Chapter 30

Two coils have mutual inductance \(M=3.25 \times 10^{-4} \mathrm{H}\) . The current \(i_{1}\) in the first coil increases at a uniform rate of 830 \(\mathrm{A} / \mathrm{s}\) . (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?

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 \(\mathrm{A}\) , what is the average flux through each turn of solenoid 1\(?\)

A toroidal solenoid has 500 turns, cross-sectional area \(6.25 \mathrm{cm}^{2},\) and mean radius \(4.00 \mathrm{cm} .\) (a) Calcualte the coil's self-inductance. (b) If the current decreases uniformly from 5.00 \(\mathrm{A}\) to 2.00 \(\mathrm{A}\) in 3.00 \(\mathrm{ms}\) , 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 0.0640 \(\mathrm{A} / \mathrm{s}\) , the magnitude of the self-induced emf is 0.0160 \(\mathrm{V}\) . (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 \(\mathrm{A} ?\)

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

Inductance of a Solenoid. A long, straight solenoid has \(N\) turms, uniform cross-sectional area \(A,\) and length \(l .\) Show that the inductance of this solenoid is given by the equation \(L=\mu_{0} A N^{2} / L\) 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.)

An inductor used in a de power supply has an inductance of 12.0 \(\mathrm{H}\) and a resistance of \(180 \Omega .\) It carries a current of 0.300 \(\mathrm{A}\) . (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 \(\mathrm{cm}\) and a cross-sectional area of \(5.00 \mathrm{cm}^{2} .\) When the current is 12.0 \(\mathrm{A}\) , the energy stored is 0.390 \(\mathrm{J}\) . How many turns does the winding have?

An air-filled toroidal solenoid has 300 turns of wire, a mean radius of \(12.0 \mathrm{cm},\) and a cross-sectional area of \(4.00 \mathrm{cm}^{2} .\) If the current is 5.00 \(\mathrm{A}\) , 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 \(\mathrm{cm}\) long and with a cross-sectional area of 0.500 \(\mathrm{cm}^{2}\) 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 fleld 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.

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 \(200-\mathrm{W}\) 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 \(\mathrm{A}\) , what is the inductance?

It is proposed to store \(1.00 \mathrm{kW} \cdot \mathrm{h}=3.60 \times 10^{6} \mathrm{J}\) of electrical energy in a uniform magnetic field with magnitude 0.600 \(\mathrm{T}\) . (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 \(\mathrm{cm}\) on a side, what magnetic field is required?

An inductor with an inductance of 2.50 \(\mathrm{H}\) and a resistance of 8.00\(\Omega\) is connected to the terminals of a battery with an emf of 6.00 \(\mathrm{V}\) and negligible intermal 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 \(0.500 \mathrm{A} ;(\mathrm{c})\) the current 0.250 \(\mathrm{s}\) after the circuit is closed; (d) the final steady-state current.

A \(15.0-\Omega\) resistor and a coil are connected in series with a 6.30-V battery with negligible internal resistance and a closed switch. (a) At 200 ms after the switch is opened the current has decayed to 0.210 A. Calculate the inductance of the coil. (b) Calculate the time constant of the circuit. (c) How long after the switch is closed will the current reach 1.00\(\%\) of its original value?

A \(35.0-\mathrm{V}\) battery with negligible internal resistance, a 50.0- \(\Omega\) resistor, and a \(1.25-\mathrm{mH}\) 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?

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

A \(7.50-\mathrm{nF}\) capacitor is charged up to \(12.0 \mathrm{V},\) 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 \(8.60 \times 10^{-5} \mathrm{s}\) . Calculate: (a) the inductance of the coil; \((\mathrm{b})\) the maximum charge on the capacitor; (c) the total energy of the circuit; (d) the maximum current in the circuit.

