In Fig. 30.11, suppose that and With switch open, switch

Two coils have mutual inductance The current in the first coil increases at a uniform rate of (a) What is the magnitude of the induced emf in the second coil? Is it constant? (b) Suppose that the current described is in the second coil rather than the first. What is the magnitude of the induced emf in the first coil?

Two coils are wound around the same cylindrical form, like the coils in Example 30.1. When the current in the first coil is decreasing at a rate of the induced emf in the second coil has magnitude (a) What is the mutual inductance of the pair of coils? (b) If the second coil has 25 turns, what is the flux through each turn when the current in the first coil equals (c) If the current in the second coil increases at a rate of what is the magnitude of the induced emf in the first coil?

A 10.0-cm-long solenoid of diameter 0.400 cm is wound uniformly with 800 turns. A second coil with 50 turns is wound around the solenoid at its center. What is the mutual inductance of the combination of the two coils?

A solenoidal coil with 25 turns of wire is wound tightly around another coil with 300 turns (see Example 30.1). The inner solenoid is 25.0 cm long and has a diameter of 2.00 cm. At a certain time, the current in the inner solenoid is 0.120 A and is increasing at a rate of For this time, calculate: (a) the average magnetic flux through each turn of the inner solenoid; (b) the mutual inductance of the two solenoids; (c) the emf induced in the outer solenoid by the changing current in the inner solenoid.

Two toroidal solenoids are wound around the same form so that the magnetic field of one passes through the turns of the other. Solenoid 1 has 700 turns, and solenoid 2 has 400 turns. When the current in solenoid 1 is 6.52 A, the average flux through each turn of solenoid 2 is 0.0320 Wb. (a) What is the mutual inductance of the pair of solenoids? (b) When the current in solenoid 2 is 2.54 A, what is the average flux through each turn of solenoid 1?

A toroidal solenoid with mean radius r and cross-sectional area A is wound uniformly with turns. A second toroidal solenoid with turns is wound uniformly on top of the first, so that the two solenoids have the same cross-sectional area and mean radius. (a) What is the mutual inductance of the two solenoids? Assume that the magnetic field of the first solenoid is uniform across the cross section of the two solenoids. (b) If turns, turns, , and , what is the value of the mutual inductance?

A 2.50-mH toroidal solenoid has an average radius of 6.00 cm and a cross-sectional area of 2.00 cm2 . (a) How many coils does it have? (Make the same assumption as in Example 30.3.) (b) At what rate must the current through it change so that a potential difference of 2.00 V is developed across its ends?

A toroidal solenoid has 500 turns, cross-sectional area and mean radius 4.00 cm. (a) Calculate the coils selfinductance. (b) If the current decreases uniformly from 5.00 A to 2.00 A in 3.00 ms, calculate the self-induced emf in the coil. (c) The current is directed from terminal a of the coil to terminal b. Is the direction of the induced emf from a to b or from b to a?

At the instant when the current in an inductor is increasing at a rate of the magnitude of the self- induced emf is 0.0160 V. (a) What is the inductance of the inductor? (b) If the inductor is a solenoid with 400 turns, what is the average magnetic flux through each turn when the current is 0.720 A?

When the current in a toroidal solenoid is changing at a rate of the magnitude of the induced emf is 12.6 mV. When the current equals 1.40 A, the average flux through each turn of the solenoid is 0.00285 Wb. How many turns does the solenoid have?

The inductor in Fig. E30.11 has inductance 0.260 H and carries a current in the direction shown that is decreasing at a uniform rate, (a) Find the self-induced emf. (b) Which end of the inductor, a or b, is at a higher potential?

The inductor shown in Fig. E30.11 has inductance 0.260 H and carries a current in the direction shown. The current is changing at a constant rate. (a) The potential between points a and b is with point a at higher potential. Is the current increasing or decreasing? (b) If the current at is 12.0 A, what is the current at

A toroidal solenoid has mean radius 12.0 cm and crosssectional area . (a) How many turns does the solenoid have if its inductance is 0.100 mH? (b) What is the resistance of the solenoid if the wire from which it is wound has a resistance per unit length of ?

A long, straight solenoid has 800 turns. When the current in the solenoid is 2.90 A, the average flux through each turn of the solenoid is \(3.25 \times 10^{-3}\mathrm{\ Wb}\text{. }\)What must be the magnitude of the rate of change of the current in order for the self-induced emf to equal 7.50 mV?

Inductance of a Solenoid. (a) A long, straight solenoid has turns, uniform cross-sectional area and length l. Show that the inductance of this solenoid is given by the equation Assume that the magnetic field is uniform inside the solenoid and zero outside. (Your answer is approximate because is actually smaller at the ends than at the center. For this reason, your answer is actually an upper limit on the inductance.) (b) A metallic laboratory spring is typically 5.00 cm long and 0.150 cm in diameter and has 50 coils. If you connect such a spring in an electric circuit, how much self-inductance must you include for it if you model it as an ideal solenoid?

An inductor used in a dc power supply has an inductance of 12.0 H and a resistance of \(\Omega\) It carries a current of 0.300 A. (a) What is the energy stored in the magnetic field? (b) At what rate is thermal energy developed in the inductor? (c) Does your answer to part (b) mean that the magnetic-field energy is decreasing with time? Explain

An air-filled toroidal solenoid has a mean radius of 15.0 cm and a cross-sectional area of When the current is 12.0 A, the energy stored is 0.390 J. How many turns does the winding have?

An air-filled toroidal solenoid has 300 turns of wire, a mean radius of 12.0 cm, and a cross-sectional area of If the current is 5.00 A, calculate: (a) the magnetic field in the solenoid; (b) the self-inductance of the solenoid; (c) the energy stored in the magnetic field; (d) the energy density in the magnetic field. (e) Check your answer for part (d) by dividing your answer to part (c) by the volume of the solenoid.

A solenoid 25.0 cm long and with a cross-sectional area of contains 400 turns of wire and carries a current of 80.0 A. Calculate: (a) the magnetic field in the solenoid; (b) the energy density in the magnetic field if the solenoid is filled with air; (c) the total energy contained in the coils magnetic field (assume the field is uniform); (d) the inductance of the solenoid.

It has been proposed to use large inductors as energy storage devices. (a) How much electrical energy is converted to light and thermal energy by a 200-W light bulb in one day? (b) If the amount of energy calculated in part (a) is stored in an inductor in which the current is 80.0 A, what is the inductance?

In a proton accelerator used in elementary particle physics experiments, the trajectories of protons are controlled by bending magnets that produce a magnetic field of 4.80 T. What is the magnetic-field energy in a \(10.0-\mathrm{cm}^{3}\) volume of space where B = 4.80 T?

It is proposed to store of electrical energy in a uniform magnetic field with magnitude 0.600 T. (a) What volume (in vacuum) must the magnetic field occupy to store this amount of energy? (b) If instead this amount of energy is to be stored in a volume (in vacuum) equivalent to a cube 40.0 cm on a side, what magnetic field is required?

An inductor with an inductance of 2.50 H and a resistance of \(8.00\ \Omega\) is connected to the terminals of a battery with an emf of 6.00 V and negligible internal resistance. Find (a) the initial rate of increase of current in the circuit; (b) the rate of increase of current at the instant when the current is 0.500 A; (c) the current 0.250 s after the circuit is closed; (d) the final steady-state current.

In Fig. 30.11, and the battery emf is 6.30 V. With switch open, switch is closed. After several minutes, is opened and is closed. (a) At 2.00 ms after is opened, the current has decayed to 0.320 A. Calculate the inductance of the coil. (b) How long after is opened will the current reach 1.00% of its original value?

A 35.0-V battery with negligible internal resistance, a resistor, and a 1.25-mH inductor with negligible resistance are all connected in series with an open switch. The switch is suddenly closed. (a) How long after closing the switch will the current through the inductor reach one-half of its maximum value? (b) How long after closing the switch will the energy stored in the inductor reach one-half of its maximum value?

In Fig. 30.11, switch is closed while switch is kept open. The inductance is and the resistance is (a) When the current has reached its final value, the energy stored in the inductor is 0.260 J. What is the emf of the battery? (b) After the current has reached its final value, is opened and is closed. How much time does it take for the energy stored in the inductor to decrease to 0.130 J, half the original value?

In Fig. 30.11, suppose that and With switch open, switch is left closed until a constant current is established. Then is closed and opened, taking the battery out of the circuit. (a) What is the initial current in the resistor, just after is closed and is opened? (b) What is the current in the resistor at (c) What is the potential difference between points b and c at Which point is at a higher potential? (d) How long does it take the current to decrease to half its initial value?

In Fig. 30.11, suppose that \(\mathcal{E}=60.0 \mathrm{\ V}\), \(R=240\ \Omega\) and L = 0.160 H. Initially there is no current in the circuit. Switch \(\mathrm{S}_{2}\) is left open, and switch \(\mathrm{S}_{1}\) is closed. (a) Just after \(\mathrm{S}_{1}\) is closed, what are the potential differences \(v_{a b}\) and \(v_{b c}\)? (b) A long time (many time constants) after \(\mathrm{S}_{1}\) is closed, what are \(v_{a b}\) and \(v_{b c}\)? (c) What are \(v_{a b}\) and \(v_{b c}\) at an intermediate time when i = 0.150 A?

Refer to the circuit in Exercise 30.23. (a) What is the power input to the inductor from the battery as a function of time if the circuit is completed at (b) What is the rate of dissipation of energy in the resistance of the inductor as a function of time? (c) What is the rate at which the energy of the magnetic field in the inductor is increasing, as a function of time? (d) Compare the results of parts (a), (b), and (c).

In Fig. 30.11 switch is closed while switch is kept open. The inductance is , the resistance is , and the emf of the battery is 18.0 V. At time t after is closed, the current in the circuit is increasing at a rate of . At this instant what is , the voltage across the resistor?

