University Physics with Modern Physics

Wolfgang Bauer, Gary D. Westfall

Chapter 29

Electromagnetic Induction - all with Video Answers

Educators

Chapter Questions

Problem 1

A solenoid with 200 turns and a cross-sectional area of $60 \mathrm{~cm}^{2}$ has a magnetic field of $0.60 \mathrm{~T}$ along its axis. If the field is confined within the solenoid and changes at a rate of $0.20 \mathrm{~T} / \mathrm{s}$, the magnitude of the induced potential difference in the solenoid will be
a) $0.0020 \mathrm{~V}$.
b) $0.02 \mathrm{~V}$.
c) $0.001 \mathrm{~V}$.
d) $0.24 \mathrm{~V}$.

Ajay Singhal

Ajay Singhal

Numerade Educator

Problem 2

The rectangular loop of wire in Figure 29.9 is pulled with a constant acceleration from a region of zero magnetic field into a region of a uniform magnetic field. During this process, the current induced in the loop
a) will be zero.
b) will be some constant value that is not zero.
c) will increase linearly with time.
d) will increase exponentially with time.
e) will increase linearly with the square of the time.

Ajay Singhal

Numerade Educator

Problem 3

Which of the following will induce a current in a loop of wire in a uniform magnetic field?
a) decreasing the strength of the field
b) rotating the loop about an axis parallel to the field
c) moving the loop within the field
d) all of the above
e) none of the above

Ajay Singhal

Numerade Educator

Problem 4

Faraday's Law of Induction states that
a) a potential difference is induced in a loop when there is a change in the magnetic flux through the loop.
b) the current induced in a loop by a changing magnetic field produces a magnetic field that opposes this change in magnetic field.
c) a changing magnetic field induces an electric field.
d) the inductance of a device is a measure of its opposition to changes in current flowing through it.
e) magnetic flux is the product of the average magnetic field and the area perpendicular to it that it penetrates.

Ajay Singhal

Numerade Educator

Problem 5

A conducting ring is moving from left to right through a uniform magnetic field, as shown in the figure. In which region(s) is there an induced current in the ring?
a) regions $\mathrm{B}$ and $\mathrm{D}$
b) regions $B, C,$ and $D$
c) region C
d) regions A through $\mathrm{E}$

Ajay Singhal

Numerade Educator

Problem 6

A circular loop of wire moving in the $x y$ -plane with a constant velocity in the negative $x$ -direction enters a uniform magnetic field, which covers the region in which $x<0,$ as shown in the figure. The surface normal vector of the loop points in the direction
of the magnetic field. Which of the following statements is correct?
a) The induced potential difference in the loop is at a maximum as the edge of the loop just enters the region with the magnetic field.
b) The induced potential difference in the loop is at a maximum when one fourth of the loop is in the region with the magnetic field.
c) The induced potential difference in the loop is at a maximum when the loop is halfway into the region with the magnetic field.
d) The induced potential difference in the loop is constant from the instant the loop starts to enter the region with the magnetic field.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 7

Which of the following statements regarding self-induction is correct?
a) Self-induction occurs only when a direct current is flowing through a circuit.
b) Self-induction occurs only when an alternating current is flowing through a circuit.
c) Self-induction occurs when either a direct current or an alternating current is flowing through a circuit.
d) Self-induction occurs when either a direct current or an alternating current is flowing through a circuit as long as the current is varying.

Bettina Hanlon

Numerade Educator

Problem 8

You have a light bulb, a bar magnet, a spool of wire that you can cut into as many pieces as you want, and nothing else. How can you get the bulb to light up?
a) You can't. The bulb needs electricity to light it, not magnetism.
b) You cut a length of wire, connect the light bulb to the two ends of the wire, and pass the magnet through the loop that is formed.
c) You cut two lengths of wire and connect the magnet and the bulb in series.

Sheh Lit Chang

University of Washington

Problem 9

Calculate the potential difference induced between the tips of the wings of a Boeing $747-400$ with a wingspan of $64.67 \mathrm{~m}$ when it is in level flight at a speed of $913 \mathrm{~km} / \mathrm{h}$. Assume that the magnitude of the downward component of the Earth's magnetic field is $B=5.00 \cdot 10^{-5} \mathrm{~T}$.
a) $0.820 \mathrm{~V}$
b) $2.95 \mathrm{~V}$
c) $10.4 \mathrm{~V}$
d) $30.1 \mathrm{~V}$
e) $225 \mathrm{~V}$

Ajay Singhal

Numerade Educator

Problem 10

A long solenoid with a circular cross section of radius $r_{1}=2.80 \mathrm{~cm}$ and $n=290$ turns/cm is inside of and coaxial with a short coil that has a circular cross section of radius $r_{2}=4.90 \mathrm{~cm}$ and $N=31$ turns. Suppose the current in the short coil is increased steadily from zero to $i=2.80 \mathrm{~A}$ in $18.0 \mathrm{~ms} .$ What is the magnitude of the potential difference induced in the solenoid while the current in the short coil is changing?
a) $0.0991 \mathrm{~V}$
b) $0.128 \mathrm{~V}$
c) $0.233 \mathrm{~V}$
d) $0.433 \mathrm{~V}$
e) $0.750 \mathrm{~V}$

