| CALC A $20\text{cm}\times 20\text{cm}$ square loop has a resistance of $0.10\text{Ω}$. A magnetic field perpendicular to the loop is $B=4t-2t^2$, where B is in tesla and t is in seconds. What is the current in the loop at $t=0.0\text{s,}$ $t=1.0\text{s,}$ and $t=2.0\text{s}?$

|| A 100-turn, 2.0-cm-diameter coil is at rest with its axis vertical. A uniform magnetic field $60\text{°}$ away from vertical increases from 0.50 T to 1.50 T in 0.60 s. What is the induced emf in the coil?

|| A 100-turn, 8.0-cm-diameter coil is made of 0.50-mm-diameter copper wire. A magnetic field is parallel to the axis of the coil. At what rate must B increase to induce a 2.0 A current in the coil?

|| CALC A $10\text{cm}\times 10\text{cm}$ square loop of wire lies in the xy-plane. The magnetic field in this region of space is $\vec{B}=(0.30tî+0.50t^2\widehat{k})\text{T,}$ where t is in s. What is the emf induced in the loop at (a) $t=0.5\text{s}$ and (b) $t=1.0\text{s}?$

|| CALC A circular loop made from a flexible, conducting wire is shrinking. Its radius as a function of time is $r=r_0{e}^{-\beta t}$. The loop is perpendicular to a steady, uniform magnetic field B. Find an expression for the induced emf in the loop at time t.

|| CALC A $4.0\text{cm}\times 4.0\text{cm}$ square loop of wire is perpendicular to a uniform magnetic field. From $t=0\text{s}$ to $t=6\text{s}$ the magnetic field strength varies with time as $B=(t^3-6t^2+14t)\text{T,}$ where t is in s. What is the maximum emf induced in the loop during this interval of time?

Hint: Use the signed version of Faraday’s law in Equation 30.27.

||| CALC An $8.0\text{cm}\times 8.0\text{cm}$ square loop is halfway into a magnetic field perpendicular to the plane of the loop. The loop’s mass is 10 g and its resistance is $0.010\text{Ω}$. A switch is closed at $t=0\text{s,}$ causing the magnetic field to increase from 0 to 1.0 T in 0.010 s.

1. What is the induced current in the square loop?

2. With what speed is the loop “kicked” away from the magnetic field?

Hint: What is the impulse on the loop?

|| CALC A $20\text{cm}\times 20\text{cm}$ square loop of wire lies in the xy-plane with its bottom edge on the x-axis. The resistance of the loop is $0.50\text{Ω}$. A magnetic field parallel to the z-axis is given by $B=0.80y^{2}t$, where B is in tesla, y in meters, and t in seconds. What is the size of the induced current in the loop at $t=0.50\text{s}?$

||| A $2.0\text{cm}\times 2.0\text{cm}$ square loop of wire with resistance $0.010\text{Ω}$ has one edge parallel to a long straight wire. The near edge of the loop is 1.0 cm from the wire. The current in the wire is increasing at the rate of $100\text{A}\text{/}\text{s}$. What is the current in the loop?

|| FIGURE P30.46 shows a 4.0-cm-diameter loop with resistance $0.10\text{Ω}$ around a 2.0-cm-diameter solenoid. The solenoid is 10 cm long, has 100 turns, and carries the current shown in the graph. A positive current is cw when seen from the left. Find the current in the loop at (a) $t=0.5\text{s,}$ (b) $t=1.5\text{s,}$ and (c) $t=2.5\text{s}$.

FIGURE P30.46

|| FIGURE P30.47 shows a 1.0-cm-diameter loop with $R=0.50\text{Ω}$ inside a 2.0-cm-diameter solenoid. The solenoid is 8.0 cm long, has 120 turns, and carries the current shown in the graph. A positive current is cw when seen from the left. Determine the current in the loop at $t=0.010\text{s}$.