A \(18.0-\mu F\) capacitor is placed across a \(22.5-\mathrm{V}\) 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 \(6.00 \times\) \(10^{-5} \mathrm{F}\) is charged by connecting it to a \(12.0-\mathrm{V}\) battery. The capacitor is disconnected from the battery and connected across an inductor with \(L=1.50 \mathrm{H}\) (a) What are the angular frequency \(\omega\) of the electrical oscillations and the period of these oscillations (the time for one oscillation \() ?(\text { b) What is the initial charge on the }\) capacitor? (c) How much energy is initially stored in the capacitor? (d) What is the charge on the capacitor 0.0230 \(\mathrm{s}\) 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 \(\mathrm{pF}\) . (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?

An \(L C\) cirruit containing an \(80.0-\mathrm{mH}\) inductor and a 1.25-nF capacitor oscillates with a maximum current of 0.750 \(\mathrm{A}\) . 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 \(\mathrm{ms}\) of oscillation.

In an \(L-\) circuit, \(L=85.0 \mathrm{mH}\) and \(C=3.20 \mu \mathrm{F}\) . During the oseillations the maximum current in the inductor is 0.850 \(\mathrm{mA}\) . (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 \(\mathrm{mA} ?\)

Show that \(\sqrt{L C}\) has units of time.

An \(L-R-C\) circuit has \(L=0.450 \mathrm{H}, C=2.50 \times 10^{-5} \mathrm{F}\) 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)?

Show that the quantity \(\sqrt{L} / C\) has units of resistance (ohms).

One solenoid is centered inside another. The outer one has a length of 50.0 \(\mathrm{cm}\) and contains 6750 coils, while the coarial inner solenoid is 3.0 \(\mathrm{cm}\) long and 0.120 \(\mathrm{cm}\) in diameter and contains 15 coils. The current in the outer solenoid is changing at 37.5 \(\mathrm{A} / \mathrm{s}\) . (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 3.50 \(\mathrm{mH}\) . The current in the coil varies with time according to \(i=\) \((680 \mathrm{mA}) \cos (\pi t)(0.0250 \mathrm{s}) \cdot(\mathrm{a})\) What is the maximum emf induced in the coil? (b) What is the maximum average flux through each turn of the coil?(c) At \(t=0.0180 \mathrm{s}\) , what is the magnitude of the induced emf?

A \(0.250-\mathrm{H}\) 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?

An inductor is connected to the terminals of a battery that has an emf of \(12.0 \mathrm{~V}\) and negligible internal resistance. The current is \(4.86 \mathrm{~mA}\) at \(0.725 \mathrm{~ms}\) after the connection is completed. After a long time the current is \(6.45 \mathrm{~mA}\). What are (a) the resistance \(R\) of the inductor and (b) the inductance \(L\) of the inductor?

A \(5.00-\mu F\) capacitor is initially charged to a potential of 16.0 \(\mathrm{V}\) . It is then connected in series with a \(3.75-\mathrm{mH}\) 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 CarAlarm. Your latest invention is a car alarm that produces sound at a particularly annoying frequency of 3500 \(\mathrm{Hz}\) . To do this, the car-alarm circuitry must produce an altermating 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 \(\mathrm{V}\) (the same voltage as the car battery). To produce a sufficiently loud sound, the capacitor must store 0.0160 \(\mathrm{J}\) of energy. What values of capacitance and inductance should you choose for your car-alarm circuit?

An \(L-C\) circuit consists of a \(60.0-\mathrm{mH}\) inductor and a \(250-\mu F\) 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?

Solar Magnetic Energy. Magnetic fields within a sunspot can be as strong as 0.4 \(\mathrm{T}\) . (By comparison, the earth's magnetic field is about \(1 / 10,000\) as strong.) Sunspots can be as large as \(25,000 \mathrm{km}\) in radius. The material in a sunspot has a density of about \(3 \times 10^{-4} \mathrm{kg} / \mathrm{m}^{3}\) . 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 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 \(6 \times 10^{3} \mathrm{m} / \mathrm{s}\) . (Hint . Calcualte the kinetic energy the magnetic field could supply to 1 \(\mathrm{m}^{3}\) of sunspot material.)