Show that the differential equation of Eq. (30.20) is satisfied by the function \(q=Q\cos(\omega t+\phi)\text{, with }\omega\text{ given by }1/\sqrt{LC}.\)

A \(20.0-\mu \mathrm{F}\) capacitor is charged by a 150.0-V power supply, then disconnected from the power and connected in series with a 0.280-mH inductor. Calculate: (a) the oscillation frequency of the circuit; (b) the energy stored in the capacitor at time t = 0 ms (the moment of connection with the inductor); (c) the energy stored in the inductor at t = 1.30 ms. Text Transcription: 20.0 - mu F

A 7.50-nF capacitor is charged up to 12.0 V, then disconnected from the power supply and connected in series through a coil. The period of oscillation of the circuit is then measured to be Calculate: (a) the inductance of the coil; (b) the maximum charge on the capacitor; (c) the total energy of the circuit; (d) the maximum current in the circuit

A capacitor is placed across a 22.5-V battery for several seconds and is then connected across a 12.0-mH inductor that has no appreciable resistance. (a) After the capacitor and inductor are connected together, find the maximum current in the circuit. When the current is a maximum, what is the charge on the capacitor? (b) How long after the capacitor and inductor are connected together does it take for the capacitor to be completely discharged for the first time? For the second time? (c) Sketch graphs of the charge on the capacitor plates and the current through the inductor as functions of time.

L-C Oscillations. A capacitor with capacitance \(6.00\times10^{-5}\mathrm{\ F}\) is charged by connecting it to a 12.0-V battery. The capacitor is disconnected from the battery and connected across an inductor with L = 1.50 H. (a) What are the angular frequency \(\omega\) of the electrical oscillations and the period of these oscillations (the time for one oscillation)? (b) What is the initial charge on the capacitor? (c) How much energy is initially stored in the capacitor? (d) What is the charge on the capacitor 0.0230 s after the connection to the inductor is made? Interpret the sign of your answer. (e) At the time given in part (d), what is the current in the inductor? Interpret the sign of your answer. (f) At the time given in part (d), how much electrical energy is stored in the capacitor and how much is stored in the inductor?

A Radio Tuning Circuit. The minimum capacitance of a variable capacitor in a radio is 4.18 pF. (a) What is the inductance of a coil connected to this capacitor if the oscillation frequency of the L-C circuit is corresponding to one end of the AM radio broadcast band, when the capacitor is set to its minimum capacitance? (b) The frequency at the other end of the broadcast band is What is the maximum capacitance of the capacitor if the oscillation frequency is adjustable over the range of the broadcast band?

An L-C circuit containing an 80.0-mH inductor and a 1.25-nF capacitor oscillates with a maximum current of 0.750 A. Calculate: (a) the maximum charge on the capacitor and (b) the oscillation frequency of the circuit. (c) Assuming the capacitor had its maximum charge at time calculate the energy stored in the inductor after 2.50 ms of oscillation

In an L-C circuit, and During the oscillations the maximum current in the inductor is 0.850 mA. (a) What is the maximum charge on the capacitor? (b) What is the magnitude of the charge on the capacitor at an instant when the current in the inductor has magnitude 0.500 mA?

An L-R-C series circuit has and resistance R. (a) What is the angular frequency of the circuit when (b) What value must R have to give a 5.0% decrease in angular frequency compared to the value calculated in part (a)?

For the circuit of Fig. 30.17, let and . (a) Calculate the oscillation frequency of the circuit once the capacitor has been charged and the switch has been connected to point a. (b) How long will it take for the amplitude of the oscillation to decay to of its original value? (c) What value of R would result in a critically damped circuit?

(a) In Eq. (14.41), substitute q for x, L for m, 1/C for k, and R for the damping constant b. Show that the result is Eq. (30.27). (b) Make these same substitutions in Eq. (14.43) and show that Eq. (30.29) results. (c) Make these same substitutions in Eq. (14.42) and show that Eq. (30.28) results.

(a) Take first and second derivatives with respect to time of q given in Eq. (30.28), and show that it is a solution of Eq. (30.27). (b) At the switch shown in Fig. 30.17 is thrown so that it connects points d and a; at this time, and Show that the constants and in Eq. (30.28) are given by

One solenoid is centered inside another. The outer one has a length of 50.0 cm and contains 6750 coils, while the coaxial inner solenoid is 3.0 cm long and 0.120 cm in diameter and contains 15 coils. The current in the outer solenoid is changing at 49.2 A/s. (a) What is the mutual inductance of these solenoids? (b) Find the emf induced in the innner solenoid.

A coil has 400 turns and self-inductance 4.80 mH. The current in the coil varies with time according to \(i=(680\mathrm{\ mA})\cos(\pi t/0.0250\mathrm{\ s})\). (a) What is the maximum emf induced in the coil? (b) What is the maximum average flux through each turn of the coil? (c) At t = 0.0180 s, what is the magnitude of the induced emf?

A Differentiating Circuit. The current in a resistanceless inductor is caused to vary with time as shown in the graph of Fig. P30.45. (a) Sketch the pattern that would be observed on the screen of an oscilloscope connected to the terminals of the inductor. (The oscilloscope spot sweeps horizontally across the screen at a constant speed, and its vertical deflection is proportional to the potential difference between the inductor terminals.) (b) Explain why a circuit with an inductor can be described as a “differentiating circuit.”

A 0.250-H inductor carries a time-varying current given by the expression \(i=(124\mathrm{\ mA})\cos[(240\pi/\mathrm{s})t]\). (a) Find an expression for the induced emf as a function of time. Graph the current and induced emf as functions of time for t = 0 to \(t=\frac{1}{60}\mathrm{\ s}\). (b) What is the maximum emf? What is the current when the induced emf is a maximum? (c) What is the maximum current? What is the induced emf when the current is a maximum?

Solar Magnetic Energy. Magnetic fields within a sunspot can be as strong as 0.4 T. (By comparison, the earth's magnetic field is about 1/10,000 as strong.) Sunspots can be as large as 25,000 km in radius. The material in a sunspot has a density of about \(3\times10^{-4}\mathrm{\ kg}/\mathrm{m}^3\). Assume \(\mu\) for the sunspot material is \(\mu_{0}\). If 100% of the magnetic-field energy stored in a sunspot could be used to eject the sunspots material away from the sun's surface, at what speed would that material be ejected? Compare to the sun's escape speed, which is about \(6\times10^5\mathrm{\ m}/\mathrm{s}\). (Hint: Calculate the kinetic energy the magnetic field could supply to \(1\mathrm{\ m}^3\) of sunspot material.)

CP CALC A Coaxial Cable. A small solid conductor with radius a is supported by insulating, nonmagnetic disks on the axis of a thin-walled tube with inner radius b. The inner and outer conductors carry equal currents i in opposite directions. (a) Use Amperes law to find the magnetic field at any point in the volume between the conductors. (b) Write the expression for the flux through a narrow strip of length l parallel to the axis, of width dr, at a distance from the axis of the cable and lying in a plane containing the axis. (c) Integrate your expression from part (b) over the volume between the two conductors to find the total flux produced by a current i in the central conductor. (d) Show that the inductance of a length l of the cable is (e) Use Eq. (30.9) to calculate the energy stored in the magnetic field for a length l of the cable

CP CALC Consider the coaxial cable of Problem 30.48. The conductors carry equal currents i in opposite directions. (a) Use Amperes law to find the magnetic field at any point in the volume between the conductors. (b) Use the energy density for a magnetic field, Eq. (30.10), to calculate the energy stored in a thin, cylindrical shell between the two conductors. Let the cylindrical shell have inner radius r, outer radius and length l. (c) Integrate your result in part (b) over the volume between the two conductors to find the total energy stored in the magnetic field for a length l of the cable. (d) Use your result in part (c) and Eq. (30.9) to calculate the inductance L of a length l of the cable. Compare your result to L calculated in part (d) of Problem 30.48.

A toroidal solenoid has a mean radius r and a crosssectional area A and is wound uniformly with turns. A second toroidal solenoid with turns is wound uniformly around the first. The two coils are wound in the same direction. (a) Derive an expression for the inductance when only the first coil is used and an expression for when only the second coil is used. (b) Show tha

(a) What would have to be the self-inductance of a solenoid for it to store 10.0 J of energy when a 2.00-A current runs through it? (b) If this solenoid’s cross-sectional diameter is 4.00 cm, and if you could wrap its coils to a density of 10 coils/mm, how long would the solenoid be? (See Exercise 30.15.) Is this a realistic length for ordinary laboratory use?

An inductor is connected to the terminals of a battery that has an emf of 12.0 V and negligible internal resistance. The current is 4.86 mA at 0.940 ms after the connection is completed. After a long time the current is 6.45 mA. What are (a) the resistance R of the inductor and (b) the inductance L of the inductor?

Continuation of Exercises 30.23 and 30.29. (a) How much energy is stored in the magnetic field of the inductor one time constant after the battery has been connected? Compute this both by integrating the expression in Exercise 30.29(c) and by using Eq. (30.9), and compare the results. (b) Integrate the expression obtained in Exercise 30.29(a) to find the total energy supplied by the battery during the time interval considered in part (a). (c) Integrate the expression obtained in Exercise 30.29(b) to find the total energy dissipated in the resistance of the inductor during the same time period. (d) Compare the results obtained in parts (a), (b), and (c).

Continuation of Exercise 30.27. (a) What is the total energy initially stored in the inductor? (b) At \(t=4.00\times10^{-4}\mathrm{\ s}\), at what rate is the energy stored in the inductor decreasing? (c) At \(t=4.00\times10^{-4}\mathrm{\ s}\), at what rate is electrical energy being converted into thermal energy in the resistor? (d) Obtain an expression for the rate at which electrical energy is being converted into thermal energy in the resistor as a function of time. Integrate this expression from t = 0 to \(t=\infty\) to obtain the total electrical energy dissipated in the resistor. Compare your result to that of part (a).