FIGURE P30.47

|| FIGURE P30.48 shows two 20-turn coils tightly wrapped on the same 2.0-cm-diameter cylinder with 1.0-mm-diameter wire. The current through coil 1 is shown in the graph. Determine the current in coil 2 at (a) $t=0.05\text{s}$ and (b) $t=0.25\text{s}$. A positive current is into the figure at the top of a loop. Assume that the magnetic field of coil 1 passes entirely through coil 2.

FIGURE P30.48

| CALC An electric generator has an 18-cm-diameter, 120-turn coil that rotates at 60 Hz in a uniform magnetic field that is perpendicular to the rotation axis. What magnetic field strength is needed to generate a peak voltage of 170 V?

|| CALC A 40-turn, 4.0-cm-diameter coil with $R=0.40\text{Ω}$ surrounds a 3.0-cm-diameter solenoid. The solenoid is 20 cm long and has 200 turns. The 60 Hz current through the solenoid is $I=I_0\text{sin}\left(2\pi ft\right)$. What is $I_0$ if the maximum induced current in the coil is 0.20 A?

|| A small, 2.0-mm-diameter circular loop with $R=0.020\text{Ω}$ is at the center of a large 100-mm-diameter circular loop. Both loops lie in the same plane. The current in the outer loop changes from $+1.0\text{A}$ to $-1.0\text{A}$ in 0.10 s. What is the induced current in the inner loop?

|| CALC The square loop shown in FIGURE P30.52 moves into a 0.80 T magnetic field at a constant speed of $10\text{m}\text{/}\text{s}$. The loop has a resistance of $0.10\text{Ω}$, and it enters the field at $t=0\text{s}$.

1. Find the induced current in the loop as a function of time. Give your answer as a graph of I versus t from $t=0\text{s}$ to $t=0.020\text{s}$.

2. What is the maximum current? What is the position of the loop when the current is maximum?

FIGURE P30.52

|| A rectangular metal loop with $0.050\text{Ω}$ resistance is placed next to one wire of the RC circuit shown in FIGURE P30.53. The capacitor is charged to 20 V with the polarity shown, then the switch is closed at $t=0\text{s}$.

1. What is the direction of current in the loop for $t>0\text{s}?$

2. What is the current in the loop at $t=5.0\mu \text{s}?$ Assume that only the circuit wire next to the loop is close enough to produce a significant magnetic field.

FIGURE P30.53

|| CALC The L-shaped conductor in FIGURE P30.54 moves at 10 m/s across and touches a stationary L-shaped conductor in a 0.10 T magnetic field. The two vertices overlap, so that the enclosed area is zero, at $t=0\text{s}$. The conductor has a resistance of 0.010 ohms per meter.

1. What is the direction of the induced current?

2. Find expressions for the induced emf and the induced current as functions of time.

3. Evaluate $\epsilon$ and I at $t=0.10\text{s}$.

FIGURE P30.54

|| INT A 20-cm-long, zero-resistance slide wire moves outward, on zero-resistance rails, at a steady speed of $10\text{m}\text{/}\text{s}$ in a 0.10 T magnetic field. (See Figure 30.26.) On the opposite side, a $1.0\text{Ω}$ carbon resistor completes the circuit by connecting the two rails. The mass of the resistor is 50 mg.

1. What is the induced current in the circuit?

2. How much force is needed to pull the wire at this speed?

3. If the wire is pulled for 10 s, what is the temperature increase of the carbon? The specific heat of carbon is $710\text{J}\text{/}\text{kg K}$.

| The 10-cm-wide, zero-resistance slide wire shown in FIGURE P30.56 is pushed toward the $2.0\text{Ω}$ resistor at a steady speed of $0.50\text{m/s}$. The magnetic field strength is 0.50 T.

1. How big is the pushing force?

2. How much power does the pushing force supply to the wire?

3. What are the direction and magnitude of the induced current?

4. How much power is dissipated in the resistor?

FIGURE P30.56

|| CALC Your camping buddy has an idea for a light to go inside your tent. He happens to have a powerful (and heavy!) horseshoe magnet that he bought at a surplus store. This magnet creates a 0.20 T field between two pole tips 10 cm apart. His idea is to build the hand-cranked generator shown in FIGURE P30.57. He thinks you can make enough current to fully light a $1.0\text{Ω}$ lightbulb rated at 4.0 W. That’s not super bright, but it should be plenty of light for routine activities in the tent.

