Review. Figure P31.31 shows a bar of mass m 5 0.200 kg

Figure OQ31.1 is a graph of the magnetic flux through a certain coil of wire as a function of time during an interval while the radius of the coil is increased, the coil is rotated through 1.5 revolutions, and the external source of the magnetic field is turned off, in that order. Rank the emf induced in the coil at the instants marked A through E from the largest positive value to the largest-magnitude negative value. In your ranking, note any cases of equality and also any instants when the emf is zero.

A flat coil of wire is placed in a uniform magnetic field that is in the y direction. (i) The magnetic flux through the coil is a maximum if the plane of the coil is where? More than one answer may be correct. (a) in the xy plane (b) in the yz plane (c) in the xz plane (d) in any orientation, because it is a constant (ii) For what orientation is the flux zero? Choose from the same possibilities as in part (i).

A rectangular conducting loop is placed near a long wire carrying a current I as shown in Figure OQ31.3. If I decreases in time, what can be said of the current induced in the loop? (a) The direction of the current depends on the size of the loop. (b) The current is clockwise. (c) The current is counterclockwise. (d) The current is zero. (e) Nothing can be said about the current in the loop without more information.

A circular loop of wire with a radius of 4.0 cm is in a uniform magnetic field of magnitude 0.060 T. The plane of the loop is perpendicular to the direction of the magnetic field. In a time interval of 0.50 s, the magnetic field changes to the opposite direction with a magnitude of 0.040 T. What is the magnitude of the average emf induced in the loop? (a) 0.20 V (b) 0.025 V (c) 5.0 mV (d) 1.0 mV (e) 0.20 mV

A square, flat loop of wire is pulled at constant velocity through a region of uniform magnetic field directed perpendicular to the plane of the loop as shown in Figure OQ31.5. Which of the following statements are correct? More than one statement may be correct. (a) Current is induced in the loop in the clockwise direction. (b) Current is induced in the loop in the counterclockwise direction. (c) No current is induced in the loop. (d) Charge separation occurs in the loop, with the top edge positive. (e) Charge separation occurs in the loop, with the top edge negative.

The bar in Figure OQ31.6 moves on rails to the right with a velocity v S, and a uniform, constant magnetic field is directed out of the page. Which of the following statements are correct? More than one statement may be correct. (a) The induced current in the loop is zero. (b) The induced current in the loop is clockwise. (c) The induced current in the loop is counterclockwise. (d) An external force is required to keep the bar moving at constant speed. (e) No force is required to keep the bar moving at constant speed.

A bar magnet is held in a vertical orientation above a loop of wire that lies in the horizontal plane as shown in Figure OQ31.7. The south end of the magnet is toward the loop. After the magnet is dropped, what is true of the induced current in the loop as viewed from above? (a) It is clockwise as the magnet falls toward the loop. (b) It is counterclockwise as the magnet falls toward the loop. (c) It is clockwise after the magnet has moved through the loop and moves away from it. (d) It is always clockwise. (e) It is first counterclockwise as the magnet approaches the loop and then clockwise after it has passed through the loop.

What happens to the amplitude of the induced emf when the rate of rotation of a generator coil is doubled? (a) It becomes four times larger. (b) It becomes two times larger. (c) It is unchanged. (d) It becomes one-half as large. (e) It becomes one-fourth as large.

Two coils are placed near each other as shown in Figure OQ31.9. The coil on the left is connected to a battery and a switch, and the coil on the right is connected to a resistor. What is the direction of the current in the resistor (i)at an instant immediately after the switch is thrown closed, (ii) after the switch has been closed for several seconds, and (iii) at an instant after the switch has then been thrown open? Choose each answer from the possibilities (a) left, (b) right, or (c) the current is zero.

A circuit consists of a conducting movable bar and a lightbulb connected to two conducting rails as shown in Figure OQ31.10. An external magnetic field is directed perpendicular to the plane of the circuit. Which of the following actions will make the bulb light up? More than one statement may be correct. (a) The bar is moved to the left. (b)The bar is moved to the right. (c) The magnitude of the magnetic field is increased. (d) The magnitude of the magnetic field is decreased. (e) The bar is lifted off the rails.