Ajay Singhal

Numerade Educator

Problem 11

A long solenoid has a circular cross section of radius $r=8.10 \mathrm{~cm},$ a length $\ell=0.540 \mathrm{~m}$, and $n=2.00 \cdot 10^{4}$ turns $/ \mathrm{m}$. The solenoid is carrying a current of magnitude $i=4.04 \cdot 10^{-3} \mathrm{~A}$. How much energy is stored in the magnetic field of the solenoid?
a) $2.11 \cdot 10^{-7} \mathrm{~J}$
b) $8.91 \cdot 10^{-6} \mathrm{~J}$
c) $4.57 \cdot 10^{-5}$ J
d) $6.66 \cdot 10^{-3}$ J.
e) $4.55 \cdot 10^{-1} \mathrm{~J}$

Ajay Singhal

Numerade Educator

Problem 12

Suppose the current in the short coil in Solved Problem 29.2 is increased steadily from zero to $i=2.80 \mathrm{~A}$ in $18.0 \mathrm{~ms} .$ What is the magnitude of the potential difference induced in the solenoid while the current in the short coil is changing?
a) $0.0991 \mathrm{~V}$
b) $0.128 \mathrm{~V}$
c) $0.233 \mathrm{~V}$
d) $0.433 \mathrm{~V}$
e) $0.750 \mathrm{~V}$

Ajay Singhal

Numerade Educator

Problem 13

Suppose the length of the rotating rod in Solved Problem 29.1 is increased by a factor of $2 .$ By what factor does the power dissipated in the resistor change?
a) $\frac{1}{2}$
b) 2
c) 4
d) 8
e) 16

Ajay Singhal

Numerade Educator

Problem 14

Suppose the resistance of the resistor in Solved Problem 29.1 is increased by a factor of $2 .$ By what factor does the power dissipated in the resistor change?
a) $\frac{1}{2}$
b) 2
c) 4
d) 8
e) 16

Ajay Singhal

Numerade Educator

Problem 15

When you plug a refrigerator into a wall socket, on occasion, a spark appears between the prongs. What causes this?

Ajay Singhal

Numerade Educator

Problem 16

People with pacemakers or other mechanical devices as implants are often warned to stay away from large machinery or motors. Why?

Ajay Singhal

Numerade Educator

Problem 17

Chapter 14 discussed damped harmonic oscillators, in which the damping force is velocity dependent and always opposes the motion of the oscillator. One way of producing this type of force is to use a piece of metal, such as aluminum, that moves through a nonuniform magnetic field. Explain why this technique is capable of producing a damping force.

Bettina Hanlon

Numerade Educator

Problem 18

In a popular lecture demonstration, a cylindrical permanent magnet is dropped down a long aluminum tube as shown in the figure. Neglecting friction of the magnet against the inner walls of the tube and assuming that the tube is very long compared to the size of the magnet, will the magnet accelerate downward with an acceleration equal to $g$ (free fall)? If not, describe the eventual motion of the magnet. Does it matter if the north pole or south pole of the magnet is on the lower side?

Bettina Hanlon

Numerade Educator

Problem 19

A popular demonstration of eddy currents involves dropping a magnet down a long metal tube and a long glass or plastic tube. As the magnet falls through a tube, the magnetic flux changes as the magnet moves toward or away from each part of the tube.
a) Which tube has the larger voltage induced in it?
b) Which tube has the larger eddy currents induced in it?

Bettina Hanlon

Numerade Educator

Problem 20

The current in a very long, tightly wound solenoid with radius $a$ and $n$ turns per unit length varies over time according to the equation $i(t)=C t^{2},$ where the current $i$ is in amps and the time $t$ is in seconds, and $C$ is a constant with appropriate units. Concentric with the solenoid is a conducting ring of radius $r,$ as shown in the figure.
a) Write an expression for the potential difference induced in the ring.
b) Write an expression for the magnitude of the electric field induced at an arbitrary point on the ring.
c) Is the ring necessary for the induced electric field to exist?

Bettina Hanlon

Numerade Educator

Problem 21

A circular wire ring experiences an increasing magnetic field in the upward direction, as shown in the figure. What is the direction of the induced current in the ring?

Ajay Singhal

Numerade Educator

Problem 22

A square conducting loop with sides of length $L$ is rotating at a constant angular speed, $\omega$, in a uniform magnetic field of magnitude $B$. At time $t=0,$ the loop is oriented so that the direction normal to the loop is aligned with the magnetic field. Find an expression for the potential difference induced in the loop as a function of time.

Ajay Singhal

Numerade Educator

Problem 23

A solid metal disk of radius $R$ is rotating around its center axis at a constant angular speed of $\omega .$ The disk is in a uniform magnetic field of magnitude $B$ that is oriented normal to the surface of the disk. Calculate the magnitude of the potential difference between the center of the disk and the outside edge.