1. Find an expression for the induced current as a function of time if you turn the crank at frequency $f$. Assume that the semicircle is at its highest point at $t=0\text{s}$.

2. With what frequency will you have to turn the crank for the maximum current to fully light the bulb? Is this feasible?

FIGURE P30.57

|| INT You’ve decided to make the magnetic projectile launcher shown in FIGURE P30.58 for your science project. An aluminum bar slides along metal rails through a magnetic field B. The switch closes at $t=0\text{s,}$ while the bar is at rest, and a battery of emf ${\epsilon }_{\text{bat}}$ starts a current flowing around the loop. The battery has internal resistance r. The resistances of the rails, which are separated by distance l, and the bar are effectively zero.

1. Show that the bar reaches a terminal speed $v_{\text{term}}$, and find an expression for $v_{\text{term}}$.

2. Evaluate $v_{\text{term}}$ for ${\epsilon }_{\text{bat}}=1.0\text{V,}$ $r=0.10\text{Ω}$, $l=6.0\text{cm,}$ and $B=0.50\text{T}$.

FIGURE P30.58

|| INT FIGURE P30.59 shows a U-shaped conducting rail that is oriented vertically in a horizontal magnetic field. The rail has no electric resistance and does not move. A slide wire with mass m and resistance R can slide up and down without friction while maintaining electrical contact with the rail. The slide wire is released from rest.

1. Show that the slide wire reaches a terminal speed $v_{\text{term}}$, and find an expression for $v_{\text{term}}$.

2. Determine the value of $v_{\text{term}}$ if $l=20\text{cm,}$ $m=10\text{g,}$ $R=0.10\text{Ω}$, and $B=0.50\text{T}$.

FIGURE P30.59

|| BIO Experiments to study vision often need to track the movements of a subject’s eye. One way of doing so is to have the subject sit in a magnetic field while wearing special contact lenses with a coil of very fine wire circling the edge. A current is induced in the coil each time the subject rotates his eye. Consider the experiment of FIGURE P30.60 in which a 20-turn, 6.0-mm-diameter coil of wire circles the subject’s cornea while a 1.0 T magnetic field is directed as shown. The subject begins by looking straight ahead. What emf is induced in the coil if the subject shifts his gaze by $5\text{°}$ in 0.20 s?

FIGURE P30.60

|| A 10-turn coil of wire having a diameter of 1.0 cm and a resistance of 0.20 Ω is in a 1.0 mT magnetic field, with the coil oriented for maximum flux. The coil is connected to a ballistic galvanometer, a device that measures the total charge passing through. The coil is quickly pulled out of the magnetic field. Afterward, what does the galvanometer read?

Hint: Use $I=dq\text{/}dt$ to relate the net change of flux to the amount of charge that flows through the galvanometer.

|| One way to measure the strength of a magnetic field is with a flip coil. Suppose a 200-turn, 4.0-cm-diameter coil with a resistance of 2.0 Ω is connected to a ballistic galvanometer, a device that measures the total charge passing through. The coil is held perpendicular to the field, then quickly flipped 180° so that the opposite side is facing the magnetic field. Afterward, the galvanometer reads 7.5 μC. What is the field strength?