Two rectangular loops of wire lie in the same plane as shown in Figure OQ31.11. If the current I in the outer loop is counterclockwise and increases with time, what is true of the current induced in the inner loop? More than one statement may be correct. (a) It is zero. (b) It is clockwise. (c) It is counterclockwise. (d) Its magnitude depends on the dimensions of the loops. (e) Its direction depends on the dimensions of the loops.

An aluminum ring of radius r1 and resistance R is placed around one end of a long air-core solenoid with n turns per meter and smaller radius r2 as shown in Figure P31.11. Assume the axial component of the field produced by the solenoid over the area of the end of the solenoid is one-half as strong as at the center of the solenoid. Also assume the solenoid produces negligible field outside its cross-sectional area. The current in the solenoid is increasing at a rate of DI/Dt. (a) What is the induced current in the ring? (b) At the center of the ring, what is the magnetic field produced by the induced current in the ring? (c) What is the direction of this field?

An aluminum ring of radius r1 and resistance R is placed around one end of a long air-core solenoid with n turns per meter and smaller radius r2 as shown in Figure P31.11. Assume the axial component of the field produced by the solenoid over the area of the end of the solenoid is one-half as strong as at the center of the solenoid. Also assume the solenoid produces negligible field outside its cross-sectional area. The current in the solenoid is increasing at a rate of DI/Dt. (a) What is the induced current in the ring? (b) At the center of the ring, what is the magnetic field produced by the induced current in the ring? (c) What is the direction of this field?

. A coil of 15 turns and radius 10.0 cm surrounds a long solenoid of radius 2.00 cm and 1.00 3 103 turns/meter (Fig. P31.14). The current in the solenoid changes as I 5 5.00 sin 120t, where I is in amperes and t is in seconds. Find the induced emf in the 15-turn coil as a function of time.

A square, single-turn wire loop , 5 1.00 cm on a side is placed inside a solenoid that has a circular cross section of radius r 5 3.00 cm as shown in the end view of Figure P31.15 (page 960). The solenoid is 20.0 cm long and wound with 100 turns of wire. (a) If the current in the solenoid is 3.00 A, what is the magnetic flux through the square loop? (b)If the current in the solenoid is reduced to zero in 3.00s, what is the magnitude of the average induced emf in the square loop?

A long solenoid has n 5 400 turns per meter and carries a current given by I 5 30.0(1 2 e21.60t ), where I is in amperes and t is in seconds. Inside the solenoid and coaxial with it is a coil that has a radius of R 5 6.00 cm and consists of a total of N 5 250 turns of fine wire (Fig. P31.16). What emf is induced in the coil by the changing current?

A coil formed by wrapping 50 turns of wire in the shape of a square is positioned in a magnetic field so that the normal to the plane of the coil makes an angle of 30.08 with the direction of the field. When the magnetic field is increased uniformly from 200 mT to 600 mT in 0.400 s, an emf of magnitude 80.0 mV is induced in the coil. What is the total length of the wire in the coil?

When a wire carries an AC current with a known frequency, you can use a Rogowski coil to determine the amplitude I max of the current without disconnecting the wire to shunt the current through a meter. The Rogowski coil, shown in Figure P31.18, simply clips around the wire. It consists of a toroidal conductor wrapped around a circular return cord. Let n represent the number of turns in the toroid per unit distance along it. Let A represent the cross-sectional area of the toroid. Let I(t) 5 I max sin vt represent the current to be measured. (a) Show that the amplitude of the emf induced in the Rogowski coil is e max 5 m0nAvI max. (b) Explain why the wire carrying the unknown current need not be at the center of the Rogowski coil and why the coil will not respond to nearby currents that it does not enclose.