CALC The equation preceding Eq. (30.27) may be converted into an energy relationship. Multiply both sides of this equation by The first term then becomes Show that the second term can be written as and that the third term can be written as What does the resulting equation say about energy conservation in the circuit?

A \(7.00-\mu \mathrm{F}\) capacitor is initially charged to a potential of 16.0 V. It is then connected in series with a 3.75-mH inductor. (a) What is the total energy stored in this circuit? (b) What is the maximum current in the inductor? What is the charge on the capacitor plates at the instant the current in the inductor is maximal?

An Electromagnetic Car Alarm. Your latest invention is a car alarm that produces sound at a particularly annoying frequency of To do this, the car-alarm circuitry must produce an alternating electric current of the same frequency. Thats why your design includes an inductor and a capacitor in series. The maximum voltage across the capacitor is to be 12.0 V (the same voltage as the car battery). To produce a sufficiently loud sound, the capacitor must store 0.0160 J of energy. What values of capacitance and inductance should you choose for your car-alarm circuit?

An L-C circuit consists of a 60.0-mH inductor and \(250-\mu \mathrm{F}\) a capacitor. The initial charge on the capacitor is \(6.00\ \mu \mathrm{C}\) and the initial current in the inductor is zero. (a) What is the maximum voltage across the capacitor? (b) What is the maximum current in the inductor? (c) What is the maximum energy stored in the inductor? (d) When the current in the inductor has half its maximum value, what is the charge on the capacitor and what is the energy stored in the inductor? Text Transcription: 250-mu F 6.00 mu C

A 84.0-nF capacitor is charged to 12.0 V, then disconnected from the power supply and connected in series with a coil that has L = 0.0420 H and negligible resistance. At an instant when the charge on the capacitor is \(0.650 \mu \mathrm{C}\), what is the magnitude of the current in the inductor and what is the magnitude of the rate of change of this current?

A charged capacitor with is connected in series to an inductor that has and negligible L = 0.330 H resistance. At an instant when the current in the inductor is , the current is increasing at a rate of . During the current oscillations, what is the maximum voltage across the capacitor?

CP In the circuit shown in Fig. P30.61, the switch has been open for a long time and is suddenly closed. Neither the battery nor the inductors have any appreciable resistance. (a) What do the ammeter and voltmeter read just after S is closed? (b) What do the ammeter and the voltmeter read after S has been closed a very long time? (c) What do the ammeter and the voltmeter read 0.115 ms after S is closed?

While studying a coil of unknown inductance and internal resistance, you connect it in series with a battery and a resistor. You then place an oscilloscope across one of these circuit elements and use the oscilloscope to measure the voltage across the circuit element as a function of time. The result is shown in Fig. P30.62. (a) Across which circuit element (coil or resistor) is the oscilloscope connected? How do you know this? (b) Find the inductance and the internal resistance of the coil. (c) Carefully make a quantitative sketch showing the voltage versus time you would observe if you put the oscilloscope across the other circuit element (resistor or coil).

In the lab, you are trying to find the inductance and internal resistance of a solenoid. You place it in series with a battery of negligible internal resistance, a resistor, and a switch. You then put an oscilloscope across one of these circuit elements to measure the voltage across that circuit element as a function of time. You close the switch, and the oscilloscope shows voltage versus time as shown in Fig. P30.63. (a) Across which circuit element (solenoid or resistor) is the oscilloscope connected? How do you know this? (b) Why doesnt the graph approach zero as (c) What is the emf of the battery? (d) Find the maximum current in the circuit. (e) What are the internal resistance and self- inductance of the solenoid?

In the circuit shown in Fig. P30.64, find the reading in each ammeter and voltmeter (a) just after switch S is closed and (b) after S has been closed a very long time.

CP In the circuit shown in Fig. P30.65, switch S is closed at time with no charge initially on the capacitor. (a) Find the reading of each ammeter and each voltmeter just after S is closed. (b) Find the reading of each meter after a long time has elapsed. (c) Find the maximum charge on the capacitor. (d) Draw a qualitative graph of the reading of voltmeter as a function of time.

In the circuit shown in Fig. P30.66 the battery and the inductor have no appreciable internal resistance and there is no current in the circuit. After the switch is closed, find the readings of the ammeter (A) and voltmeters and (a) the instant after the switch is closed and (b) after the switch has been closed for a very long time. (c) Which answers in parts (a) and (b) would change if the inductance were 24.0 mH instead?

In the circuit shown in Fig. P30.67, switch S is closed at time (a) Find the reading of each meter just after S is closed. (b) What does each meter read long after S is closed?

In the circuit shown in Fig. P30.68, switch S1 has been closed for a long enough time so that the current reads a steady 3.50 A. Suddenly, switch is closed and is opened at the same instant. (a) What is the maximum charge that the capacitor will receive? (b) What is the current in the inductor at this time?

In the circuit shown in Fig. P30.69, \(\mathcal{E}=60.0 \mathrm{~V}\), \(R_{1}=40.0 \Omega, R_{2}=25.0 \Omega\) and L=0.300H. Switch S is closed at t=0. Just after the switch is closed, (a) what is the potential difference \(v_{a b}\) across the resistor (b) which point, a or b, is at a higher potential; (c) what is the potential difference \(v_{c d}\) across the inductor L; (d) which point, c or d, is at a higher potential? The switch is left closed a long time and then opened. Just after the switch is opened, (e) what is the potential difference \(v_{a b}\) across the resistor \(R_{1}\) (f) which point, a or b, is at a higher potential; (g) what is the potential difference \(v_{c d}\) across the inductor L; (h) which point, c or d, is at a higher potential?

In the circuit shown in Fig. P30.69, and (a) Switch S is closed. At some time t afterward, the current in the inductor is increasing at a rate of At this instant, what are the current through and the current through (Hint: Analyze two separate loops: one containing and and the other containing and L.) (b) After the switch has been closed a long time, it is opened again. Just after it is opened, what is the current through

CALC Consider the circuit shown in Fig. P30.71. Let and (a) Switch is closed and switch is left open. Just after is closed, what are the current through and the potential differences and (b) After has been closed a long time is still open) so that the current has reached its final, steady value, what are and (c) Find the expressions for and as functions of the time t since was closed. Your results should agree with part (a) when and with part (b) when Graph and versus time.

After the current in the circuit of Fig. P30.71 has reached its final, steady value with switch closed and open, switch is closed, thus short-circuiting the inductor. (Switch remains closed. See Problem 30.71 for numerical values of the circuit elements.) (a) Just after is closed, what are and and what are the currents through R, and (b) A long time after is closed, what are and and what are the currents through R, and (c) Derive expressions for the currents through R, and as functions of the time t that has elapsed since was closed. Your results should agree with part (a) when and with part (b) when Graph these three currents versus time.

CP CALC We have ignored the variation of the magnetic field across the cross section of a toroidal solenoid. Lets now examine the validity of that approximation. A certain toroidal solenoid has a rectangular cross section (Fig. P30.73). It has N uniformly spaced turns, with air inside. The magnetic field at a point inside the toroid is given by the equation derived in Example 28.10 (Section 28.7). Do not assume the field is uniform over the cross section. (a) Show that the magnetic flux through a cross section of the toroid is (b) Show that the inductance of the toroidal solenoid is given by L = m0N2 h 2p lna b a b (c) The fraction may be written as Use the power series expansion valid for to show that when is much less than a, the inductance is approximately equal to Compare this result with the result given in Example 30.3 (Section 30.2).

CP In the circuit shown in Fig. P30.74, neither the battery nor the inductors have any appreciable resistance, the capacitors are initially uncharged, and the switch S has been in position 1 for a very long time. (a) What is the current in the circuit? (b) The switch is now suddenly flipped to position 2. Find the maximum charge that each capacitor will receive, and how much time after the switch is flipped it will take them to acquire this charge.

Demonstrating Inductance. A common demonstration of inductance employs a circuit such as the one shown in Fig. P30.69. Switch S is closed, and the light bulb (represented by resistance \(R_{1}\)) just barely glows. After a period of time, switch S is opened, and the bulb lights up brightly for a short period of time. To understand this effect, think of an inductor as a device that imparts an “inertia” to the current, preventing a discontinuous change in the current through it. (a) Derive, as explicit functions of time, expressions for \(i_{1}\) (the current through the light bulb) and \(i_{2}\) (the current through the inductor) after switch S is closed. (b) After a long period of time, the currents \(i_{1}\) and \(i_{2}\) reach their steady-state values. Obtain expressions for these steady-state currents. (c) Switch S is now opened. Obtain an expression for the current through the inductor and light bulb as an explicit function of time. (d) You have been asked to design a demonstration apparatus using the circuit shown in Fig. P30.69 with a 22.0-H inductor and a 40.0-W light bulb. You are to connect a resistor in series with the inductor, and \(R_{2}\) represents the sum of that resistance plus the internal resistance of the inductor. When switch S is opened, a transient current is to be set up that starts at 0.600 A and is not to fall below 0.150 A until after 0.0800 s. For simplicity, assume that the resistance of the light bulb is constant and equals the resistance the bulb must have to dissipate 40.0 W at 120 V. Determine \(R_{2}\) and \(\mathcal{E}\) for the given design considerations. (e) With the numerical values determined in part (d), what is the current through the light bulb just before the switch is opened? Does this result confirm the qualitative description of what is observed in the demonstration?

