Numerade Educator
05:12

Problem 85

Figure 30.54 shows a uniform magnetic field $\vec{B}$ confined to a cylindrical volume of radius $R$. The magnitude of $\vec{B}$ is decreasing at a constant rate of $10 \mathrm{mT} / \mathrm{s}$. In unit-vector notation, what is the initial acceleration of an electron released at (a) point $a$ (radial distance $r=5.0 \mathrm{~cm}$ ), (b) point $b(r=0)$, and (c) point $c(r=5.0 \mathrm{~cm}) ?$

Keshav Singh
Keshav Singh
Numerade Educator
02:49

Problem 86

In Fig. 30.55a, switch $\mathrm{S}$ has been closed on $A$ long enough to establish a steady current in the inductor of inductance $L_1=5.00 \mathrm{mH}$ and the resistor of resistance $R_1=25.0 \Omega$. Similarly, in Fig. $30.55 b$, switch $\mathrm{S}$ has been closed on $A$ long enough to establish a steady current in the inductor of inductance $L_2=3.00 \mathrm{mH}$ and the resistor of resistance $R_2=30.0 \Omega$. The ratio $\Phi_{02} / \Phi_{01}$ of the magnetic flux through a turn in inductor 2 to that in inductor 1 is 1.50 . At time $t=0$, the two switches are closed on $B$. At what time $t$ is the flux through a turn in the two inductors equal?

Keshav Singh
Keshav Singh
Numerade Educator
03:55

Problem 87

A square wire loop $20 \mathrm{~cm}$ on a side, with resistance $20 \mathrm{~m} \Omega$, has its plane normal to a uniform magnetic field of magnitude $B=2.0 \mathrm{~T}$. If you pull two opposite sides of the loop away from each other, the other two sides automatically draw toward each other, reducing the area enclosed by the loop. If the area is reduced to zero in time $\Delta t=0.20 \mathrm{~s}$, what are (a) the average emf and (b) the average current induced in the loop during $\Delta r$ ?

Vishal Gupta
Vishal Gupta
Numerade Educator
03:22

Problem 88

A coil with 150 turns has a magnetic flux of $50.0 \mathrm{nT} \cdot \mathrm{m}^2$ through each turn when the current is $2.00 \mathrm{~mA}$. (a) What is the inductance of the coil? What are the (b) inductance and (c) flux through each turn when the current is increased to $4.00 \mathrm{~mA}$ ? (d) What is the maximum emf $\mathscr{E}$ across the coil when the current through it is given by $i=(3.00 \mathrm{~mA}) \cos (377 t)$, with $t$ in seconds?

Narayan Hari
Narayan Hari
Numerade Educator
01:29

Problem 89

A coil with an inductance of $2.0 \mathrm{H}$ and a resistance of $10 \Omega$ is suddenly connected to an ideal battery with $\mathscr{E}=100 \mathrm{~V}$. (a) What is the equilibrium current? (b) How much energy is stored in the magnetic field when this current exists in the coil?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:40

Problem 90

How long would it take, following the removal of the battery, for the potential difference across the resistor in an $R L$ circuit (with $L=2.00 \mathrm{H}, R=3.00 \Omega$ ) to decay to $10.0 \%$ of its initial value?

Averell Hause
Averell Hause
Carnegie Mellon University
05:50

Problem 91

In the circuit of Fig. $30.56, \quad R_1=20 \mathrm{k} \Omega, \quad R_2=20 \Omega$, $L=50 \mathrm{mH}$, and the ideal battery has $\mathscr{E}=40 \mathrm{~V}$. Switch S has been open for a long time when it is closed at time $t=0$. Just after the switch is closed, what are (a) the current $i_{\text {bat }}$ through the battery and (b) the rate $d i_{\text {bat }} / d r$ ? At $t=3.0 \mu \mathrm{s}$, what are (c) $i_{\text {bat }}$ and (d) $d i_{\text {bat }} / d r$ ? A long time later, what are (e) $i_{\text {bat }}$ and (f) $d i_{\text {bat }} / d t$ ?

Keshav Singh
Keshav Singh
Numerade Educator
02:04

Problem 92

The flux linkage through a certain coil of $0.75 \Omega$ resistance would be $26 \mathrm{mWb}$ if there were a current of $5.5 \mathrm{~A}$ in it. (a) Calculate the inductance of the coil. (b) If a $6.0 \mathrm{~V}$ ideal battery were suddenly connected across the coil, how long would it take for the current to rise from 0 to $2.5 \mathrm{~A}$ ?

Averell Hause
Averell Hause
Carnegie Mellon University
02:41

Problem 93

Fringing in a capacitor. Prove that the electric field $\vec{E}$ in a charged parallel capacitor cannot drop abruptly to zero as is suggested at point $a$ in Fig. 30.57 as we move perpendicular to the field along the horizontal arrow in the figure. To do this, apply Faraday's law to the rectangular path shown by the dashed lines. In actual capacitors fringing of the field lines always occurs, which means that the field approaches zero in a continuous and gradual way.

Amit Srivastava
Amit Srivastava
Numerade Educator
05:09

Problem 94

Coaxial cable. A long coaxial cable consists of two thin-walled concentric conducting cylinders with radii $a$ and $b$. The inner cylinder $A$ carries a steady current $i$, the outer cylinder $B$ providing the return path. (a) Calculate the energy stored in the magnetic field between the cylinders for length $l$ of the cable. (b) What is the stored energy per unit length of the cable if $a=1.2 \mathrm{~mm}, b=3.5 \mathrm{~mm}$, and $i=2.7 \mathrm{~A}$ ?

Keshav Singh
Keshav Singh
Numerade Educator
01:52

Problem 95

Ballistic galvanometer. Circuit 1 in Fig. 30.58 consists of an ammeter in series with a battery and coil 1 . Circuit 2 consists of coil 2 and a ballistic galvanometer of resistance $R$; the galvanometer can measure the charge that moves through itself. When switch $S$ is closed, the equilibrium current reading on the ammeter is $i_f$. The total charge sent through the galvanometer while the current circuit 2 reaches equilibrium is $Q$. Find the mutual inductance $M$ between coils 1 and 2 .

Dading Chen
Dading Chen
Numerade Educator
02:03

Problem 96

Straight and triangle wires. In Fig. 30.59, a long straight wire lies in the same plane as an equilateral triangle formed from a wire of length $3 S$. The long wire is parallel to one side of the triangle and at distance $d$ from the nearest vertex. What is the mutual inductance $M$ of the wire and triangle?

Jacob Shpiece
Jacob Shpiece
Numerade Educator
07:51

Problem 97

Induction, large loop, small 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 $200 \mathrm{~A}$ to $-200 \mathrm{~A}$ (a change in direction) in a time of $1.00 \mathrm{~s}$, beginning at $t=0$. (a) What is the magnetic field at the center of the small circular loop due to the current in the large loop at $t=0, t=0.500 \mathrm{~s}$, and $t=1.00 \mathrm{~s}$ ?
(b) What is the magnitude of the emf induced in the small loop at $t=0.500 \mathrm{~s}$ ? Because the inner loop is small, assume the magnetic field due to the outer loop is uniform over the area of the smaller loop.

Vishal Gupta
Vishal Gupta
Numerade Educator
02:01

Problem 98

Currents first equal. Switch S in Fig. 30.60 is closed for time $t<0$ and is opened at $t=0$. When current $i_1$ through $L_1$ and $R_1$ and current $i_2$ through $L_2$ and $R_2$ are first equal to each other, what is their common value?

Ze-Han Lee
Ze-Han Lee
Numerade Educator