A toroid having a rectangular cross section (a 5 2.00 cm by b 5 3.00 cm) and inner radius R 5 4.00 cm consists of N 5 500 turns of wire that carry a sinusoidal current I 5 I max sin vt, with I max 5 50.0 A and a frequency f 5 v/2p 5 60.0 Hz. A coil that consists of N9 5 20 turns of wire is wrapped around one section of the toroid as shown in Figure P31.19. Determine the emf induced in the coil as a function of time.

A piece of insulated wire is shaped into a figure eight as shown in Figure P31.20. For simplicity, model the two halves of the figure eight as circles. The radius of the upper circle is 5.00 cm and that of the lower circle is 9.00 cm. The wire has a uniform resistance per unit length of 3.00 V/m. A uniform magnetic field is applied perpendicular to the plane of the two circles, in the direction shown. The magnetic field is increasing at a constant rate of 2.00 T/s. Find (a) the magnitude and (b) the direction of the induced current in the wire.

A helicopter (Fig. P31.21) has blades of length 3.00 m, extending out from a central hub and rotating at 2.00 rev/s. If the vertical component of the Earths magnetic field is 50.0 mT, what is the emf induced between the blade tip and the center hub?

Use Lenzs law to answer the following questions concerning the direction of induced currents. Express your answers in terms of the letter labels a and b in each part of Figure P31.22. (a) What is the direction of the induced current in the resistor R in Figure P31.22a when the bar magnet is moved to the left? (b) What is the direction of the current induced in the resistor R immediately after the switch S in Figure P31.22b is closed? (c) What is the direction of the induced current in the resistor R when the current I in Figure P31.22c decreases rapidly to zero?

A truck is carrying a steel beam of length 15.0 m on a freeway. An accident causes the beam to be dumped off the truck and slide horizontally along the ground at a speed of 25.0 m/s. The velocity of the center of mass of the beam is northward while the length of the beam maintains an eastwest orientation. The vertical component of the Earths magnetic field at this location has a magnitude of 35.0 mT. What is the magnitude of the induced emf between the ends of the beam?

A small airplane with a wingspan of 14.0 m is flying due north at a speed of 70.0 m/s over a region where the vertical component of the Earths magnetic field is 1.20 mT downward. (a) What potential difference is developed between the airplanes wingtips? (b) Which wingtip is at higher potential? (c) What If? How would the answers to parts (a) and (b) change if the plane turned to fly due east? (d) Can this emf be used to power a lightbulb in the passenger compartment? Explain your answer.

A 2.00-m length of wire is held in an eastwest direction and moves horizontally to the north with a speed of 0.500 m/s. The Earths magnetic field in this region is of magnitude 50.0 mT and is directed northward and 53.08 below the horizontal. (a) Calculate the magnitude of the induced emf between the ends of the wire and (b) determine which end is positive.

Consider the arrangement shown in Figure P31.26. Assume that R 5 6.00 V, , 5 1.20 m, and a uniform 2.50-T magnetic field is directed into the page. At what speed should the bar be moved to produce a current of 0.500 A in the resistor?

Figure P31.26 shows a top view of a bar that can slide on two frictionless rails. The resistor is R 5 6.00 V, and a 2.50-T magnetic field is directed perpendicularly downward, into the paper. Let , 5 1.20 m. (a) Calculate the applied force required to move the bar to the right at a constant speed of 2.00 m/s. (b) At what rate is energy delivered to the resistor?

A metal rod of mass m slides without friction along two parallel horizontal rails, separated by a distance , and connected by a resistor R, as shown in Figure P31.26. A uniform vertical magnetic field of magnitude B is applied perpendicular to the plane of the paper. The applied force shown in the figure acts only for a moment, to give the rod a speed v. In terms of m, ,, R, B, and v, find the distance the rod will then slide as it coasts to a stop.

A conducting rod of length , moves on two horizontal, frictionless rails as shown in Figure P31.26. If a constant force of 1.00 N moves the bar at 2.00 m/s through a magnetic field B S that is directed into the page, (a) what is the current through the 8.00-V resistor R? (b) What is the rate at which energy is delivered to the resistor? (c) What is the mechanical power delivered by the force F S app ?