CP CALC Consider the circuit shown in Fig. P30.76. The circuit elements are as follows: and At time switch S is closed. The current through the inductor is the current through the capacitor branch is and the charge on the capacitor is (a) Using Kirchhoffs rules, verify the circuit equations (b) What are the initial values of and (c) Show by direct substitution that the following solutions for and satisfy the circuit equations from part (a). Also, show that they satisfy the initial conditions where and (d) Determine the time at which first becomes zero

A Volume Gauge. A tank containing a liquid has turns of wire wrapped around it, causing it to act like an inductor. The liquid content of the tank can be measured by using its inductance to determine the height of the liquid in the tank. The inductance of the tank changes from a value of \(L_{0}\) corresponding to a relative permeability of 1 when the tank is empty to a value of \(L_{f}\) corresponding to a relative permeability of \(K_{m}\) (the relative permeability of the liquid) when the tank is full. The appropriate electronic circuitry can determine the inductance to five significant figures and thus the effective relative permeability of the combined air and liquid within the rectangular cavity of the tank. The four sides of the tank each have width W and height D (Fig. P30.77). The height of the liquid in the tank is d. You can ignore any fringing effects and assume that the relative permeability of the material of which the tank is made can be ignored. (a) Derive an expression for d as a function of L, the inductance corresponding to a certain fluid height, \(L_{0}\), \(L_{f}\) and D. (b) What is the inductance (to five significant figures) for a tank \(\frac{1}{4}\) full, \(\frac{1}{2}\) full, \(\frac{3}{4}\) full, and completely full if the tank contains liquid oxygen? Take \(L_0=0.63000\ \mathrm{H}\). The magnetic susceptibility of liquid oxygen is \(\chi_{\mathrm{m}}=1.52 \times 10^{-3}\). (c) Repeat part (b) for mercury. The magnetic susceptibility of mercury is given in Table 28.1. (d) For which material is this volume gauge more practical?

Two coils are wrapped around each other as shown in Fig. 30.3. The current travels in the same sense around each coil. One coil has self-inductance and the other coil has selfinductance The mutual inductance of the two coils is M. (a) Show that if the two coils are connected in series, the equivalent inductance of the combination is (b) Show that if the two coils are connected in parallel, the equivalent inductance of the combination is

CP CALC Consider the circuit shown in Fig. P30.79. Switch S is closed at time causing a current through the inductive branch and a current through the capaci i tive branch. The initial charge on the capacitor is zero, and the charge at time t is (a) Derive expressions for and as functions of time. Express your answers in terms of L, C, and t. For the remainder of the problem let the circuit elements have the following values: and (b) What is the initial current through the inductive branch? What is the initial current 5000 . C = 20 mF, R1 = 25 , R2 = E = 48 V, L = 8.0 H, R2 R , 1, E, q2 i2, i1 q , 2. through the capacitive branch? (c) What are the currents through the inductive and capacitive branches a long time after the switch has been closed? How long is a long time? Explain. (d) At what time (accurate to two significant figures) will the currents and be equal? (Hint: You might consider using series expansions for the exponentials.) (e) For the conditions given in part (d), determine (f) The total current through the battery is At what time (accurate to two significant figures) will i equal one-half of its final value? (Hint: The numerical work is greatly simplified if one makes suitable approximations. A sketch of and versus t may help you decide what approximations are valid.)

Problem 1DQ In an electric trolley or bus system, the vehicle’s motor draws current from an overhead wire by means of a long arm with an attachment at the end that slides along the overhead wire. A brilliant electric spark is often seen when the attachment crosses a junction in the wires where contact is momentarily lost. Explain this phenomenon.

Problem 1E Two coils have mutual inductance M = 3.25 × 10-4 H. The current i1 in the first coil increases at a uniform rate of 830 A/s. (a) What is the magnitude of the induced emf in the second coil? Is it constant? (b) Suppose that the current described is in the second coil rather than the first. What is the magnitude of the induced emf in the first coil?

From Eq. (30.5) 1 H = 1 Wb/A , and from Eq. (30.4) 1 H = 1 ??s. Show that these two definitions are equivalent.

Two coils are wound around the same cylindrical form, like the coils in Example 30.1. When the current in the first coil is decreasing at a rate of \(-0.242 \mathrm{~A} / \mathrm{s}\), the induced emf in the second coil has magnitude \(1.65 \times 10^{-3}\) V (a) What is the mutual inductance of the pair of coils? (b) If the second coil has 25 turns, what is the flux through each turn when the current in the first coil equals 1.20 A? (c) If the current in the second coil increases at a rate of what is the magnitude of \(0.360 \mathrm{~A} / \mathrm{s}\), the induced emf in the first coil? Equation Transcription: Text Transcription: -0.242 A/s 1.65x10^-3 0.360 A/s

Problem 3DQ In Fig. 30.1, if coil 2 is turned 90° so that its axis is vertical, does the mutual inductance increase or decrease? Explain.

Problem 3E A 10.0-cm-long solenoid of diameter 0.400 cm is wound uniformly with 800 turns. A second coil with 50 turns is wound around the solenoid at its center. What is the mutual inductance of the combination of the two coils?

Problem 4DQ The tightly wound toroidal solenoid is one of the few configurations for which it is easy to calculate self-inductance. What features of the toroidal solenoid give it this simplicity?

A solenoidal coil with 25 turns of wire is wound tightly around another coil with 300 turns (see Example 30.1). The inner solenoid is 25.0 cm long and has a diameter of 2.00 cm. At a certain time, the current in the inner solenoid is 0.120 A and is increasing at a rate of \(1.75 \times 10^{3} \mathrm{~A} / \mathrm{s}\). For this time, calculate: (a) the average magnetic flux through each turn of the inner solenoid; (b) the mutual inductance of the two solenoids; (c) the emf induced in the outer solenoid by the changing current in the inner solenoid. Equation Transcription: Text Transcription: 1.75x10^3 A/s

Problem 5DQ Two identical, closely wound, circular coils, each having self-inductance L, are placed next to each other, so that they are coaxial and almost touching. If they are connected in series, what is the self-inductance of the combination? What if they are connected in parallel? Can they be connected so that the total inductance is zero? Explain.

Problem 5E Two toroidal solenoids are wound around the same form so that the magnetic field of one passes through the turns of the other. Solenoid 1 has 700 turns, and solenoid 2 has 400 turns. When the current in solenoid 1 is 6.52 A, the average flux through each turn of solenoid 2 is 0.0320 Wb. (a) What is the mutual inductance of the pair of solenoids? (b) When the current in solenoid 2 is 2.54 A, what is the average flux through each turn of solenoid 1?

Problem 6DQ Two closely wound circular coils have the same number of turns, but one has twice the radius of the other. How are the self-inductances of the two coils related? Explain your reasoning.

A toroidal solenoid with mean radius r and cross-sectional area A is wound uniformly with \(N_{1}\) turns. A second toroidal solenoid with \(N_{2}\) turns is wound uniformly on top of the first, so that the two solenoids have the same cross-sectional area and mean radius. (a) What is the mutual inductance of the two solenoids? Assume that the magnetic field of the first solenoid is uniform across the cross section of the two solenoids. (b) If \(N_{1}\) = 500 turns, \(N_{2}\) = 300 turns, r = 10.0 cm, and A = 0.800 cm\(^2\), what is the value of the mutual inductance?

Problem 7DQ You are to make a resistor by winding a wire around a cylindrical form. To make the inductance as small as possible, it is proposed that you wind half the wire in one direction and the other half in the opposite direction. Would this achieve the desired result? Why or why not?

Problem 13E A toroidal solenoid has mean radius 12.0 cm and cross-sectional area 0.600 cm2. (a) How many turns does the solenoid have if its inductance is 0.100 mH? (b) What is the resistance of the solenoid if the wire from which it is wound has a resistance per unit length of 0.0760 ?/m?

Problem 14DQ In an L-R-C series circuit, what criteria could be used to decide whether the system is overdamped or underdamped? For example, could we compare the maximum energy stored during one cycle to the energy dissipated during one cycle? Explain.

Problem 14E long, straight solenoid has 800 turns. When the current in the solenoid is 2.90 A, the average flux through each turn of the solenoid is 3.25 × 10-3 Wb. What must be the magnitude of the rate of change of the current in order for the self-induced emf to equal 6.20 mV?

Problem 15E Inductance of a Solenoid. (a) A long, straight solenoid has N turns, uniform cross-sectional area A, and length l. Show that the inductance of this solenoid is given by the equation L = µ0AN2/l. Assume that the magnetic field is uniform inside the solenoid and zero outside. (Your answer is approximate because B is actually smaller at the ends than at the center. For this reason, your answer is actually an upper limit on the inductance.) (b) A metallic laboratory spring is typically 5.00 cm long and 0.150 cm in diameter and has 50 coils. If you connect such a spring in an electric circuit, how much self-inductance must you include for it if you model it as an ideal solenoid?

Problem 16E An inductor used in a dc power supply has an inductance of 12.0 H and a resistance of 180 ?. It carries a current of 0.300 A. (a) What is the energy stored in the magnetic field? (b) At what rate is thermal energy developed in the inductor? (c) Does your answer to part (b) mean that the magnetic-field energy is decreasing with time? Explain.

Problem 17E An air-filled toroidal solenoid has a mean radius of 15.0 cm and a cross-sectional area of 5.00 cm2. When the current is 12.0 A, the energy stored is 0.390 J. How many turns does the winding have?

Problem 18E An air-filled toroidal solenoid has 300 turns of wire, a mean radius of 12.0 cm, and a cross-sectional area of 4.00 cm2. If the current is 5.00 A, calculate: (a) the magnetic field in the solenoid; (b) the self-inductance of the solenoid; (c) the energy stored in the magnetic field; (d) the energy density in the magnetic field. (e) Check your answer for part (d) by dividing your answer to part (c) by the volume of the solenoid.