Ajay Singhal

Numerade Educator

Problem 24

Large electric fields are certainly a hazard to the human body, as they can produce dangerous currents, but what about large magnetic fields? A man $1.80 \mathrm{~m}$ tall walks at $2.00 \mathrm{~m} / \mathrm{s}$ perpendicular to a horizontal magnetic field of $5.0 \mathrm{~T}$; that is, he walks between the pole faces of a very big magnet. (Such a magnet can, for example, be found in the National Superconducting Cyclotron Laboratory at Michigan State University.) Given that his body is full of conducting fluids, estimate the potential difference induced between his head and feet.

Bettina Hanlon

Numerade Educator

Problem 25

At Los Alamos National Laboratories, one means of producing very large magnetic fields is the EPFCG (explosively-pumped flux compression generator), which is used to study the effects of a high-power electromagnetic pulse (EMP) in electronic warfare. Explosives are packed and detonated in the space between a solenoid and a small copper cylinder coaxial with and inside the solenoid, as shown in the figure. The explosion occurs in a very short time and collapses the cylinder rapidly. This rapid change creates inductive currents that keep the magnetic flux constant while the cylinder's radius shrinks by a factor of $r_{\mathrm{i}} / r_{\mathrm{f}}$. Estimate the magnetic field produced, assuming that the radius is compressed by a factor of 14 and the initial magnitude of the magnetic field, $B_{i}$, is $1.0 \mathrm{~T}$.

Ajay Singhal

Numerade Educator

Problem 26

A metal hoop is laid flat on the ground. A magnetic field that is directed upward, out of the ground, is increasing in magnitude. As you look down on the hoop from above, what is the direction of the induced current in the hoop?

Ajay Singhal

Numerade Educator

Problem 27

The wire of a tightly wound solenoid is unwound and then rewound to form another solenoid with double the diameter of the first solenoid. By what factor will the inductance change?

Ajay Singhal

Numerade Educator

Problem 28

A circular coil of wire with 20 turns and a radius of $40.0 \mathrm{~cm}$ is laying flat on a horizontal tabletop as shown in the figure. There is a uniform magnetic field extending over the entire table with a magnitude of $5.00 \mathrm{~T}$ and directed to the north and downward, making an angle of $25.8^{\circ}$ with the horizontal. What is the magnitude of the magnetic flux through the coil?

Ajay Singhal

Numerade Educator

Problem 29

When a magnet in an MRI is abruptly shut down, the magnet is said to be quenched. Quenching can occur in as little as $20.0 \mathrm{~s}$. Suppose a magnet with an initial field of $1.20 \mathrm{~T}$ is quenched in $20.0 \mathrm{~s},$ and the final field is approximately zero. Under these conditions, what is the average induced potential difference around a conducting loop of radius $1.00 \mathrm{~cm}$ (about the size of a wedding ring) oriented perpendicular to the field?

Ajay Singhal

Numerade Educator

Problem 30

An 8 -turn coil has square loops measuring $0.200 \mathrm{~m}$ along a side and a resistance of $3.00 \Omega$. It is placed in a magnetic field that makes an angle of $40.0^{\circ}$ with the plane of each loop. The magnitude of this field varies with time according to $B=1.50 t^{3},$ where $t$ is measured in seconds and $B$ in teslas. What is the induced current in the coil at $t=2.00 \mathrm{~s} ?$

Ajay Singhal

Numerade Educator

Problem 31

A metal loop has an area of $0.100 \mathrm{~m}^{2}$ and is placed flat on the ground. There is a uniform magnetic field pointing due west, as shown in the figure. This magnetic field initially has a magnitude of $0.123 \mathrm{~T}$ which decreases steadily to $0.075 \mathrm{~T}$ during a period of $0.579 \mathrm{~s}$. Find the potential difference induced in the loop during this time.

Ajay Singhal

Numerade Educator

Problem 32

A respiration monitor has a flexible loop of copper wire, which wraps about the chest. As the wearer breathes, the radius of the loop of wire increases and decreases. When a person in the Earth's magnetic field (assume $0.426 \cdot 10^{-4} \mathrm{~T}$ ) inhales, what is the average current in the loop, assuming that it has a resistance of $30.0 \Omega$ and increases in radius from $20.0 \mathrm{~cm}$ to $25.0 \mathrm{~cm}$ over 1.00 s? Assume that the magnetic field is perpendicular to the plane of the loop.

Ajay Singhal

Numerade Educator

Problem 33

A circular conducting loop with radius $a$ and resistance $R_{2}$ is concentric with a circular conducting loop with radius $b \gg a(b$ much greater than $a$ ) and resistance $R_{1}$. A time-dependent voltage is applied to the larger loop; its slow sinusoidal variation in time is given by $V(t)=V_{0} \sin \omega t$ where $V_{0}$ and $\omega$ are constants with dimensions of voltage and inverse time, respectively. Assuming that the magnetic field throughout the inner loop is uniform (constant in space) and equal to the field at the center of the loop, derive expressions for the potential difference induced in the inner loop and the current $i$ through that loop.

Manik Pulyani

Numerade Educator

Problem 34

A long solenoid with cross-sectional area $A_{1}$ surrounds another long solenoid with cross-sectional area $A_{2}<A_{1}$ and resistance $R .$ Both solenoids have the same length and the same number of turns. A current given by $i=i_{0} \cos \omega t$ is flowing through the outer solenoid. Find an expression for the magnetic field in the inner solenoid due to the induced current.