Why is the following situation impossible? An automobile has a vertical radio antenna of length , 5 1.20 m. The automobile travels on a curvy, horizontal road where the Earths magnetic field has a magnitude of B 5 50.0 mT and is directed toward the north and downward at an angle of u 5 65.08 below the horizontal. The motional emf developed between the top and bottom of the antenna varies with the speed and direction of the automobiles travel and has a maximum value of 4.50 mV.

Review. Figure P31.31 shows a bar of mass m 5 0.200 kg that can slide without friction on a pair of rails separated by a distance , 5 1.20 m and located on an inclined plane that makes an angle u 5 25.08 with respect to the ground. The resistance of the resistor is R 5 1.00 V and a uniform magnetic field of magnitude B 5 0.500 T is directed downward, perpendicular to the ground, over the entire region through which the bar moves. With what constant speed v does the bar slide along the rails?

Review. Figure P31.31 shows a bar of mass m that can slide without friction on a pair of rails separated by a distance , and located on an inclined plane that makes an angle u with respect to the ground. The resistance of the resistor is R, and a uniform magnetic field of magnitude B is directed downward, perpendicular to the ground, over the entire region through which the bar moves. With what constant speed v does the bar slide along the rails?

The homopolar generator, also called the Faraday disk, is a low-voltage, high-current electric generator. It consists of a rotating conducting disk with one stationary brush (a sliding electrical contact) at its axle and another at a point on its circumference as shown in Figure P31.33. A uniform magnetic field is applied perpendicular to the plane of the disk. Assume the field is 0.900 T, the angular speed is 3.20 3 103 rev/min, and the radius of the disk is 0.400 m. Find the emf generated between the brushes. When superconducting coils are used to produce a large magnetic field, a homopolar generator can have a power output of several megawatts. Such a generator is useful, for example, in purifying metals by electrolysis. If a voltage is applied to the output terminals of the generator, it runs in reverse as a homopolar motor capable of providing great torque, useful in ship propulsion.

A conducting bar of length , moves to the right on two frictionless rails as shown in Figure P31.34. A uniform magnetic field directed into the page has a magnitude of 0.300 T. Assume R 5 9.00 V and , 5 0.350 m. (a) At what constant speed should the bar move to produce an 8.50-mA current in the resistor? (b) What is the direction of the induced current? (c) At what rate is energy delivered to the resistor? (d) Explain the origin of the energy being delivered to the resistor.

Review. After removing one string while restringing his acoustic guitar, a student is distracted by a video game. His experimentalist roommate notices his inattention and attaches one end of the string, of linear density m 5 3.00 3 1023 kg/m, to a rigid support. The other end passes over a pulley, a distance , 5 64.0 cm from the fixed end, and an object of mass m 5 27.2 kg is attached to the hanging end of the string. The roommate places a magnet across the string as shown in Figure P31.35. The magnet does not touch the string, but produces a uniform field of 4.50 mT over a 2.00-cm length of the string and negligible field elsewhere. Strumming the string sets it vibrating vertically at its fundamental (lowest) frequency. The section of the string in the magnetic field moves perpendicular to the field with a uniform amplitude of 1.50 cm. Find (a) the frequency and (b) the amplitude of the emf induced between the ends of the string

A rectangular coil with resistance R has N turns, each of length , and width w as shown in Figure P31.36. The coil moves into a uniform magnetic field B S with constant velocity v S. What are the magnitude and direction of the total magnetic force on the coil (a) as it enters the magnetic field, (b) as it moves within the field, and (c) as it leaves the field?

Two parallel rails with negligible resistance are 10.0 cm apart and are connected by a resistor of resistance R3 5 5.00 V. The circuit also contains two metal rods having resistances of R1 5 10.0 V and R2 5 15.0 V sliding along the rails (Fig. P31.37). The rods are pulled away from the resistor at constant speeds of v1 5 4.00 m/s and v2 5 2.00 m/s, respectively. A uniform magnetic field of magnitude B 5 0.010 0 T is applied perpendicular to the plane of the rails. Determine the current in R3.