Problem 19E A solenoid 25.0 cm long and with a cross-sectional area of 0.500 cm2 contains 400 turns of wire and carries a current of 80.0 A. Calculate: (a) the magnetic field in the solenoid; (b) the energy density in the magnetic field if the solenoid is filled with air; (c) the total energy contained in the coil’s magnetic field (assume the field is uniform); (d) the inductance of the solenoid.

Problem 20E It has been proposed to use large inductors as energy storage devices. (a) How much electrical energy is convened to light End thermal energy by a 200-W light bulb in one day? (b) If the amount of energy calculated in part. (a) is stored in an inductor in -which the current is 80.0 A, what is the inductance?

In Fig. 30.11, \(R=15.0\) \(\Omega\) and the battery emf is 6.30 V. With switch open \(\mathrm{S}_{2}\), switch \(\mathrm{S}_{1}\) is closed. After several minutes, \(\mathrm{S}_{1}\) is opened and \(\mathrm{S}_{2}\) is closed. (a) At 2.00 ms after \(\mathrm{S}_{1}\) is opened, the current has decayed to 0.320 A. Calculate the inductance of the coil. (b) How long after \(\mathrm{S}_{1}\) is opened will the current reach 1.00% of its original value? Equation Transcription: Text Transcription: R=15.0 Ohms S_2 S_1 S_1 S_2 S_1 S_1

Problem 25E A 35.0-V battery with negligible internal resistance, a 50.0-? resistor, and a 1.25-mH inductor with negligible resistance are all connected in series with an open switch. The switch is suddenly closed. (a) How long after closing the switch will the current through the inductor reach one-half of its maximum value? (b) How long after closing the switch will the energy stored in the inductor reach one-half of its maximum value?

In Fig. 30.11, switch \(\mathrm{S}_{1}\) is closed while switch \(\mathrm{S}_{2}\) is kept open. The inductance is \(L=0.115\) H, and the resistance is \(R=120\) \(\Omega\). (a) When the current has reached its final value, the energy stored in the inductor is 0.260 J. What is the emf \(\varepsilon\) of the battery? (b) After the current has reached its final value, \(\mathrm{S}_{1}\) is opened and \(\mathrm{S}_{2}\) is closed. How much time does it take for the energy stored in the inductor to decrease to 0.130 J, half the original value? Equation Transcription: Text Transcription: S_2 L=0.115 R=120 Ohms Epsilon S_1 S_2

In Fig. 30.11, suppose that \(\varepsilon=60.0\) V, \(R=240\) \(\Omega\), and L = 0.160 H. With switch \(\mathrm{S}_{2}\) open, switch \(\mathrm{S}_{1}\) is left closed until a constant current is established. Then \(\mathrm{S}_{2}\) is closed and \(\mathrm{S}_{1}\) opened, taking the battery out of the circuit. (a) What is the initial current in the resistor, just after \(\mathrm{S}_{1}\) is closed and \(\mathrm{S}_{2}\) is opened? (b) What is the current in the resistor at \(t=4.00 \times 10^{-4}\) s? (c) What is the potential difference between points b and c at \(t=4.00 \times 10^{-4}\) s? Which point is at a higher potential? (d) How long does it take the current to decrease to half its initial value? Equation Transcription: Text Transcription: epsilon=60.0 R=240 Ohms L=0.160 S_2 S_1 S_2 S_1 S_1 S_2 t=4.00x10^-4 t=4.00x10^-4

In Fig. 30.11, suppose that \(\varepsilon=60.0\) V, \(R=240\) \(\Omega\), and \(L=0.160\) H. Initially there is no current in the circuit. Switch \(\mathrm{S}_{2}\) is left open, and switch \(\mathrm{S}_{1}\) is closed. (a) Just after \(\mathrm{S}_{1}\) is closed, what are the potential differences \(v_{a b}\) and \(v_{b c}\)? (b) A long time (many time constants) after \(\mathrm{S}_{1}\) is closed, what are \(v_{a b}\) and \(v_{b c}\)? (c) What are \(v_{a b}\) and \(v_{b c}\) at an intermediate time when \(i=0.150\) A? Equation Transcription: Text Transcription: epsilon=60.0 R=240 Ohms L=0.160 S_2 S_1 S_1 v_ab v_bc S_1 v_ab v_bc v_ab v_bc i=0.150

Problem 29E (a) What is the power input to the inductor from the battery as a function of time if the circuit is completed at t = 0? (b) What is the rate of dissipation of energy in the resistance of the inductor as a function of time? (c)What is the rate at which the energy of the magnetic field in the inductor is increasing, as a function of time? (d) Compare the results of parts (a), (b), and (c).

(a) Take first and second derivatives with respect to time of \(q\) given in Eq. (30.28), and show that it is a solution of Eq. (30.27). (b) At \(t=0\) the switch shown in Fig. 30.17 is thrown so that it connects points \(d\) and \(a\); at this time, \(q=Q\) and \(i=d q / d t=0\).Show that the constants and in Eq. (30.28) are given by \(\tan \phi=-\frac{R}{2 L \sqrt{(1 / L C)-\left(R^{2} 4 L^{2}\right)}} \text { and } A=\frac{Q}{\cos \phi}\) Equation Transcription: Text Transcription: q t=0 d a q=Q i=dq/dt=0 phi tanphi=-R over 2L sqrt (1/LC)-(R^2/4L^2) and A=Q over cosphi

Problem 43P One solenoid is centered inside another. The outer one has a length of 50.0 cm and contains 6750 coils, while the coaxial inner solenoid is 3.0 cm long and 0.120 cm in diameter and contains 15 coils. The current in the outer solenoid is changing at 49.2 A/s. (a) What is the mutual inductance of these solenoids? (b) Find the emf induced in the inner solenoid.

Problem 44P A coil has 400 turns and self-inductance 4.80 mH. The current in the coil varies with time according to i = (680 mA) cos (?t/0.0250 s). (a) What is the maximum emf induced in the coil? (b) What is the maximum average flux through each turn of the coil? (c)At t = 0.0180 s, what is the magnitude of the induced emf?

A Differentiating Circuit. The current in a resistanceless inductor is caused to vary with time as shown in the graph of Fig. P30.45. (a) Sketch the pattern that would be observed on the screen of an oscilloscope connected to the terminals of the inductor. (The oscilloscope spot sweeps horizontally across the screen at a constant speed, and its vertical deflection is proportional to the potential difference between the inductor terminals.) (b) Explain why a circuit with an inductor can be described as a “differentiating circuit.”

Problem 46P A 0.250-H inductor carries a time-varying current given by the expression i = (124 mA) cos[(240?/s)t] (a) Find an expression for the induced emf as a functions of time. Graph the current and induced emf as functions of time for t = 0 and t = 1/60s. (b)What is the maximum emf? What is the current when the induced emf is a maximum? (c) What is the maximum current? What is the induced emf when the current is a maximum?

Problem 47P Solar Magnetic Energy. Magnetic fields within a sunspot can be as strong as 0.4 T. (By comparison, the earth’s magnetic field is about 1/10,000 as strong.) Sunspots can be as large as 25,000 km in radius. The material in a sunspot has a density of about 3 × 10-4 kg/m3. Assume µ for the sunspot material is µ0. If 100% of the magnetic-field energy stored in a sunspot could be used to eject the sunspot’s material away from the sun’s surface, at what speed would that material be ejected? Compare to the sun’s escape speed, which is about 6 × 105 m/s. (Hint: Calculate the kinetic energy the magnetic field could supply to 1 m3 of sunspot material.)

CP CALC A Coaxial Cable. A small solid conductor with radius a is supported by insulating, nonmagnetic disks on the axis of a thin-walled tube with inner radius \(b\). The inner and outer conductors carry equal currents \(i\) in opposite directions. (a) Use Ampere’s law to find the magnetic field at any point in the volume between the conductors. (b) Write the expression for the flux \(d \Phi_{\mathrm{B}}\) through a narrow strip of length l parallel to the axis, of width dr, at a distance from the axis of the cable and lying in a plane containing the axis. (c) Integrate your expression from part (b) over the volume between the two conductors to find the total flux produced by a current i in the central conductor. (d) Show that the inductance of a length l of the cable is \(L=l \frac{\mu_{0}}{2 \pi} \ln \left(\frac{b}{\alpha}\right)\) (e) Use Eq. (30.9) to calculate the energy stored in the magnetic field for a length l of the cable. Equation Transcription: dB Text Transcription: b i d phi B L=l mu_0/2 pi ln(b/alpha)

CP CALC Consider the coaxial cable of Problem 30.48. The conductors carry equal currents i in opposite directions. (a) Use Ampere’s law to find the magnetic field at any point in the volume between the conductors. (b) Use the energy density for a magnetic field, Eq. (30.10), to calculate the energy stored in a thin, cylindrical shell between the two conductors. Let the cylindrical shell have inner radius r, outer radius \(r+d r\), and length l. (c) Integrate your result in part (b) over the volume between the two conductors to find the total energy stored in the magnetic field for a length l of the cable. (d) Use your result in part (c) and Eq. (30.9) to calculate the inductance L of a length l of the cable. Compare your result to L calculated in part (d) of Problem 30.48. Equation Transcription: Text Transcription: r+dr

Problem 50P A toroidal solenoid has a mean radius r and a crosssectional area A and is wound uniformly with N1 turns. A second toroidal solenoid with N2 turns is wound uniformly around the first. The two coils are wound in the same direction. (a) Derive an expression for the inductance L1 when only the first coil is used and an expression for L2 when only the second coil is used. (b) Show that M2 = L1L2

CALC Continuation of Exercise 30.27. (a) What is the total energy initially stored in the inductor? (b) At \(t=4.00 \times 10^{4}\) s, at what rate is the energy stored in the inductor decreasing? (c) At \(t=4.00 \times 10^{4}\) s, at what rate is electrical energy being converted into thermal energy in the resistor? (d) Obtain an expression for the rate at which electrical energy is being converted into thermal energy in the resistor as a function of time. Integrate this expression from \(t=0\) to \(t=\infty\) to obtain the total electrical energy dissipated in the resistor. Compare your result to that of part (a) Equation Transcription: Text Transcription: t=4.00x10^4 t=4.00x10^4 t=0 t=infinity

CALC The equation preceding Eq. (30.27) may be converted into an energy relationship. Multiply both sides of this equation by \(-i=-d q / d t\). The first term then becomes \(i^{2} R\). Show that the second term can be written as \(d\left(\frac{1}{2} L i^{2}\right) / d t\) and that the third term can be written as \(d\left(q^{2} 2 C\right) / d t\). What does the resulting equation say about energy conservation in the circuit? Equation Transcription: Text Transcription: -i=-dq/dt i^2R d(1 over 2 Li^2)/dt d(q^22C)/dt

Problem 56P A 7.00-?F capacitor is initially charged to a potential of 16.0 V. It is then connected in series with a 3.75-mH inductor. (a) What is the total energy stored in this circuit? (b) What is the maximum current in the inductor? What is the charge on the capacitor plates at the instant the current in the inductor is maximal?

