Chapter 21

A circular area with a radius of 6.50 \(\mathrm{cm}\) lies in the \(x\) -y plane. What is the magnitude of the magnetic flux through this circle due to a uniform magnetic field \(B=0.230 \mathrm{T}\) that points (a) in the \(+z\) direction? (b) at an angle of \(53.1^{\circ}\) from the \(+z\) direction? (c) in the \(+y\) direction?

A single loop of wire with an area of 0.0900 \(\mathrm{m}^{2}\) is in a uniform magnetic field that has an initial value of 3.80 \(\mathrm{T}\) , is perpendicular to the plane of the loop, and is decreasing at a constant rate of 0.190 \(\mathrm{T} / \mathrm{s}\) . (a) What emf is induced in this loop? (b) If the loop has a resistance of \(0.600 \Omega,\) find the current induced in the loop.

A coil of wire with 200 circular turns of radius 3.00 \(\mathrm{cm}\) is in a uniform magnetic field along the axis of the coil. The coil has \(R=40.0 \Omega\) . At what rate, in teslas per second, must the magnetic field be changing to induce a current of 0.150 \(\mathrm{A}\) in the coil?

In a physics laboratory experiment, a coil with 200 turns enclosing an area of 12 \(\mathrm{cm}^{2}\) is rotated from a position where its plane is perpendicular to the earth's magnetic field to one where its plane is parallel to the field. The rotation takes 0.040 s. The earth's magnetic field at the location of the laboratory is \(6.0 \times 10^{-5} \mathrm{T.}\) (a) What is the total magnetic flux through the coil before it is rotated? After it is rotated? (b) What is the average emf induced in the coil?

A closely wound rectangular coil of 80 turns has dimensions of 25.0 \(\mathrm{cm}\) by 40.0 \(\mathrm{cm}\) . The plane of the coil is rotated from a position where it makes an angle of \(37.0^{\circ}\) with a magnetic field of 1.10 \(\mathrm{T}\) to a position perpendicular to the field. The rotation takes 0.0600 s. What is the average emf induced in the coil?

A very long, straight solenoid with a cross-sectional area of 6.00 \(\mathrm{cm}^{2}\) is wound with 40 turns of wire per centimeter, and the windings carry a current of 0.250 A. A secondary winding of 2 turns encircles the solenoid at its center. When the primary circuit is opened, the magnetic field of the solenoid becomes zero in 0.0500 s. What is the average induced emf in the secondary coil?

\(\bullet\) A circular loop of wire with a radius of 12.0 \(\mathrm{cm}\) is lying flat on a tabletop. A magnetic field of 1.5 \(\mathrm{T}\) is directed vertically upward through the loop (Figure 21.49 ). (a) If the loop is removed from the field region in a time interval of 2.0 \(\mathrm{ms}\) , find the average emf that will be induced in the wire loop during the extraction process. (b) If the loop is viewed looking down on it from above, is the induced current in the loop clockwise or counterclockwise?

A flat, square coil with 15 turns has sides of length 0.120 \(\mathrm{m}\) . The coil rotates in a magnetic field of 0.0250 \(\mathrm{T}\) (a) What is the angular velocity of the coil if the maximum emf produced is 20.0 \(\mathrm{mV}^{\prime} ?\) (Hint: Look at the motional emf induced across the ends of the segments of the coil.) (b) What is the average emf at this angular velocity?

A circular loop of wire is in a spatially uniform magnetic field, as shown in Figure \(21.51 .\) The magnetic field is directed into the plane of the figure. Determine the direction (clockwise or counterclockwise) of the induced current in the loop when (a) \(B\) is increasing; (b) \(B\) is decreasing; (c) \(B\) is constant with a value of \(B_{0}\) . Explain your reasoning.

A very thin 15.0 \(\mathrm{cm}\) copper bar is aligned horizontally along the east-west direction. If it moves horizontally from south to north at 11.5 \(\mathrm{m} / \mathrm{s}\) in a vertically upward magnetic field of \(1.22 \mathrm{T},\) (a) what potential difference is induced across its ends, and (b) which end (east or west) is at a higher potential? (c) What would be the potential difference if the bar moved from east to west instead?