Hint: Use $I=dq/dt$ to relate the net change of flux to the amount of charge that flows through the galvanometer.

|| CALC Equation 30.26 is an expression for the induced electric field inside a solenoid $(r<R)$. Find an expression for the induced electric field outside a solenoid $(r>R)$ in which the magnetic field is changing at the rate $dB\text{/}dt$.

|| A solenoid inductor has an emf of 0.20 V when the current through it changes at the rate $10.0\text{A}\text{/}\text{s}$. A steady current of 0.10 A produces a flux of $5.0\mu \text{Wb}$ per turn. How many turns does the inductor have?

| BIO One possible concern with MRI (see Exercise 28) is turning the magnetic field on or off too quickly. Bodily fluids are conductors, and a changing magnetic field could cause electric currents to flow through the patient. Suppose a typical patient has a maximum cross-section area of $0.060{\text{m}}^2$. What is the smallest time interval in which a 5.0 T magnetic field can be turned on or off if the induced emf around the patient’s body must be kept to less than 0.10 V?

|| FIGURE P30.66 shows the current through a 10 mH inductor. Draw a graph showing the potential difference $\text{Δ}{V}_{\text{L}}$ across the inductor for these 6 ms.

FIGURE P30.66

|| CALC FIGURE P30.67 shows the potential difference across a 20 mH inductor. The current through the inductor at $t=0\text{ms}$ is $0.25\text{A}$. What is the current at $t=10\text{ms}?$

FIGURE P30.67

||| A 3.6 mH inductor with negligible resistance has a 1.0 A current through it. The current starts to increase at $t=0\text{s}$, creating a constant 5.0 mV voltage across the inductor. How much charge passes through the inductor between $t=0\text{s}$ and $t=5.0\text{s}$?

|| A 50 cm solenoid with 1000 turns has an inductance of 20 mH.

1. What is the magnetic field strength inside the inductor when the current is 75 mA?

2. Flipping a switch disconnects the inductor from the battery and connects it to a resistor. What is the value of the resistance if the magnetic field decreases by 50% in $150\mu \text{s}?$

| CALC The current through inductance L is given by $I=I_0{e}^{-t\text{/}\tau }$.

1. Find an expression for the potential difference $\text{Δ}{V}_{\text{L}}$ across the inductor.

2. Evaluate $\text{Δ}{V}_{\text{L}}$ at $t=0$, 1.0, and 3.0 ms if $L=20\text{mH,}$ $I_0=$ $50\text{mA,}$ and $\tau =1.0\text{ms}$.

||| An $LC$ circuit is built with a 20 mH inductor and an 8.0 pF capacitor. The capacitor voltage has its maximum value of 25 V at $t=0\text{s}$.

1. How long is it until the capacitor is first fully discharged?

2. What is the inductor current at that time?

| An electric oscillator is made with a $0.10\mu \text{F}$ capacitor and a 1.0 mH inductor. The capacitor is initially charged to 5.0 V. What is the maximum current through the inductor as the circuit oscillates?

|| For your final exam in electronics, you’re asked to build an LC circuit that oscillates at 10 kHz. In addition, the maximum current must be 0.10 A and the maximum energy stored in the capacitor must be $1.0\times {10}^{-5}\text{J}$. What values of inductance and capacitance must you use?

|| The inductor in FIGURE P30.74 is a 9.0-cm-long, 2.0-cm-diameter solenoid wrapped with 300 turns. What is the current in the circuit $10\mu \text{s}$ after the switch is moved from 1 to 2?

FIGURE P30.74

|| The $300\mu \text{F}$ capacitor in FIGURE P30.75 is initially charged to 100 V, the $1200\mu \text{F}$ capacitor is uncharged, and the switches are both open.

1. What is the maximum voltage to which you can charge the $1200\mu \text{F}$ capacitor by the proper closing and opening of the two switches?

2. How would you do it? Describe the sequence in which you would close and open switches and the times at which you would do so. The first switch is closed at $t=0\text{s}$.

FIGURE P30.75

| The switch in FIGURE P30.76 has been open for a long time. It is closed at $t=0\text{s}$. What is the current through the $20\text{Ω}$ resistor

1. Immediately after the switch is closed?

2. After the switch has been closed a long time?

3. Immediately after the switch is reopened?

FIGURE P30.76

|| The switch in FIGURE P30.77 has been open for a long time. It is closed at $t=0\text{s}$.

