Problems & Exercises

What is the value of the magnetic flux through the coil in Figure 23.53(b) due to the wire?

Referring to Figure 23.54(b), what is the direction of the current induced in the coil: (a) If the current in the wire increases? (b) If the current in the wire decreases? (c) If the current in the wire suddenly changes direction? Explicitly show how you follow the steps in the Problem-Solving Strategy for Lenz’s Law.

Repeat the previous problem with the battery reversed.

Verify that the units of $\mathrm{\Delta }\Phi$/ $\mathrm{\Delta }t$ are volts. That is, show that $1\text{T}\cdot {\text{m}}^2/\text{s}=\mathrm{1\; V}$.

Suppose a 50-turn coil lies in the plane of the page in a uniform magnetic field that is directed into the page. The coil originally has an area of $0.250{\text{m}}^{\text{2}}$. It is stretched to have no area in 0.100 s. What is the direction and magnitude of the induced emf if the uniform magnetic field has a strength of 1.50 T?

Integrated Concepts

Referring to the situation in the previous problem: (a) What current is induced in the ring if its resistance is 0.0100 $\Omega$? (b) What average power is dissipated? (c) What magnetic field is induced at the center of the ring? (d) What is the direction of the induced magnetic field relative to the MRI’s field?

A 0.250 m radius, 500-turn coil is rotated one-fourth of a revolution in 4.17 ms, originally having its plane perpendicular to a uniform magnetic field. (This is 60 rev/s.) Find the magnetic field strength needed to induce an average emf of 10,000 V.

Integrated Concepts

(a) A lightning bolt produces a rapidly varying magnetic field. If the bolt strikes the earth vertically and acts like a current in a long straight wire, it will induce a voltage in a loop aligned like that in Figure 23.54(b). What voltage is induced in a 1.00 m diameter loop 50.0 m from a $2\text{.}\text{00}\times {\text{10}}^6\text{A}$ lightning strike, if the current falls to zero in $\mathrm{25.0\; \mu s}$? (b) Discuss circumstances under which such a voltage would produce noticeable consequences.

Use Faraday’s law, Lenz’s law, and RHR-1 to show that the magnetic force on the current in the moving rod in Figure 23.10 is in the opposite direction of its velocity.

If a current flows in the Satellite Tether shown in Figure 23.11, use Faraday’s law, Lenz’s law, and RHR-1 to show that there is a magnetic force on the tether in the direction opposite to its velocity.

(a) A nonferrous screwdriver is being used in a 2.00 T magnetic field. What maximum emf can be induced along its 12.0 cm length when it moves at 6.00 m/s? (b) Is it likely that this emf will have any consequences or even be noticed?

The 12.0 cm long rod in Figure 23.10 moves at 4.00 m/s. What is the strength of the magnetic field if a 95.0 V emf is induced?

Prove that when $B$, $\ell$, and $v$ are not mutually perpendicular, motional emf is given by $\text{emf}=\mathrm{B\ell v}\text{sin}\theta$. If $v$ is perpendicular to $B$, then $\theta$ is the angle between $\ell$ and $B$. If $\ell$ is perpendicular to $B$, then $\theta$ is the angle between $v$ and $B$.

In the August 1992 space shuttle flight, only 250 m of the conducting tether considered in Example 23.2 could be let out. A 40.0 V motional emf was generated in the Earth’s $5\text{.}\text{00}\times {\text{10}}^{-5}\text{T}$ field, while moving at $7\text{.}\text{80}\times {\text{10}}^3\text{m/s}$. What was the angle between the shuttle’s velocity and the Earth’s field, assuming the conductor was perpendicular to the field?

Integrated Concepts

Derive an expression for the current in a system like that in Figure 23.10, under the following conditions. The resistance between the rails is $R$, the rails and the moving rod are identical in cross section $A$ and have the same resistivity $\rho$. The distance between the rails is l, and the rod moves at constant speed $v$ perpendicular to the uniform field $B$. At time zero, the moving rod is next to the resistance $R$.

Integrated Concepts

The Tethered Satellite in Figure 23.11 has a mass of 525 kg and is at the end of a 20.0 km long, 2.50 mm diameter cable with the tensile strength of steel. (a) How much does the cable stretch if a 100 N force is exerted to pull the satellite in? (Assume the satellite and shuttle are at the same altitude above the Earth.) (b) What is the effective force constant of the cable? (c) How much energy is stored in it when stretched by the 100 N force?

Make a drawing similar to Figure 23.13, but with the pendulum moving in the opposite direction. Then use Faraday’s law, Lenz’s law, and RHR-1 to show that magnetic force opposes motion.

At what angular velocity in rpm will the peak voltage of a generator be 480 V, if its 500-turn, 8.00 cm diameter coil rotates in a 0.250 T field?