Problem 57P An Electromagnetic Car Alarm. Your latest invention is a car alarm that produces sound at a particularly annoying frequency of 3500 Hz. To do this, the car-alarm circuitry must produce an alternating electric current of the same frequency. That’s why your design includes an inductor and a capacitor in series. The maximum voltage across the capacitor is to be 12.0 V. To produce a sufficiently loud sound, the capacitor must store 0.0160 J of energy. What values of capacitance and inductance should you choose for your car-alarm circuit?

Problem 58P An L-C circuit consists of a 60.0-mH inductor and a 250-?F capacitor. The initial charge on the capacitor is 6.00 ?C, and the initial current in the inductor is zero. (a) What is the maximum voltage across the capacitor? (b) What is the maximum current in the inductor? (c) What is the maximum energy stored in the inductor? (d) When the current in the inductor has half its maximum value, what is the charge on the capacitor and what is the energy stored in the inductor?

Problem 59P A 84.0 -nF capacitor is charged to 12.0 V, then disconnected from the power supply and connected in series with a coil that has L = 0.0420 H and negligible resistance. At an instant when the charge on the capacitor is 0.650 µC, what is the magnitude of the current in the inductor and what is the magnitude of the rate of change of this current?

CP In the circuit shown in Fig. P30.74, neither the battery nor the inductors have any appreciable resistance, the capacitors are initially uncharged, and the switch S has been in position 1 for a very long time. (a) What is the current in the circuit? (b) The switch is now suddenly flipped to position 2. Find the maximum charge that each capacitor will receive, and how much time after the switch is flipped it will take them to acquire this charge. Equation Transcription: Text Transcription: 75 V 125.0 ohms 15.0 mH 25.0 muF 5.0 mH 35.0 muF

Demonstrating Inductance. A common demonstration of inductance employs a circuit such as the one shown in Fig. P30.69. Switch S is closed, and the light bulb (represented by resistance R1) just barely glows. After a period of time, switch S is opened, and the bulb lights up brightly for a short period of time. To understand this effect, think of an inductor as a device that imparts an “inertia” to the current, preventing a discontinuous change in the current through it. (a) Derive, as explicit functions of time, expressions for i1 (the current through the light bulb) and i2 (the current through the inductor) after switch S is closed. (b) After a long period of time, the currents i1 and i2 reach their steady-state values. Obtain expressions for these steady-state currents. (c) Switch S is now opened. Obtain an expression for the current through the inductor and light bulb as an explicit function of time. (d) You have been asked to design a demonstration apparatus using the circuit shown in Fig. P30.69 with a 22.0-H inductor and a 40.0-W light bulb. You are to connect a resistor in series with the inductor, and R2 represents the sum of that resistance plus the internal resistance of the inductor. When switch S is opened, a transient current is to be set up that starts at 0.600 A and is not to fall below 0.150 A until after 0.0800 s. For simplicity, assume that the resistance of the light bulb is constant and equals the resistance the bulb must have to dissipate 40.0 W at 120 V. Determine R2 and ? for the given design considerations. (e) With the numerical values determined in part (d), what is the current through the light bulb just before the switch is opened? Does this result confirm the qualitative description of what is observed in the demonstration?

Consider the circuit shown in Fig. P30.76. The circuit elements are as follows: \(\mathcal{E}=32.0 \mathrm{~V}, L=0.640 \mathrm {~H}, C=2.00 ~\mu \mathrm{F} \text {, and } R=400 ~\Omega\). At time t = 0, switch S is closed. The current through the inductor is \(i_{1}\), the current through the capacitor branch is \(i_{2}\), and the charge on the capacitor is \(q_{2}\). (a) Using Kirchhoff’s rules, verify the circuit equations \(R\left(i_{1}+i_{2}\right)+L\left(\frac{d i_{1}}{d t}\right)=\mathcal{E}\) \(R\left(i_{1}+i_{2}\right)+\frac{q_{2}}{C}=\mathcal{E}\) (b) What are the initial values of \(i_{1}, i_{2} \text {, and } q_{2} ?\) (c) Show by direct substitution that the following solutions for \(i_{1} \text { and } q_{2}\) satisfy the circuit equations from part (a). Also, show that they satisfy the initial conditions \(i_{1}=\left(\frac{\mathcal{E}}{R}\right)\left[1-e^{-\beta t}\left\{(2 \omega R C)^{-1} \sin (\omega t)+\cos (\omega t)\right\}\right]\) \(q_{2}=\left(\frac{\mathcal{E}}{\omega R}\right) e^{-\beta t} \sin (\omega t)\) Where \(\beta=(2 R C)^{-1} \text { and } \omega=\left[(L C)^{-1}-(2 R C)^{-2}\right]^{1 / 2}\). (d) Determine the time \(t_{1}\) at which \(i_{2}\) first becomes zero.

A Volume Gauge. A tank containing a liquid has turns of wire wrapped around it, causing it to act like an inductor. The liquid content of the tank can be measured by using its inductance to determine the height of the liquid in the tank. The inductance of the tank changes from a value of \(L_{0}\) corresponding to a relative permeability of 1 when the tank is empty to a value of \(L_{f}\) corresponding to a relative permeability of \(K_{m}\) (the relative permeability of the liquid) when the tank is full. The appropriate electronic circuitry can determine the inductance to five significant figures and thus the effective relative permeability of the combined air and liquid within the rectangular cavity of the tank. The four sides of the tank each have width W and height D (Fig. P30.77). The height of the liquid in the tank is d. You can ignore any fringing effects and assume that the relative permeability of the material of which the tank is made can be ignored. (a) Derive an expression for d as a function of L, the inductance corresponding to a certain fluid height \(L_{0}\), \(L_{f}\), and D. (b) What is the inductance (to five significant figures) for a tank \(\frac{1}{4}\) full, \(\frac{1}{2}\) full, \(\frac{3}{4}\) full, and completely full if the tank contains liquid oxygen? Take \(L_{0}=0.63000\) H. The magnetic susceptibility of liquid oxygen is \(X_{m}=1.52 \times 10^{-3}\). (c) Repeat part (b) for mercury. The magnetic susceptibility of mercury is given in Table 28.1. (d) For which material is this volume gauge more practical? Equation Transcription: Text Transcription: L_0 L_f K_m L_0 L_f 1/4 1/2 3/4 L_0=0.63000 X_m=1.5210-3

Two coils are wrapped around each other as shown in Fig. 30.3. The current travels in the same sense around each coil. One coil has self-inductance \(L_{1}\), and the other coil has self-inductance \(L_{2}\). The mutual inductance of the two coils is M. (a) Show that if the two coils are connected in series, the equivalent inductance of the combination is \(L_{e q}=L_{1}+L_{2}+2 M\). (b) Show that if the two coils are connected in parallel, the equivalent inductance of the combination is \(L_{\mathrm{eq}}=\frac{L_{1} L_{2}-M^{2}}{L_{1}+L_{2}-2 M}\) Equation Transcription: Text Transcription: L_1 L_2 L_eq=L_1+L_2+2M L_eq=L_1L_2-M^2 over L_1+L_2-2M

Consider the circuit shown in Fig. P30.79. Switch S is closed at time \(t=0\), causing a current \(i_{1}) through the inductive branch and a current \(i_{2}) through the capacitive branch. The initial charge on the capacitor is zero, and the charge at time is \(q_{2}) (a) Derive expressions for \(i_{1}), \(i_{2}) and \(q_{2}) as functions of time. Express your answers in terms of \(\varepsilon\),\(R_{1}\), \(R_{2}\) and . For the remainder of the problem let the circuit elements have the following values: \(\varepsilon=48\) V, \(L=8.0\) H, \(C=20\) \(\mu \mathrm{F}\), \(R_{1}=25\) \(\Omega\), and \(R_{2}=5000\) \(\Omega\).(b) What is the initial current through the inductive branch? What is the initial current through the capacitive branch? (c) What are the currents through the inductive and capacitive branches a long time after the switch has been closed? How long is a “long time”? Explain. (d) At what time \(t_{1}) (accurate to two significant figures) will the currents \(i_{1}) and \(i_{2}) be equal? (Hint: You might consider using series expansions for the exponentials.) (e) For the conditions given in part (d), determine \(i_{1}). (f) The total current through the battery is \(i=i_{1}+i_{2}\). At what time \(t_{2}) (accurate to two significant figures) will i equal one-half of its final value? (Hint: The numerical work is greatly simplified if one makes suitable approximations. A sketch of \(i_{1}) and \(i_{2}) versus may help you decide what approximations are valid.) Equation Transcription: Text Transcription: t=0 i_1 i_2 q_2 i_1 i_2 q_2 R_1 R_2 epsilon=48 L=8.0 C=20 muF R_1=25 R_2=5000 t_1 i_1 i_2 i_1 i=i_1+i_2 t_2 i_1 i_2

A 2.50-mH toroidal solenoid has an average radius of 6.00 cm and a cross-sectional area of 2.00 cm\(^{2}\). (a) How many coils does it have? (Make the same assumption as in Example 30.3.) (6) At what rate must the current through it change so that a potential difference of 2.00 V is developed across its ends?