Ajay Singhal

Numerade Educator

Problem 35

The conducting loop in the shape of a quarter-circle shown in the figure has a radius of $10.0 \mathrm{~cm}$ and a resistance of $0.200 \Omega$. The magnetic field strength within the dotted circle of radius $3.00 \mathrm{~cm}$ is initially $2.00 \mathrm{~T}$. The magnetic field strength then decreases from $2.00 \mathrm{~T}$ to $1.00 \mathrm{~T}$ in $2.00 \mathrm{~s}$. Find (a) the magnitude and
(b) the direction of the induced current in the loop.

Ajay Singhal

Numerade Educator

Problem 36

A supersonic aircraft with a wingspan of $10.0 \mathrm{~m}$ is flying over the north magnetic pole (in a magnetic field of magnitude 0.500 G oriented perpendicular to the ground) at a speed of three times the speed of sound (Mach 3). What is the potential difference between the tips of the wings? Assume that the wings are made of aluminum.

Ajay Singhal

Numerade Educator

Problem 37

A helicopter hovers above the north magnetic pole in a magnetic field of magnitude $0.426 \mathrm{G}$ and oriented perpendicular to the ground. The helicopter rotors are $10.0 \mathrm{~m}$ long, are made of aluminum, and rotate about the hub with a rotational speed of $1.00 \cdot 10^{4} \mathrm{rpm}$. What is the potential difference from the hub to the end of a rotor?

Ajay Singhal

Numerade Educator

Problem 38

An elastic circular conducting loop expands at a constant rate over time such that its radius is given by $r(t)=r_{0}+v t,$ where $r_{0}=0.100 \mathrm{~m}$ and $v=0.0150 \mathrm{~m} / \mathrm{s}$. The loop has a constant resistance of $R=12.0 \Omega$ and is placed in a uniform magnetic field of magnitude $B_{0}=0.750 \mathrm{~T}$, perpendicular to the plane of the loop, as shown in the figure. Calculate the direction and the magnitude of the induced current, $i$ at $t=5.00 \mathrm{~s}$.

Ajay Singhal

Numerade Educator

Problem 39

A rectangular frame of conducting wire has negligible resistance and width $w$ and is held vertically in a magnetic field of magnitude $B$, as shown in the figure. A metal bar with mass $m$ and resistance $R$ is placed across the frame, maintaining contact with the frame. Derive an expression for the terminal velocity of the bar if it is allowed to fall freely along this frame starting from rest. Neglect friction between the wires and the metal bar.

Ajay Singhal

Numerade Educator

Problem 40

Two parallel conducting rails with negligible resistance are connected at one end by a resistor of resistance $R$, as shown in the figure. The rails are placed in a magnetic field $\vec{B}_{\text {ext }}$, which is perpendicular to the plane of the rails. This magnetic field is uniform and time independent. The distance between the rails is $\ell$. A conducting rod slides without friction on top of the two rails at constant velocity $\vec{v}$.
a) Using Faraday's Law of Induction, calculate the magnitude of the potential difference induced in the moving rod.
b) Calculate the magnitude of the induced current in the rod, $i_{\text {ind }}$
c) Show that for the rod to move at a constant velocity as shown, it must be pulled with an external force, $\vec{F}_{\text {ext }}$ and calculate the magnitude of this force.
d) Calculate the work done, $W_{\text {ext }},$ and the power generated, $P_{\text {ext }}$, by the external force in moving the rod.
e) Calculate the power used (dissipated) by the resistor, $P_{\mathrm{R}}$. Explain the correlation between this result and those of part (d).

Ajay Singhal

Numerade Educator

Problem 41

A long, straight wire runs along the $y$ -axis. The wire carries a current in the positive $y$ -direction that is changing as a function of time according to $i=2.00 \mathrm{~A}+$ $(0.300 \mathrm{~A} / \mathrm{s})$ t. A loop of wire is located in the $x y$ -plane near the $y$ -axis, as shown in the figure. The loop has dimensions $7.00 \mathrm{~m}$ by $5.00 \mathrm{~m}$ and is $1.00 \mathrm{~m}$ away from the wire. What is the induced potential difference in the wire loop at $t=10.0 \mathrm{~s} ?$

Ajay Singhal

Numerade Educator

Problem 42

The long, straight wire in the figure has a current $i=1.00 \mathrm{~A}$ flowing in it. A square loop with 10.0 -cm sides and a resistance of $0.0200 \Omega$ is positioned $10.0 \mathrm{~cm}$ away from the wire. The loop is then moved in the positive $x$ -direction with a speed $v=10.0 \mathrm{~cm} / \mathrm{s}$
a) Find the direction of the induced current in the loop.
b) Identify the directions of the magnetic forces acting on all sides of the square loop.
c) Calculate the direction and the magnitude of the net force acting on the loop at the instant it starts to move.