What is the peak emf generated by a 0.250 m radius, 500-turn coil is rotated one-fourth of a revolution in 4.17 ms, originally having its plane perpendicular to a uniform magnetic field. (This is 60 rev/s.)

Integrated Concepts

This problem refers to the bicycle generator considered in the previous problem. It is driven by a 1.60 cm diameter wheel that rolls on the outside rim of the bicycle tire. (a) What is the velocity of the bicycle if the generator’s angular velocity is 1875 rad/s? (b) What is the maximum emf of the generator when the bicycle moves at 10.0 m/s, noting that it was 18.0 V under the original conditions? (c) If the sophisticated generator can vary its own magnetic field, what field strength will it need at 5.00 m/s to produce a 9.00 V maximum emf?

Show that if a coil rotates at an angular velocity $\omega$, the period of its AC output is $\mathrm{2\pi /\omega }$.

(a) If the emf of a coil rotating in a magnetic field is zero at $t=0$, and increases to its first peak at $t=0\text{.}\text{100}\text{ms}$, what is the angular velocity of the coil? (b) At what time will its next maximum occur? (c) What is the period of the output? (d) When is the output first one-fourth of its maximum? (e) When is it next one-fourth of its maximum?

A motor operating on 240 V electricity has a 180 V back emf at operating speed and draws a 12.0 A current. (a) What is its resistance? (b) What current does it draw when it is first started?

The motor in a toy car operates on 6.00 V, developing a 4.50 V back emf at normal speed. If it draws 3.00 A at normal speed, what current does it draw when starting?

An American traveler in New Zealand carries a transformer to convert New Zealand’s standard 240 V to 120 V so that she can use some small appliances on her trip. (a) What is the ratio of turns in the primary and secondary coils of her transformer? (b) What is the ratio of input to output current? (c) How could a New Zealander traveling in the United States use this same transformer to power her 240 V appliances from 120 V?

(a) What is the voltage output of a transformer used for rechargeable flashlight batteries, if its primary has 500 turns, its secondary 4 turns, and the input voltage is 120 V? (b) What input current is required to produce a 4.00 A output? (c) What is the power input?

A multipurpose transformer has a secondary coil with several points at which a voltage can be extracted, giving outputs of 5.60, 12.0, and 480 V. (a) The input voltage is 240 V to a primary coil of 280 turns. What are the numbers of turns in the parts of the secondary used to produce the output voltages? (b) If the maximum input current is 5.00 A, what are the maximum output currents (each used alone)?

If the power output in the previous problem is 1000 MW and line resistance is $\mathrm{2.00\; \Omega }$, what were the old and new line losses?

Construct Your Own Problem

Consider a double transformer to be used to create very large voltages. The device consists of two stages. The first is a transformer that produces a much larger output voltage than its input. The output of the first transformer is used as input to a second transformer that further increases the voltage. Construct a problem in which you calculate the output voltage of the final stage based on the input voltage of the first stage and the number of turns or loops in both parts of both transformers (four coils in all). Also calculate the maximum output current of the final stage based on the input current. Discuss the possibility of power losses in the devices and the effect on the output current and power.

If two coils placed next to one another have a mutual inductance of 5.00 mH, what voltage is induced in one when the 2.00 A current in the other is switched off in 30.0 ms?

A device is turned on and 3.00 A flows through it 0.100 ms later. What is the self-inductance of the device if an induced 150 V emf opposes this?

Starting with ${\text{emf}}_2=-M\frac{\mathrm{\Delta }{I}_1}{\mathrm{\Delta }t}$, show that the units of inductance are $(\text{V}\cdot \text{s})\text{/A}=\Omega \cdot \text{s}$.

Camera flashes charge a capacitor to high voltage by switching the current through an inductor on and off rapidly. In what time must the 0.100 A current through a 2.00 mH inductor be switched on or off to induce a 500 V emf?

(a) Calculate the self-inductance of a 50.0 cm long, 10.0 cm diameter solenoid having 1000 loops. (b) How much energy is stored in this inductor when 20.0 A of current flows through it? (c) How fast can it be turned off if the induced emf cannot exceed 3.00 V?

The heating coils in a hair dryer are 0.800 cm in diameter, have a combined length of 1.00 m, and a total of 400 turns. (a) What is their total self-inductance assuming they act like a single solenoid? (b) How much energy is stored in them when 6.00 A flows? (c) What average emf opposes shutting them off if this is done in 5.00 ms (one-fourth of a cycle for 50 Hz AC)?

How fast can the 150 A current through a 0.250 H inductor be shut off if the induced emf cannot exceed 75.0 V?

Unreasonable Results

A 25.0 H inductor has 100 A of current turned off in 1.00 ms. (a) What voltage is induced to oppose this? (b) What is unreasonable about this result? (c) Which assumption or premise is responsible?