An astronaut is connected to her spacecraft by a 25.0-m-long tether cord as she and the spacecraft orbit the Earth in a circular path at a speed of 7.80 3 103 m/s. At one instant, the emf between the ends of a wire embedded in the cord is measured to be 1.17 V. Assume the long dimension of the cord is perpendicular to the Earths magnetic field at that instant. Assume also the tethers center of mass moves with a velocity perpendicular to the Earths magnetic field. (a) What is the magnitude of the Earths field at this location? (b) Does the emf change as the system moves from one location to another? Explain. (c) Provide two conditions under which the emf would be zero even though the magnetic field is not zero.

Within the green dashed circle shown in Figure P31.39, the magnetic field changes with time according to the expression B 5 2.00t 3 2 4.00t 2 1 0.800, where B is in teslas, t is in seconds, and R 5 2.50 cm. When t 5 2.00 s, calculate (a) the magnitude and (b) the direcS Q/C tion of the force exerted on an electron located at point P1, which is at a distance r1 5 5.00 cm from the center of the circular field region. (c) At what instant is this force equal to zero?

A magnetic field directed into the page changes with time according to B 5 0.030 0t 2 1 1.40, where B is in teslas and t is in seconds. The field has a circular cross section of radius R 5 2.50 cm (see Fig. P31.39). When t 5 3.00 s and r2 5 0.020 0 m, what are (a) the magnitude and (b) the direction of the electric field at point P2?

A long solenoid with 1.00 3 103 turns per meter and radius 2.00 cm carries an oscillating current I 5 5.00 sin 100pt, where I is in amperes and t is in seconds. (a) What is the electric field induced at a radius r 5 1.00 cm from the axis of the solenoid? (b) What is the direction of this electric field when the current is increasing counterclockwise in the solenoid?

A 100-turn square coil of side 20.0 cm rotates about a vertical axis at 1.50 3 103 rev/min as indicated in Figure P31.42. The horizontal component of the Earths magnetic field at the coils location is equal to 2.00 3 1025 T. (a) Calculate the maximum emf induced in the coil by this field. (b) What is the orientation of the coil with respect to the magnetic field when the maximum emf occurs?

A generator produces 24.0 V when turning at 900 rev/min. What emf does it produce when turning at 500 rev/min?

Figure P31.44 (page 964) is a graph of the induced emf versus time for a coil of N turns rotating with angular speed v in a uniform magnetic field directed perpendicular to the coils axis of rotation. What If? Copy this sketch (on a larger scale) and on the same set of axes show the graph of emf versus t (a) if the number of turns in the coil is doubled, (b) if instead the angular speed is doubled, and (c) if the angular speed is doubled while the number of turns in the coil is halved.

In a 250-turn automobile alternator, the magnetic flux in each turn is FB 5 2.50 3 1024 cos vt, where FB is in webers, v is the angular speed of the alternator, and t is in seconds. The alternator is geared to rotate three times for each engine revolution. When the engine is running at an angular speed of 1.00 3 103 rev/min, determine (a) the induced emf in the alternator as a function of time and (b) the maximum emf in the alternator.

In Figure P31.46, a semicircular conductor of radius R 5 0.250 m is rotated about the axis AC at a constant rate of 120 rev/min. A uniform magnetic field of magnitude 1.30 T fills the entire region below the axis and is directed out of the page. (a) Calculate the maximum value of the emf induced between the ends of the conductor. (b) What is the value of the average induced emf for each complete rotation? (c) What If? How would your answers to parts (a) and (b) change if the magnetic field were allowed to extend a distance R above the axis of rotation? Sketch the emf versus time (d) when the field is as drawn in Figure P31.46 and (e) when the field is extended as described in part (c).