You're driving at 95 \(\mathrm{km} / \mathrm{h}\) in a direction \(35^{\circ}\) east of north, in a region where the earth's magnetic field of \(5.5 \times 10^{-5} \mathrm{T}\) is horizontal and points due north. If your car measures 1.5 \(\mathrm{m}\) from its underbody to its roof, calculate the induced emf between roof and underbody. (You can assume the sides of the car are straight and vertical.) Is the roof of the car at a higher or lower potential than the underbody?

Measuring blood flow. Blood contains positive and negative ions and therefore is a conductor. A blood vessel, therefore, can be viewed as an electrical wire. We can even picture the flowing blood as a series of parallel conducting slabs whose thickness is the diameter \(d\) of the vessel moving with speed \(v\) .(See Figure \(21.62 . )\) (a) If the blood vessel is placed in a magnetic field \(B\) perpendicular to the vessel, as in the figure, show that the motional potential difference induced across it is \(\mathcal{E}=v B d .\) (b) If you expect that the blood will be flowing at 15 \(\mathrm{cm} / \mathrm{s}\) for a vessel 5.0 \(\mathrm{mm}\) in diameter, what strength of magnetic field will you need to produce a potential difference of 1.0 \(\mathrm{mV} ?\) (c) Show that the volume rate of flow \((R)\) of theblood is equal to \(R=\pi \mathcal{E} d / 4 B .\) (Note: Although the method developed here is useful in measuring the rate of blood flow in a vessel, it is limited to use in surgery because measurement of the potential \(\mathcal{E}\) must be made directly across the vessel.)

A toroidal solenoid has a mean radius of 10.0 \(\mathrm{cm}\) and a cross- sectional area of 4.00 \(\mathrm{cm}^{2}\) and is wound uniformly with 100 turns. A second coil with 500 turns is wound uniformly on top of the first. What is the mutual inductance of these coils?

A 10.0 -cm-long solenoid of diameter 0.400 \(\mathrm{cm}\) 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?

Two coils are wound around the same cylindrical form, like the coils in Example \(21.8 .\) 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 \(\mathrm{mV}\) . (a) What is the mutual inductance of the pair of coils? (b) If the second coil has 25 turns, what is the average magnetic flux through each turn when the current in the first coil equals 1.20 \(\mathrm{A} ?(\mathrm{c})\) If the current in the second coil increases at a rate of \(0.360 \mathrm{A} / \mathrm{s},\) what is the magnitude of the induced emf in the first coil?

One solenoid is centered inside another. The outer one has a length of 50.0 \(\mathrm{cm}\) and contains 6750 coils, while the coaxial 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 inner solenoid.

A 4.5 \(\mathrm{mH}\) toroidal inductor has 125 identical equally spaced coils. (a) If it carries an 11.5 A current, how much magnetic flux passes through each of its coils? (b) If the potential difference across its ends is \(1.16 \mathrm{V},\) at what rate is the current in it changing?

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 \(\mathrm{V} .\) What is the inductance of the inductor?

An inductor has inductance of 0.260 \(\mathrm{H}\) and carries a current that is decreasing at a uniform rate of 18.0 \(\mathrm{mA} / \mathrm{s} .\) Find the self-induced emf in this inductor.

Self-inductance of a solenoid. A long, straight solenoid has \(N\) turns, a uniform cross-sectional area \(A,\) and length \(l .\) Use the definition of self- inductance expressed by Equation 21.13 to show that the inductance of this solenoid is given approximately 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 of the solenoid. For this reason, your answer is actually an upper limit on the inductance.)

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 \(\mathrm{mV}\) . 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?

A transformer consists of 275 primary windings and 834 secondary windings. If the potential difference across the primary coil is \(25.0 \mathrm{V},\) (a) what is the voltage across the secondary coil, and (b) what is the effective load resistance of the secondary coil if it is connected across a \(125 . \Omega\) resistor?