Dominador Tan

Numerade Educator

Problem 43

A simple generator consists of a loop rotating inside a constant magnetic field (see Figure 29.19 ). If the loop is rotating with frequency $f$, the magnetic flux is given by $\Phi(t)=B A \cos (2 \pi f t)$. If $B=1.00 \mathrm{~T}$ and $A=1.00 \mathrm{~m}^{2}$, what must the value of $f$ be for the maximum induced potential difference to be $110 .$ V?

Ajay Singhal

Numerade Educator

Problem 44

A motor has a single loop inside a magnetic field of magnitude $0.870 \mathrm{~T}$. If the area of the loop is $300 . \mathrm{cm}^{2}$, find the maximum angular speed possible for this motor when connected to a source of emf providing $170 .$ V.

Ajay Singhal

Numerade Educator

Problem 45

Your friend decides to produce electrical power by turning a coil of $1.00 \cdot 10^{5}$ circular loops of wire around an axis perpendicular to the Earth's magnetic field, which has a local magnitude of $0.300 \mathrm{G}$. The loops have a radius of $25.0 \mathrm{~cm} .$
a) If your friend turns the coil at a frequency of $150 . \mathrm{Hz}$, what peak current will flow in a resistor, $R=1.50 \mathrm{k} \Omega$, connected to the coil?
b) The average current flowing in the coil will be 0.7071 times the peak current. What will be the average power obtained from this device?

Ajay Singhal

Numerade Educator

Problem 46

Find the mutual inductance of the solenoid and the coil described in Example 29.1 and the induced potential difference in the coil at $t=2.0 \mathrm{~s}$ using the techniques described in Section $29.7 .$ How do the results for the induced potential difference compare?

Eduard Sanchez

Numerade Educator

Problem 47

The figure shows the current through a $10.0-\mathrm{mH}$ inductor over a time interval of $8.00 \mathrm{~ms}$. Draw a graph showing the self-induced potential difference, $\Delta V_{\text {ind }, L}$, for the inductor over the same interval.

Ajay Singhal

Numerade Educator

Problem 48

A short coil with radius $R=10.0 \mathrm{~cm}$ contains $N=30.0$ turns and surrounds a long solenoid with radius $r=8.00\mathrm{~cm}$ containing $n=60$ turns per centimeter. The current in the short coil is increased at a constant rate from zero to $i=2.00 \mathrm{~A}$ in a time of $t=12.0 \mathrm{~s}$. Calculate the induced potential difference in the long solenoid while the current is increasing in the short coil.

Ajay Singhal

Numerade Educator

Problem 49

Consider an $\mathrm{RL}$ circuit with resistance $R=1.00 \mathrm{M} \Omega$ and inductance $L=1.00 \mathrm{H},$ which is powered by a $10.0-\mathrm{V}$ battery.
a) What is the time constant of the circuit?
b) If the switch is closed at time $t=0$, what is the current just after that time? After $2.00 \mu$ s? When a long time has passed?

Ajay Singhal

Numerade Educator

Problem 50

In the circuit in the figure, $R=120 . \Omega, L=3.00 \mathrm{H},$ and $V_{\mathrm{emf}}=40.0 \mathrm{~V}$
After the switch is closed, how long will it take the current in the inductor to reach $300 . \mathrm{mA} ?$

Ajay Singhal

Numerade Educator

Problem 51

The current is increasing at a rate of $3.60 \mathrm{~A} / \mathrm{s}$ in an $\mathrm{RL}$ circuit with $R=3.25 \Omega$ and $L=440 . \mathrm{mH}$. What is the potential difference across the circuit at the moment when the current in the circuit is $3.00 \mathrm{~A} ?$

Ajay Singhal

Numerade Educator

Problem 52

In the circuit in the figure, a battery supplies $V_{\mathrm{emf}}=18.0 \mathrm{~V}$ and $R_{1}=6.00 \Omega, R_{2}=6.00 \Omega,$ and $L=5.00 \mathrm{H} .$ Calculate each of the following immediately after the switch is closed:
a) the current flowing out of the battery
b) the current through $R_{1}$
c) the current through $R_{2}$
d) the potential difference across $R_{1}$
e) the potential difference across $R_{2}$
f) the potential difference across $L$
g) the rate of current change across $R_{1}$

Ajay Singhal

Numerade Educator

Problem 53

In the circuit in the figure, a battery supplies $V_{\mathrm{cmf}}=18.0 \mathrm{~V}$ and $R_{1}=6.00 \Omega, R_{2}=6.00 \Omega,$ and $L=5.00 \mathrm{H} .$ Calculate each of the following a long time after the switch is closed:
a) the current flowing out of the battery
b) the current through $R_{1}$
c) the current through $R_{2}$
d) the potential difference across $R_{1}$
e) the potential difference across $R_{2}$
f) the potential difference across $L$
g) the rate of current change across $R_{1}$

Ajay Singhal

Numerade Educator

Problem 54

A circuit contains a battery, three resistors, and an inductor, as shown in the figure. What will be the current through each resistor (a) immediately after the switch is closed and (b) a long time after the switch is closed? (c) Suppose the switch is reopened a long time after it has been closed. What is the current in each resistor? After a long time?