A long solenoid, with its axis along the x axis, consists of 200 turns per meter of wire that carries a steady current of 15.0 A. A coil is formed by wrapping 30 turns of thin wire around a circular frame that has a radius of 8.00 cm. The coil is placed inside the solenoid and mounted on an axis that is a diameter of the coil and coincides with the y axis. The coil is then rotated with an angular speed of 4.00p rad/s. The plane of the coil is in the yz plane at t 5 0. Determine the emf generated in the coil as a function of time.

A motor in normal operation carries a direct current of 0.850 A when connected to a 120-V power supply. The resistance of the motor windings is 11.8 V. While in normal operation, (a) what is the back emf generated by the motor? (b) At what rate is internal energy produced in the windings? (c) What If? Suppose a malfunction stops the motor shaft from rotating. At what rate will internal energy be produced in the windings in this case? (Most motors have a thermal switch that will turn off the motor to prevent overheating when this stalling occurs.)

The rotating loop in an AC generator is a square 10.0 cm on each side. It is rotated at 60.0 Hz in a uniform field of 0.800 T. Calculate (a) the flux through the loop as a function of time, (b) the emf induced in the loop, (c) the current induced in the loop for a loop resistance of 1.00 V, (d) the power delivered to the loop, and (e) the torque that must be exerted to rotate the loop

Figure P31.50 represents an electromagnetic brake that uses eddy currents. An electromagnet hangs from a railroad car near one rail. To stop the car, a large current is sent through the coils of the electromagnet. The moving electromagnet induces eddy currents in the rails, whose fields oppose the change in the electromagnets field. The magnetic fields of the eddy currents exert force on the current in the electromagnet, thereby slowing the car. The direction of the cars motion and the direction of the current in the electromagnet are shown correctly in the picture. Determine which of the eddy currents shown on the rails is correct. Explain your answer.

Consider a transcranial magnetic stimulation (TMS) device (Figure P31.3) containing a coil with several turns of wire, each of radius 6.00 cm. In a circular area of the brain of radius 6.00 cm directly below and coaxial with the coil, the magnetic field changes at the rate of 1.00 3 104 T/s. Assume that this rate of change is the same everywhere inside the circular area. (a) What is the emf induced around the circumference of this circular area in the brain? (b) What electric field is induced on the circumference of this circular area?

Suppose you wrap wire onto the core from a roll of cellophane tape to make a coil. Describe how you can use a bar magnet to produce an induced voltage in the coil. What is the order of magnitude of the emf you generate? State the quantities you take as data and their values.

A circular coil enclosing an area of 100 cm2 is made of 200 turns of copper wire (Figure P31.53). The wire making up the coil has no resistance; the ends of the wire are connected across a 5.00-V resistor to form a closed circuit. Initially, a 1.10-T uniform magnetic field points perpendicularly upward through the plane of the coil. The direction of the field then reverses so that the final magnetic field has a magnitude of 1.10 T and points downward through the coil. If the time interval required for the field to reverse directions is 0.100 s, what is the average current in the coil during that time?

A circular loop of wire of resistance R 5 0.500 V and radius r 5 8.00 cm is in a uniform magnetic field directed out of the page as in Figure P31.54. If a clockwise current of I 5 2.50 mA is induced in the loop, (a) is the magnetic field increasing or decreasing in time? (b) Find the rate at which the field is changing with time.

A rectangular loop of area A 5 0.160 m2 is placed in a region where the magnetic field is perpendicular to the plane of the loop. The magnitude of the field is allowed to vary in time according to B 5 0.350 e2t/2.00, where B is in teslas and t is in seconds. The field has the constant value 0.350 T for t , 0. What is the value for e at t 5 4.00 s?

A rectangular loop of area A is placed in a region where the magnetic field is perpendicular to the plane of the loop. The magnitude of the field is allowed to vary in time according to B 5 Bmaxe2t/t , where Bmax and t are constants. The field has the constant value Bmax for t , 0. Find the emf induced in the loop as a function of time.