Your RL circuit has a characteristic time constant of 20.0 ns, and a resistance of $\text{5.00 MΩ}$. (a) What is the inductance of the circuit? (b) What resistance would give you a 1.00 ns time constant, perhaps needed for quick response in an oscilloscope?

Verify that after a time of 10.0 ms, the current for the situation considered in Example 23.9 will be 0.183 A as stated.

(a) What is the characteristic time constant of a 25.0 mH inductor that has a resistance of $\text{4.00 Ω}$? (b) If it is connected to a 12.0 V battery, what is the current after 12.5 ms?

The 5.00 A current through a 1.50 H inductor is dissipated by a $\text{2.00 Ω}$ resistor in a circuit like that in Figure 23.42 with the switch in position 2. (a) What is the initial energy in the inductor? (b) How long will it take the current to decline to 5.00% of its initial value? (c) Calculate the average power dissipated, and compare it with the initial power dissipated by the resistor.

(a) Using the exact exponential treatment, find the time required for the current through a 2.00 H inductor in series with a $\text{0.500 Ω}$ resistor to be reduced to 0.100% of its original value. (b) Compare your answer to the approximate treatment using integral numbers of $\tau$. (c) Discuss how significant the difference is.

What value of inductance should be used if a $\text{20.0 kΩ}$ reactance is needed at a frequency of 500 Hz?

At what frequency will an 80.0 mF capacitor have a reactance of $\text{0.250 Ω}$?

(a) What current flows when a 60.0 Hz, 480 V AC source is connected to a $\text{0.250 μF}$ capacitor? (b) What would the current be at 25.0 kHz?

A 20.0 Hz, 16.0 V source produces a 2.00 mA current when connected to a capacitor. What is the capacitance?

The capacitor in Figure 23.52(a) is designed to filter low-frequency signals, impeding their transmission between circuits. (a) What capacitance is needed to produce a $\text{100 kΩ}$ reactance at a frequency of 120 Hz? (b) What would its reactance be at 1.00 MHz? (c) Discuss the implications of your answers to (a) and (b).

Unreasonable Results

In a recording of voltages due to brain activity (an EEG), a 10.0 mV signal with a 0.500 Hz frequency is applied to a capacitor, producing a current of 100 mA. Resistance is negligible. (a) What is the capacitance? (b) What is unreasonable about this result? (c) Which assumption or premise is responsible?

Construct Your Own Problem

Consider the use of an inductor in series with a computer operating on 60 Hz electricity. Construct a problem in which you calculate the relative reduction in voltage of incoming high frequency noise compared to 60 Hz voltage. Among the things to consider are the acceptable series reactance of the inductor for 60 Hz power and the likely frequencies of noise coming through the power lines.

An RC circuit consists of a $\mathrm{40.0\; \Omega }$ resistor and a $\text{5.00 μF}$ capacitor. (a) Find its impedance at 60.0 Hz and 10.0 kHz. (b) Compare these values of $Z$ with those found in Example 23.12, in which there was also an inductor.

What is the resonant frequency of a 0.500 mH inductor connected to a $\text{40.0 μF}$ capacitor?

Suppose you have a supply of inductors ranging from 1.00 nH to 10.0 H, and capacitors ranging from 1.00 pF to 0.100 F. What is the range of resonant frequencies that can be achieved from combinations of a single inductor and a single capacitor?

What inductance do you need to produce a resonant frequency of 60.0 Hz, when using a $\mathrm{2.00\; \mu F}$ capacitor?

An RLC series circuit has a $\mathrm{2.50\; \Omega }$ resistor, a $\mathrm{100\; \mu H}$ inductor, and an $\mathrm{80.0\; \mu F}$ capacitor.(a) Find the circuit’s impedance at 120 Hz. (b) Find the circuit’s impedance at 5.00 kHz. (c) If the voltage source has $V_{\text{rms}}=5\text{.}\text{60}\text{V}$, what is $I_{\text{rms}}$ at each frequency? (d) What is the resonant frequency of the circuit? (e) What is $I_{\text{rms}}$ at resonance?

An RLC series circuit has a $\mathrm{2.50\; \Omega }$ resistor, a $\mathrm{100\; \mu H}$ inductor, and an $\mathrm{80.0\; \mu F}$ capacitor. (a) Find the power factor at $f=\mathrm{120\; Hz}$. (b) What is the phase angle at 120 Hz? (c) What is the average power at 120 Hz? (d) Find the average power at the circuit’s resonant frequency.

An RLC series circuit has a $\mathrm{200\; \Omega }$ resistor and a 25.0 mH inductor. At 8000 Hz, the phase angle is $\mathrm{45.0º}$. (a) What is the impedance? (b) Find the circuit’s capacitance. (c) If $V_{\text{rms}}=\text{408}\text{V}$ is applied, what is the average power supplied?