Ralph Maestre

Numerade Educator

Problem 55

Having just learned that there is energy associated with magnetic fields, an inventor sets out to tap the energy associated with the Earth's magnetic field. What volume of space near Earth's surface contains 1.00 J of energy, assuming the strength of the magnetic field to be $5.00 \cdot 10^{-5} \mathrm{~T} ?$

Ajay Singhal

Numerade Educator

Problem 56

A clinical MRI (magnetic resonance imaging) superconducting magnet can be approximated as a solenoid with a diameter of $1.00 \mathrm{~m}$ a length of $1.50 \mathrm{~m},$ and a uniform magnetic field of $3.00 \mathrm{~T}$. Determine
(a) the energy density of the magnetic field and (b) the total energy in the solenoid.

Ajay Singhal

Numerade Educator

Problem 57

A magnetar (magnetic neutron star) has a magnetic field near its surface of magnitude $4.00 \cdot 10^{10} \mathrm{~T}$
a) Calculate the energy density of this magnetic field.
b) The Special Theory of Relativity associates energy with any mass $m$ at rest according to $E_{0}=m c^{2}$ (more on this in Chapter 35 ). Find the rest mass density associated with the energy density of part (a).

Ajay Singhal

Numerade Educator

Problem 58

An emf of $20.0 \mathrm{~V}$ is applied to a coil with an inductance of $40.0 \mathrm{mH}$ and a resistance of $0.500 \Omega$.
a) Determine the energy stored in the magnetic field when the current reaches $\frac{1}{4}$ of its maximum value.
b) How long does it take for the current to reach this value?

Ajay Singhal

Numerade Educator

Problem 59

A student wearing a $15.0-\mathrm{g}$ gold band with radius $0.750 \mathrm{~cm}$ (and with a resistance of $61.9 \mu \Omega$ and a specific heat capacity of $c=129 \mathrm{~J} / \mathrm{kg}^{\circ} \mathrm{C}$ ) on her finger moves her finger from a region having a magnetic field of $0.0800 \mathrm{~T}$ pointing along her finger, to a region with zero magnetic field in $40.0 \mathrm{~ms}$. As a result of this action, thermal energy is added to the band due to the induced current, which raises the temperature of the band. Calculate the temperature rise in the band, assuming that all the energy produced is used in raising the temperature.

Ajay Singhal

Numerade Educator

Problem 60

A coil with $N$ turns and area $A,$ carrying a constant current, $i$, flips in an external magnetic field, $\vec{B}_{\text {ext }},$ so that its dipole moment switches from opposition to the field to alignment with the field. During this process, induction produces a potential difference that tends to reduce the current in the coil. Calculate the work done by the coil's power supply to maintain the constant current.

Ajay Singhal

Numerade Educator

Problem 61

An electromagnetic wave propagating in vacuum has electric and magnetic fields given by $\vec{E}(\vec{x}, t)=\vec{E}_{0} \cos (\vec{k} \cdot \vec{x}-\omega t)$ and $\vec{B}(\vec{x}, t)=\vec{B}_{0} \cos (k \cdot \vec{x}-\omega t),$ where $\vec{B}_{0}$ is given by $\vec{B}_{0}=\vec{k} \times \vec{E}_{0} / \omega$ and the wave vector $\vec{k}$ is perpendicular to both $\vec{E}_{0}$ and $\vec{B}_{0}$. The magnitude of $\vec{k}$ and the angular frequency $\omega$ satisfy the dispersion relation, $\omega / \vec{k} \mid=\left(\mu_{0} \epsilon_{0}\right)^{-1 / 2}$ where $\mu_{0}$ and $\epsilon_{0}$ are the permeability and permittivity of free space, respectively. Such a wave transports energy in both its electric and magnetic fields. Calculate the ratio of the energy densities of the magnetic and electric fields, $u_{B} / u_{E}$, in this wave. Simplify your final answer as much as possible.

Manik Pulyani

Numerade Educator

Problem 62

A wire of length $\ell=10.0 \mathrm{~cm}$ is moving with constant velocity in the $x y$ -plane; the wire is parallel to the $y$ -axis and moving along the $x$ -axis. If a magnetic field of magnitude $1.00 \mathrm{~T}$ is pointing along the positive $z$ -axis, what must the velocity of the wire be in order for a potential difference of $2.00 \mathrm{~V}$ to be induced across it?

Ajay Singhal

Numerade Educator

Problem 63

The magnetic field inside the solenoid in the figure changes at the rate of $1.50 \mathrm{~T} / \mathrm{s}$. A conducting coil with 2000 turns surrounds the solenoid, as shown. The radius of the solenoid is $4.00 \mathrm{~cm},$ and the radius of the coil is $7.00 \mathrm{~cm} .$ What is the potential difference induced in the coil?

Ajay Singhal

Numerade Educator

Problem 64

An ideal battery (with no internal resistance) supplies $V_{\text {emf }}$ and is connected to a superconducting (no resistance!) coil of inductance $L$ at time $t=0 .$ Find the current in the coil as a function of time, $i(t) .$ Assume that all connections also have zero resistance.

Ajay Singhal

Numerade Educator

Problem 65

A 100 -turn solenoid of length $8.00 \mathrm{~cm}$ and radius $6.00 \mathrm{~mm}$ carries a current of 0.400 A from right to left. The current is then reversed so that it flows from left to right. By how much does the energy stored in the magnetic field inside the solenoid change?