For the same magnetic field strength , is the energy density greater in vacuum or in a magnetic material? Explain. Does Eq. (30.11) imply that for a long solenoid in which the current is the energy stored is proportional to \(1 / \mu ?\) And does this mean that for the same current less energy is stored when the solenoid is filled with a ferromagnetic material rather than with air? Explain. Equation transcription: Text transcription: 1 / mu ?

Problem 8E A toroidal solenoid has 500 turns, cross-sectional area 6.25 cm2, and mean radius 4.00 cm. (a) Calculate the coil’s self-inductance. (b) If the current decreases uniformly from 5.00 A to 2.00 A in 3.00 ms, calculate the self-induced emf in the coil. (c) The current is directed from terminal a of the coil to terminal b. Is the direction of the induced emf from a to b or from b to a?

In Section 30.5 Kirchhoff’s loop rule is applied to an L-C circuit where the capacitor is initially fully charged and the equation -L (di/dt) – (q/C) = 0 is derived. But as the capacitor starts to discharge, the current increases from zero. The equation says L di/dt = -q/C, so it says L di/dt is negative. Explain how L di/dt can be negative when the current is increasing.

Problem 9E At the instant when the current in an inductor is increasing at a rate of 0.0640 A/s, the magnitude of the self-induced emf is 0.0160 V. (a) What is the inductance of the inductor? (b) If the inductor is a solenoid with 400 turns, what is the average magnetic flux through each turn when the current is 0.720 A?

In Section the relationship is used in deriving Eq. (30.20). But a flow of current corresponds to a decrease in the charge on the capacitor. Explain, therefore, why this is the correct equation to use in the derivation, rather than \(i=-d q / d t\). Equation transcription: Text transcription: i=-d q / d t

Problem 10E When the current in a toroidal solenoid is changing at a rate of 0.0260 A/s, the magnitude of the induced emf is 12.6 mV. When the current equals 1.40 A, the average flux through each turn of the solenoid is 0.00285 Wb. How many turns does the solenoid have?

In the circuit shown in Fig. , when switch \(S_{1}\) is closed, the potential \(V_{a b}\) changes suddenly and discontinuously, but the current does not. Explain why the voltage can change suddenly but the current can't. Equation transcription: Text transcription: S{1} V{a b}

The inductor in Fig. E30.11 has inductance 0.260H and carries a current in the direction shown that is decreasing at a uniform rate, \(d i / d t=-0.0180 \mathrm{~A} / \mathrm{s}\) (a) Find the self-induced emf. (b) Which end of the inductor, or , is at a higher potential?

In the R-L circuit shown in Fig. 30.11, is the current in the resistor always the same as the current in the inductor? How do you know?

The inductor shown in Fig. E30.11 has inductance and carries a current in the direction shown. The current is changing at a constant rate. (a) The potential between points and is \(V_{a b}=1.04 V\), with point at higher potential. Is the current increasing or decreasing? (b) If the current at is , what is the current at ? Equation transcription: Text transcription: V{a b}=1.04 V

Suppose there is a steady current in an inductor. If you attempt to reduce the current to zero instantaneously by quickly opening a switch, an arc can appear at the switch contacts. Why? Is it physically possible to stop the current instantaneously? Explain.

Problem 21E In a proton accelerator used in elementary particle physics experiments, the trajectories of protons are controlled by bending magnets that produce a magnetic field of 4.80 T. What is the magnetic-field energy in a 10.0-cm3 volume of space where B = 4.80 T?

It is proposed to store 1.00 kW ? h = 3.60 \(\times\) 106 J of electrical energy in a uniform magnetic field with magnitude 0.600 T. (a) What volume (in vacuum) must the magnetic field occupy to store this amount of energy? (b) If instead this amount of energy is to be stored in a volume (in vacuum) equivalent to a cube 40.0 cm on a side, what magnetic field is required?

Problem 23E An inductor with an inductance of 2.50 H and a resistance of 8.00 ? is connected to the terminals of a battery with an emf of 6.00 V and negligible internal resistance. Find (a) the initial rate of increase of current in the circuit; (b) the rate of increase of current at the instant when the current is 0.500 A; (c) the current 0.250 s after the circuit is closed; (d) the final steady-state current.

In Fig. switch \(S_{1}\) is closed while switch \(S_{2}\) is kept open. The inductance is , the resistance is \(R=-48.0 \Omega\), and the emf of the battery is . At time after \(S_{1}\) is closed, the current in the circuit is increasing at a rate of \(d i / d t=7.20 \mathrm{~A} / \mathrm{s}\). At this instant what is \(V_{a b}\), the voltage across the resistor? Equation transcription: Text transcription: S{1} S{2} R=-48.0 Omega d i / d t=7.20{~A} /{s} V{a b}

CALC Show that the differential equation of Eq. (30.20) is satisfied by the function \(q=Q \cos (\omega t+\varphi)\), with \(\omega\) given by \(1 / \sqrt{L C}\). Equation Transcription: Text Transcription: q = Qcos(omega t + phi) omega 1/sqrt LC

Problem 32E A 20.0-µF capacitor is charged by a 150.0 V power supply, then disconnected from the power and connected in series with a 0.280-mH inductor. Calculate: (a) the oscillation frequency of the circuit; (b) the energy stored in the capacitor at time t = 0 ms (the moment of connection with the inductor); (c) the energy stored in the inductor at t = 1.30 ms.

Problem 33E A 7.50-nF capacitor is charged up to 12.0 V, then disconnected from the power supply and connected in series through a coil. The period of oscillation of the circuit is then measured to be 8.60 × 10-5 s. Calculate: (a) the inductance of the coil; (b) the maximum charge on the capacitor; (c) the total energy of the circuit; (d) the maximum current in the circuit.

Problem 34E A 18.0-µF capacitor is placed across a 22.5-V battery for several seconds and is then connected across a 12.0-mH inductor that has no appreciable resistance. (a) After the capacitor and inductor are connected together, find the maximum current in the circuit. When the current is a maximum, what is the charge on the capacitor? (b) How long after the capacitor and inductor are connected together does it take for the capacitor to be completely discharged for the first time? For the second time? (c) Sketch graphs of the charge on the capacitor plates and the current through the inductor as functions of time.

Problem 35E L-C Oscillations. A capacitor with capacitance 6.00 × 10-5 F is charged by connecting it to a 12.0-V battery. The capacitor is disconnected from the battery and connected across an inductor with L = 1.50 H. (a) What are the angular frequency v of the electrical oscillations and the period of these oscillations (the time for one oscillation)? (b) What is the initial charge on the capacitor? (c) How much energy is initially stored in the capacitor? (d) What is the charge on the capacitor 0.0230 s after the connection to the inductor is made? Interpret the sign of your answer. (e) At the time given in part (d), what is the current in the inductor? Interpret the sign of your answer. (f) At the time given in part (d), how much electrical energy is stored in the capacitor and how much is stored in the inductor?

A Radio Tuning Circuit. The minimum capacitance of a variable capacitor in a radio is 4.18 pF. (a) What is the inductance of a coil connected to this capacitor if the oscillation frequency of the L-C circuit is \(1600 \times 10^3\mathrm{\ Hz}\), corresponding to one end of the AM radio broadcast band, when the capacitor is set to its minimum capacitance? (b) The frequency at the other end of the broadcast band is \(540 \times 10^3\mathrm{\ Hz}\). What is the maximum capacitance of the capacitor if the oscillation frequency is adjustable over the range of the broadcast band? Text Transcription: 1600 times 10^3 Hz 540 times 10^3 Hz

An L-C circuit containing an 80.0-mH inductor and a 1.25-nF capacitor oscillates with a maximum current of 0.750 A. Calculate: (a) the maximum charge on the capacitor and (b) the oscillation frequency of the circuit. (c) Assuming the capacitor had its maximum charge at time t = 0, calculate the energy stored in the inductor after 2.50 ms of oscillation.

Problem 38E In an L-C circuit, L = 85.0 mH and C = 3.20 µF. During the oscillations the maximum current in the inductor is 0.850 mA. (a) What is the maximum charge on the capacitor? (b) What is the magnitude of the charge on the capacitor at an instant when the current in the inductor has magnitude 0.500 mA?

Problem 39E An L-R-C series circuit has L = 0.450 H, C = 2.50 × 10-5 F, and resistance R. (a) What is the angular frequency of the circuit when R = 0? (b) What value must R have to give a 5.0% decrease in angular frequency compared to the value calculated in part (a)?

For the circuit of Fig. 30.17, let \(C=15.0 n F, L=22 \mathrm{mH}\), and \(R=75.0 \Omega\) (a) Calculate the oscillation frequency of the circuit once the capacitor has been charged and the switch has been connected to point . (b) How long will it take for the amplitude of the oscillation to decay to of its original value? (c) What value of would result in a critically damped circuit? Equation transcription: Text transcription: C=15.0 n F, L=22{mH} R=75.0 Omega

CP (a) In Eq. (14.41), substitute for for for , and for the damping constant . Show that the result is Eq. (30.27). (b) Make these same substitutions in Eq. (14.43) and show that Eq. (30.29) results. (c) Make these same substitutions in Eq. (14.42) and show that Eq. ( ) results.