Off to Europe! You plan to take your hair blower to Europe, where the electrical outlets put out 240 \(\mathrm{V}\) instead of the 120 \(\mathrm{V}\) seen in the United States. The blower puts out 1600 \(\mathrm{W}\) at 120 \(\mathrm{V}\) . (a) What could you do to operate your blower via the 240 \(\mathrm{V}\) line in Europe? (b) What current will your blower draw from a European outlet? (c) What resistance will your blower appear to have when operated at 240 \(\mathrm{V} ?\)

You need a transformer that will draw 15 \(\mathrm{W}\) of power from a 220 \(\mathrm{V}\) power line, stepping the voltage down to \(6.0 \mathrm{V}(\mathrm{rms}),\) (a) What will be the current in the secondary coil? (b) What should be the resistance of the secondary circuit? (c) What will be the equivalent resistance of the input circuit?

A step-up transformer. A transformer connected to a 120 \(\mathrm{V}\) (rms) ac line is to supply \(13,000 \mathrm{V}\) (rms) for a neon sign. To reduce the shock hazard, a fuse is to be inserted in the primary circuit and is to blow when the rms current in the secondary circuit exceeds 8.50 \(\mathrm{mA}\) (a) What is the ratio of secondary to primary turns of the transformer? (b) What power must be supplied to the transformer when the rms secondary current is 8.50 \(\mathrm{mA}\) ? (c) What current rating should the fuse in the primary circuit have?

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?

Energy in a typical inductor. (a) How much energy is stored in a 10.2 \(\mathrm{mH}\) inductor carrying a 1.15 A current? (b) How much current would such an inductor have to carry to store 1.0 \(\mathrm{J}\) of energy? Is this a reasonable amount of current for ordinary laboratory circuit elements?

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 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.

Large inductors have been proposed as energy-storage devices. (a) How much electrical energy is converted to light and thermal energy by a 200 \(\mathrm{W}\) lightbulb 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?

When a certain inductor carries a current \(I,\) it stores 3.0 \(\mathrm{mJ}\) of magnetic energy. How much current (in terms of \(I )\) would it have to carry to store 9.0 \(\mathrm{mJ}\) of energy?

A \(12.0 \mathrm{~V}\) dc battery having no appreciable internal resistance, a \(150.0 \Omega\) resistor, an \(11.0 \mathrm{mH}\) inductor, and an open switch are all connected in series. After the switch is closed, what are (a) the time constant for this circuit, (b) the maximum current that flows through it, (c) the current \(73.3 \mu\) s after the switch is closed, and (d) the maximum energy stored in the inductor?

An inductor with an inductance of 2.50 \(\mathrm{H}\) and a resistor with a resistance of 8.00\(\Omega\) are connected to the terminals of a battery with an emf of 6.00 \(\mathrm{V}\) and negligible internal resistance. Find (a) the initial rate of increase of the current in the circuit, (b) the initial potential difference across the inductor, (c) the current 0.313 s after the circuit is closed, and (d) the maximum current.

A 1.50 \(\mathrm{mH}\) inductor is connected in series with a dc battery of negligible internal resistance, a 0.750 \(\mathrm{k} \Omega\) resistor, and an open switch. How long after the switch is closed will it take for (a) the current in the circuit to reach half of its maximum value, (b) the energy stored in the inductor to reach half of its maximum value? (Hint: You will have to solve an exponential equation.)

A 12.0\(\mu \mathrm{F}\) capacitor and a 5.25 \(\mathrm{mH}\) inductor are connected in series with an open switch. The capacitor is initially charged to 6.20\(\mu \mathrm{C}\) . What is the angular frequency of the charge oscillations in the capacitor after the switch is closed?

A 5.00\(\mu \mathrm{F}\) 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?

A 15.0\(\mu \mathrm{F}\) capacitor is charged to 175\(\mu \mathrm{C}\) and then connected across the ends of a 5.00 \(\mathrm{mH}\) inductor. (a) Find the maximum current in the inductor. At the instant the current in the inductor is maximal, how much charge is on the capacitor At this instant, what is the current in the inductor? (c) Find the maximum energy stored in the inductor. At this instant, what is the current in the circuit?

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?

An electromagnetic car alarm. 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 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 \(\mathrm{V}\) (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?