1. After the switch has been closed for a long time, what is the current in the circuit? Call this current $I_0$.

2. Find an expression for the current I as a function of time. Write your expression in terms of $I_0$, R, and L.

3. Sketch a current-versus-time graph from $t=0\text{s}$ until the current is no longer changing.

FIGURE P30.77

|| To determine the inductance of an unmarked inductor, you set up the circuit shown in FIGURE P30.78. After moving the switch from 1 to 2 at $t=0\text{s,}$ you monitor the resistor voltage with an oscilloscope. Your data are shown in the table:

FIGURE P30.78

Use an appropriate graph of the data to determine the inductance.

|| $\mathrm{5.0 }\mu \text{s}$ after the switch of FIGURE P30.79 is moved from 1 to 2, the magnetic energy stored in the inductor has decreased by half. What is the value of the inductance L?

FIGURE P30.79

||| In recent years it has been possible to buy a 1.0 F capacitor. This is an enormously large amount of capacitance. Suppose you want to build a 1.0 Hz oscillator with a 1.0 F capacitor. You have a spool of 0.25-mm-diameter wire and a 4.0-cm-diameter plastic cylinder. How long must your inductor be if you wrap it with 2 layers of closely spaced turns?

||| CALC The rectangular loop in FIGURE CP30.81 has $0.020\text{Ω}$ resistance. What is the induced current in the loop at this instant?

FIGURE CP30.81

||| CALC The metal wire in FIGURE CP30.82 moves with speed $v$ parallel to a straight wire that is carrying current I. The distance between the two wires is d. Find an expression for the potential difference between the two ends of the moving wire.

FIGURE CP30.82

||| CALC Let’s look at the details of eddy-current braking. A square loop, length l on each side, is shot with velocity $v_0$ into a uniform magnetic field B. The field is perpendicular to the plane of the loop. The loop has mass m and resistance R, and it enters the field at $t=0\text{s}$. Assume that the loop is moving to the right along the x-axis and that the field begins at $x=0\text{m}$.

1. Find an expression for the loop’s velocity as a function of time as it enters the magnetic field. You can ignore gravity, and you can assume that the back edge of the loop has not entered the field.

2. Calculate and draw a graph of $v$ over the interval $0\text{s}\le t\le 0.04\text{s}$ for the case that $v_0=10\text{m/s,}$ $l=10\text{cm,}$ $m=1.0\text{g,}$ $R=0.0010\text{Ω}$, and $B=0.10\text{T}$. The back edge of the loop does not reach the field during this time interval.

||| CALC A 3.0-cm-diameter, 10-turn coil of wire, located at $z=0$ in the xy-plane, carries a current of 2.5 A. A 2.0-mm-diameter conducting loop with $2.0\times {10}^{-4}\text{Ω}$ resistance is also in the xy-plane at the center of the coil. At $t=0\text{ s,}$ the loop begins to move along the z-axis with a constant speed of 75 m/s. What is the induced current in the conducting loop at $t=200\mu \text{s}?$ The diameter of the conducting loop is much smaller than that of the coil, so you can assume that the magnetic field through the loop is everywhere the on-axis field of the coil.

||| CALC A 2.0-cm-diameter solenoid is wrapped with 1000 turns per meter. 0.50 cm from the axis, the strength of an induced electric field is $5.0\times {10}^{-4}\text{V/m}$. What is the rate $dI\text{/}dt$ with which the current through the solenoid is changing?

||| CALC High-frequency signals are often transmitted along a coaxial cable, such as the one shown in FIGURE CP30.86. For example, the cable TV hookup coming into your home is a coaxial cable. The signal is carried on a wire of radius $r_1$ while the outer conductor of radius $r_2$ is grounded. A soft, flexible insulating material fills the space between them, and an insulating plastic coating goes around the outside.

1. Find an expression for the inductance per meter of a coaxial cable. To do so, consider the flux through a rectangle of length l that spans the gap between the inner and outer conductors.

2. Evaluate the inductance per meter of a cable having $r_1=0.50\text{mm}$ and $r_2=3.0\text{mm}$.