(a) What would have to be the self-inductance of a solenoid for it to store \(10.0 J\) of energy when a - A current runs through it? (b) If this solenoid's cross-sectional diameter is \(4.00 \mathrm{~cm}\), and if you could wrap its coils to a density of \(10 \text { coils } / \mathrm{mm}\), how long would the solenoid be? (See Exercise Is this a realistic length for ordinary laboratory use? Equation transcription: Text transcription: 10.0 J 10 text { coils } /{mm} 4.00{~cm}

Problem 52P An inductor is connected to the terminals of a battery that has an emf of 12.0 V and negligible internal resistance. The current is 4.86 mA at 0.940 ms after the connection is completed. After a long time the current is 6.45 mA. What are (a) the resistance R of the inductor and (b) the inductance L of the inductor?

CALC Continuation of Exercises 30.23 and 30.29. (a) How much energy is stored in the magnetic field of the inductor one time constant after the battery has been connected? Compute this both by integrating the expression in Exercise 30.29(c) and by using Eq. (30.9), and compare the results. (b) Integrate the expression obtained in Exercise 30.29( a) to find the total energy supplied by the battery during the time interval considered in part (a). (c) Integrate the expression obtained in Exercise 30.29 (b) to find the total energy dissipated in the resistance of the inductor during the same time period. (d) Compare the results obtained in parts (a), (b), (c).

A charged capacitor with \(C = 590 \mu F\) is connected in series to an inductor that has L = 0.330 H and negligible resistance. At an instant when the current in the inductor is , the current is increasing at a rate of di/dt = 89.0 A/s. During the current oscillations, what is the maximum voltage across the capacitor? Text Transcription: C = 590 mu F

CP In the circuit shown Figure P30.61 in the switch has been open for a long time and is suddenly closed. Neither the battery nor the inductors have any appreciable resistance. (a) What do the ammeter and voltmeter read just after is closed? (b) What do the ammeter and the voltmeter read after has been closed a very long time? (c) What do the ammeter and the voltmeter read after is closed? \(50.0 \Omega\) \(12.0 \mathrm{mH}\) \(20.0 \mathrm{~V}\) \(18.0 \mathrm{mH}\) \(25.0 \Omega\) \(15.0 \mathrm{mH}\) Equation transcription: Text transcription: 50.0 Omega 12.0{mH} 20.0{~V} 18.0{mH} 25.0 Omega 15.0{mH}

While studying a coil of unknown inductance and internal resistance, you connect it in series with a \(25.0-V\) battery and a \(150-\Omega\) resistor. You then place an oscilloscope across one of these circuit elements and use the oscilloscope to measure the voltage across the circuit element as a function of time. The result is shown in Fig. P30.62. (a) Across which circuit element (coil or resistor) is the oscilloscope connected? How do you know this? (b) Find the inductance and the internal resistance of the coil. (c) Carefully make a quantitative sketch showing the voltage versus time you would observe if you put the oscilloscope across the other circuit element (resistor or coil). Equation transcription: Text transcription: 25.0-V 150-Omega

In the lab, you are trying to find the inductance and internal resistance of a solenoid. You place it in series with a battery of negligible internal resistance, a \(10.0-\Omega\) resistor, and a switch. You then put an oscilloscope across one of these circuit elements to measure the voltage across that circuit element as a function of time. You close the switch, and the oscilloscope shows voltage versus time as shown in Fig. P30.63. (a) Across which circuit element (solenoid or resistor) is the oscilloscope connected? How do you know this? (b) Why doesn’t the graph approach zero as \(t \rightarrow \infty\)? (c) What is the emf of the battery? (d) Find the maximum current in the circuit. (e) What are the internal resistance and self-inductance of the solenoid?

CP In the circuit shown in Fig. P30.64, find the reading in each ammeter and voltmeter (a) just after switch is closed and (b) after has been closed a very long time. \(100.0 \Omega\) \(15.0 \mathrm{mH}\) \(50.0 \mathrm{~V}\) \(75.0 \Omega\) \(50.0 \Omega\) Equation transcription: Text transcription: 100.0 Omega 15.0{mH} 50.0{~V} 75.0 Omega 50.0 Omega

CP In the circuit shown in Fig. , switch is closed at time with no charge initially on the capacitor. (a) Find the reading of each ammeter and each voltmeter just after is closed. (b) Find the reading of each meter after a long time has elapsed. (c) Find the maximum charge on the capacitor. (d) Draw a qualitative graph of the reading of voltmeter \(V_{2}\) as a function of time. \(50.0 \Omega\) \(5.00 \mathrm{mH}\) \(40.0 \mathrm{~V}\) \(100.0 \Omega\) \(12.0 \mu F\) \)50.0 \Omega\) Equation transcription: Text transcription: V_{2} 50.0 Omega 5.00{mH} 40.0{~V} 100.0 Omega 12.0 mu F 50.0 Omega

In the circuit shown in Fig. P30.66 the battery and the inductor have no appreciable internal resistance and there is no current in the circuit. After the switch is closed, find the readings of the ammeter (A) and voltmeters \(\left(V_1\ and\ V_2\right)\) (a) the instant after the switch is closed and (b) after the switch has been closed for a very long time. (c) Which answers in parts (a) and (b) would change if the inductance were 24.0 mH instead?

CP In the circuit shown in Fig. P30.67, switch is closed at time . (a) Find the reading of each meter just after is closed. (b) What does each meter read long after is closed? \(40.0 \Omega\) \(5.0 \Omega\) \(20.0 \mathrm{mH}\) \(25.0 \mathrm{~V}\) \(10.0 \Omega\) \(10.0 \mathrm{mH}\) \(15.0 \Omega\) Equation transcription: Text transcription: 40.0 Omega 5.0 Omega 20.0{mH} 25.0{~V} 10.0 Omega 10.0{mH} 15.0 Omega

In the circuit shown in Fig. P30.68, switch S1 has been closed for a long enough time so that the current reads a steady 3.50 A. Suddenly, switch \(S_{2}\) is closed and \(S_{1}\) is opened at the same instant. (a) What is the maximum charge that the capacitor will receive? (b) What is the current in the inductor at this time? \(S_{1}\) \(S_{2}\) \(2.0 \mathrm{mH}\) \(5.0 \mu F\) Equation transcription: Text transcription: S{1} S{2} 2.0{mH} 5.0 mu F

CP In the circuit shown in Fig. P30.69, \(\mathcal{E}=60.0 \mathrm{~V}, \ R_{1}=40.0 \Omega, \ R_{2}=25.0 \Omega, \text { and } L=0.300 \mathrm{H}\). (a) Switch S is closed. At some time t afterward, the current in the inductor is increasing at a rate of di/dt = 50.0 A/s. At this instant, what are the current \(i_{1}\) through \(R_{1}\) and the current \(i_{2}\) through \(R_{2}\)? (Hint: Analyze two separate loops: one containing \(\mathcal{\varepsilon}\) and \(R_{1}\) and the other containing \(\mathcal{\varepsilon}, \ R_{2}\), and L.) (b) After the switch has been closed a long time, it is opened again. Just after it is opened, what is the current through \(R_{1}\)?

CALC Consider the circuit shown in Fig. P30.71. Let E = 36.0 V, \(R_0=50.0\ \Omega\), \(R=150\ \Omega\) and L = 4.00H. (a) Switch \(S_{1}\) is closed and switch \(S_{2}\) is left open. Just after \(S_{1}\) is closed, what are the current \(i_{0}\) through \(R_{0}\) and the potential differences \(v_{ac}\) and \(v_{cb}\)? (b) After \(S_{1}\) has been closed a long time (\(S_{2}\) is still open) so that the current has reached its final, steady value, what are \(i_{0}\), \(v_{ac}\), and \(v_{cb}\)? (c) Find the expressions for \(i_{0}\), \(v_{ac}\), and \(v_{cb}\) as functions of the time \(t\) since \(S_{1}\) was closed. Your results should agree with part (a) when t = 0 and with part (b) when \(t \rightarrow \infty\). Graph \(i_{0}\), \(v_{ac}\), and \(v_{cb}\) versus time. Equation Transcription: Text Transcription: R_0=50.0 omega R = 150 omega S_1 S_2 i_0 R_0 v_ac v_cb t t rightarrow infty

CP CALC We have ignored the variation of the magnetic field across the cross of a toroidal solenoid. Let's examine the validity of proximation. A certain solenoid has a rectangular section (Fig. P30.73). It uniformly spaced turns, inside. The magnetic field point inside the toroid is given by the equation derived in Example 28.10 (Section 28.7). Do not assume the field is uniform over the cross section. (a) Show that the magnetic flux through a cross section of the toroid is $$\Phi_{B}=\frac{\mu_{1} N i h}{2 \pi} \ln \left(\begin{array}{l}b \\a\end{array}\right)$$ (b) Show that the inductance of the toroidal solenoid is given by $$L=\frac{\mu_{N} N^{2} h}{2 \pi} \ln \left(\frac{b}{a}\right)$$ $$\Phi_{B}=\frac{\mu_{1} N i h}{2 \pi} \ln \left(\begin{array}{l}b \\a\end{array}\right)$$ (c) The fraction b/a may be written as $$\frac{b}{a}=\frac{a+b-\theta}{a}=1+\frac{b-a}{a}$$ Use the power series expansion \(\ln (1+z)=z+z^{2} / 2+\cdots\), valid for \(|z|<1\), to show that when b-a is much less than a, the inductance is approximately equal to $$L=\frac{\mu_{0} N^{2} h(b-a)}{2 \pi \alpha}$$ Compare this result with the result given in Example 30.3 (Section 30.2).. Equation Transcription: Text Transcription: Phi_B = mu_0 Nih/2 pi ln(b/a) L = mu_0 N^2 h/2 pi ln(b/2) b/2 = a + b - a/a = 1 + b-a/a ln(1+z) = z + z^2/2 + times times times |z| < 1 L = mu_0 N^2 ln(b-a)/2 pi a