Ajay Singhal

Numerade Educator

Problem 66

The electric field near the Earth's surface has a magnitude of $150 . \mathrm{N} / \mathrm{C}$ and the magnitude of the Earth's magnetic field near the surface is typically $50.0 \mu \mathrm{T}$. Calculate and compare the energy densities associated with these two fields. Assume that the electric and magnetic properties of air are essentially those of a vacuum.

Ajay Singhal

Numerade Educator

Problem 67

What is the inductance in a series $\mathrm{RL}$ circuit in which $R=3.00 \mathrm{k} \Omega$ if the current increases to $\frac{1}{2}$ of its final value in $20.0 \mu \mathrm{s} ?$

Ajay Singhal

Numerade Educator

Problem 68

A $100 .-V$ battery is connected in series with a $500 .-\Omega$ resistor. According to Faraday's Law of Induction, current can never change instantaneously, so there is always some "stray" inductance. Suppose the stray inductance is $0.200 \mu \mathrm{H} .$ How long will it take the current to build up to within $0.500 \%$ of its final value of 0.200 A after the resistor is connected to the battery?

Ajay Singhal

Numerade Educator

Problem 69

A single loop of wire with an area of $5.00 \mathrm{~m}^{2}$ is located in the plane of the page, as shown in the figure. A time-varying magnetic field in the region of the loop is directed into the page, and its magnitude is given by $B=3.00 \mathrm{~T}+(2.00 \mathrm{~T} / \mathrm{s}) t .$ At $t=2.00 \mathrm{~s},$ what are the induced potential difference in the loop and the direction of the induced current?

Bettina Hanlon

Numerade Educator

Problem 70

A $9.00-V$ battery is connected through a switch to two identical resistors and an ideal inductor, as shown in the figure. Each of the resistors has a resistance of $100 . \Omega$, and the inductor has an inductance of $3.00 \mathrm{H}$. The switch is initially open.
a) Immediately after the switch is closed, what is the current in resistor $R_{1}$ and in resistor $R_{2} ?$
b) At $50.0 \mathrm{~ms}$ after the switch is closed, what is the current in resistor $R_{1}$ and in resistor $R_{2} ?$
c) At 500 . ms after the switch is closed, what is the current in resistor $R_{1}$ and in resistor $R_{2} ?$
d) After a long time $(>10.0 \mathrm{~s})$, the switch is opened again. Immediately after the switch is opened, what is the current in resistor $R_{1}$ and in resistor $R_{2} ?$
e) At $50.0 \mathrm{~ms}$ after the switch is opened, what is the current in resistor $R$ and in resistor $R_{2} ?$
f) At $500 . \mathrm{ms}$ after the switch is opened, what is the current in resistor $R$ and in resistor $R_{2} ?$

Ralph Maestre

Numerade Educator

Problem 71

A long solenoid with length $3.00 \mathrm{~m}$ and $n=290 .$ turns $/ \mathrm{m}$ carries a current of $3.00 \mathrm{~A}$. It stores $2.80 \mathrm{~J}$ of energy. What is the cross-sectional area of the solenoid?

Ajay Singhal

Numerade Educator

Problem 72

A rectangular conducting loop with dimensions $a$ and $b$ and resistance $R$ is placed in the $x y$ -plane. A magnetic field of magnitude $B$ passes through the loop. The magnetic field is in the positive $z$ -direction and varies in time according to $B=B_{0}\left(1+c_{1} t^{3}\right),$ where $c_{1}$ is a constant with units of $1 / \mathrm{s}^{3}$. What is the direction of the current induced in the loop, and what is its value at $t=1 \mathrm{~s}$ (in terms of $a, b, R, B_{0},$ and $\left.c_{1}\right) ?$

Prashant Bana

Numerade Educator

Problem 73

A circuit contains a 12.0 -V battery, a switch, and a light bulb connected in series. When the light bulb has a current of 0.100 A flowing in it, it just starts to glow. This bulb draws $2.00 \mathrm{~W}$ when the switch has been closed for a long time. The switch is opened, and an inductor is added to the circuit, in series with the bulb. If the light bulb begins to glow $3.50 \mathrm{~ms}$ after the switch is closed again, what is the magnitude of the inductance? Ignore any time needed to heat the filament, and assume that you are able to observe a glow as soon as the current in the filament reaches the 0.100 -A threshold.

Vishal Gupta

Numerade Educator

Problem 74

A circular loop of area $A$ is placed perpendicular to a time-varying magnetic field of magnitude $B(t)=B_{0}+a t+b t^{2},$ where $B_{0}, a,$ and $b$ are constants.
a) What is the magnetic flux through the loop at $t=0 ?$
b) Derive an equation for the induced potential difference in the loop as a function of time.
c) What are the magnitude and the direction of the induced current if the resistance of the loop is $R ?$

Ajay Singhal

Numerade Educator

Problem 75

A conducting rod of length $50.0 \mathrm{~cm}$ slides over two parallel metal bars placed in a magnetic field with a magnitude of $1.00 \mathrm{kG},$ as shown in the figure. The ends of the rods are connected by two resistors $R_{1}=100 . \Omega$ and $R_{2}=200 . \Omega$
The conducting rod moves with a constant speed of $8.00 \mathrm{~m} / \mathrm{s}$.
a) What are the currents flowing through the two resistors?
b) What power is delivered to the resistors?
c) What force is needed to keep the rod moving with constant velocity?

Vishal Gupta

Numerade Educator

Problem 76

A rectangular wire loop (dimensions of $h=15.0 \mathrm{~cm}$ and $w=8.00 \mathrm{~cm}$ ) with resistance $R=5.00 \Omega$ is mounted on a door, as shown in the figure. The Earth's magnetic field, $B_{\mathrm{E}}=2.60 \cdot 10^{-5} \mathrm{~T}$, is uniform and perpendicular to the surface of the closed door (the surface is in the $x z$ -plane). At time $t=0$, the door is opened (right edge moves toward the $y$ -axis) at a constant rate, with an opening angle of $\theta(t)=\omega t,$ where $\omega=3.50 \mathrm{rad} / \mathrm{s}$. Calculate the direction and the magnitude of the current induced in the loop, $i(t=0.200 \mathrm{~s})$.

Ajay Singhal

Numerade Educator

Problem 77

A steel cylinder with radius $2.50 \mathrm{~cm}$ and length $10.0 \mathrm{~cm}$ rolls without slipping down a ramp that is inclined at $15.0^{\circ}$ above the horizontal and has a length (along the ramp) of $3.00 \mathrm{~m}$. What is the induced potential difference between the ends of the cylinder as the cylinder leaves the bottom of the ramp, if the downward slope of the ramp points in the direction of the Earth's magnetic field at that location? (Use $0.426 \mathrm{G}$ for the local strength of the Earth's magnetic field.)

Eduard Sanchez

Numerade Educator

Problem 78

The figure shows a circuit in which a battery is connected to a resistor and an inductor in series.
a) What is the current in the circuit at any time $t$ after the switch is closed?
b) Calculate the total energy provided by the battery from $t=0$ to $t=L / R$.
c) Calculate the total energy dissipated in the resistor over the same time period.
d) Is energy conserved in this circuit?

Manik Pulyani

Numerade Educator

Problem 79

As shown in the figure, a rectangular $(60.0 \mathrm{~cm}$ long by $15.0 \mathrm{~cm}$ wide) circuit loop with resistance $35.0 \Omega$ is held parallel to the $x y$ -plane with one half inside a uniform magnetic field. A magnetic field given by $\vec{B}=2.00 \hat{z} \mathrm{~T}$ is directed along the positive $z$ -axis to the right of the dashed line; there is no external magnetic field to the left of the dashed line.
a) Calculate the magnitude of the force required to move the loop to the left at a constant speed of $10.0 \mathrm{~cm} / \mathrm{s}$ while the right end of the loop is still in the magnetic field.
b) What power is expended to pull the loop out of the magnetic field at this speed?
c) What is the power dissipated by the resistor?

Ajay Singhal

Numerade Educator

Problem 80

What is the resistance in an $\mathrm{RL}$ circuit with $L=33.03 \mathrm{mH}$ if the time required for the current to reach $75 \%$ of its maximum value is $3.350 \mathrm{~ms} ?$

Ajay Singhal

Numerade Educator

Problem 81

What is the inductance in an $\mathrm{RL}$ circuit with $R=17.88 \Omega$ if the time required for the current to reach $75 \%$ of its maximum value is $3.450 \mathrm{~ms} ?$

Ajay Singhal

Numerade Educator

Problem 82

For an RL circuit with $R=21.84 \Omega$ and $L=55.93 \mathrm{mH}$, how long does it take the current to reach $75 \%$ of its maximum value?

Ajay Singhal

Numerade Educator

Problem 83

A wedding ring (of diameter $1.95 \mathrm{~cm}$ ) is tossed into the air and given a spin, resulting in an angular velocity of 13.3 rev/s. The rotation axis is a diameter of the ring. If the magnitude of the Earth's magnetic field at the ring's location is $4.77 \cdot 10^{-5} \mathrm{~T}$, what is the maximum induced potential difference in the ring?

Ajay Singhal

Numerade Educator

Problem 84

A wedding ring is tossed into the air and given a spin, resulting in an angular velocity of 13.5 rev/s. The rotation axis is a diameter of the ring. The magnitude of the Earth's magnetic field is $4.97 \cdot 10^{-5} \mathrm{~T}$ at the ring's location. If the maximum induced voltage in the ring is $1.446 \cdot 10^{-6} \mathrm{~V},$ what is the diameter of the ring?

Ajay Singhal

Numerade Educator

Problem 85

A wedding ring of diameter $1.63 \mathrm{~cm}$ is tossed into the air and given a spin, resulting in an angular velocity of $13.7 \mathrm{rev} / \mathrm{s} .$ The rotation axis is a diameter of the ring. If the maximum induced voltage in the ring is $6.556 \cdot 10^{-7} \mathrm{~V},$ what is the magnitude of the Earth's magnetic field at this location?

Ajay Singhal

Numerade Educator