Strong magnetic fields are used in such medical procedures as magnetic resonance imaging, or MRI. A technician wearing a brass bracelet enclosing area 0.005 00 m2 places her hand in a solenoid whose magnetic field is 5.00 T directed perpendicular to the plane of the bracelet. The electrical resistance around the bracelets circumference is 0.020 0 V. An unexpected power failure causes the field to drop to 1.50 T in a time interval of 20.0 ms. Find (a) the current induced in the bracelet and (b) the power delivered to the bracelet. Note: As this problem implies, you should not wear any metal objects when working in regions of strong magnetic fields.

Consider the apparatus shown in Figure P31.58 in which a conducting bar can be moved along two rails connected to a lightbulb. The whole system is immersed in a magnetic field of magnitude B 5 0.400 T perpendicular and into the page. The distance between the horizontal rails is , 5 0.800 m. The resistance of the lightbulb is R 5 48.0 V, assumed to be constant. The bar and rails have negligible resistance. The bar is moved toward the right by a constant force of magnitude F 5 0.600 N. We wish to find the maximum power delivered to the lightbulb. (a) Find an expression for the current in the lightbulb as a function of B, ,, R, and v, the speed of the bar. (b) When the maximum power is delivered to the lightbulb, what analysis model properly describes the moving bar? (c) Use the analysis model in part (b) to find a numerical value for the speed v of the bar when the maximum power is being delivered to the lightbulb. (d) Find the current in the lightbulb when maximum power is being delivered to it. (e) Using P 5 I 2R, what is the maximum power delivered to the lightbulb? (f) What is the maximum mechanical input power delivered to the bar by the force F ? (g) We have assumed the resistance of the lightbulb is constant. In reality, as the power delivered to the lightbulb increases, the filament temperature increases and the resistance increases. Does the speed found in part (c) change if the resistance increases and all other quantities are held constant? (h) If so, does the speed found in part (c) increase or decrease? If not, explain. (i) With the assumption that the resistance of the lightbulb increases as the current increases, does the power found in part (f) change? ( j) If so, is the power found in part (f) larger or smaller? If not, explain.

A guitars steel string vibrates (see Fig. 31.5). The component of magnetic field perpendicular to the area of a pickup coil nearby is given by B 5 50.0 1 3.20 sin 1 046pt where B is in milliteslas and t is in seconds. The circular pickup coil has 30 turns and radius 2.70 mm. Find the emf induced in the coil as a function of time.

Why is the following situation impossible? A conducting rectangular loop of mass M 5 0.100 kg, resistance R 5 1.00 V, and dimensions w 5 50.0 cm by , 5 90.0 cm is held with its lower edge just above a region with a uniform magnetic field of magnitude B 5 1.00 T as shown in Figure P31.60. The loop is released from rest. Just as the top edge of the loop reaches the region containing the field, the loop moves with a speed 4.00 m/s.

The circuit in Figure P31.61 is located in a magnetic field whose magnitude varies with time according to the expression B 5 1.00 3 1023 t, where B is in teslas and t is in seconds. Assume the resistance per length of the wire is 0.100 V/m. Find the current in section PQ of length a 5 65.0 cm.

Magnetic field values are often determined by using a device known as a search coil. This technique depends on the measurement of the total charge passing through a coil in a time interval during which the magnetic flux linking the windings changes either because of the coils motion or because of a change in the value of B. (a) Show that as the flux through the coil changes from F1 to F2, the charge transferred through the coil is given by Q 5 N(F2 2 F1)/R, where R is the resistance of the coil and N is the number of turns. (b) As a specific example, calculate B when a total charge of 5.00 3 1024 C passes through a 100-turn coil of resistance 200 V and cross-sectional area 40.0 cm2 as it is rotated in a uniform field from a position where the plane of the coil is perpendicular to the field to a position where it is parallel to the field.

A conducting rod of length , 5 35.0 cm is free to slide on two parallel conducting bars as shown in Figure P31.63. Two resistors R1 5 2.00 V and R2 5 5.00 V are connected across the ends of the bars to form a loop. A constant magnetic field B 5 2.50 T is directed perpendicularly into the page. An external agent pulls the rod to the left with a constant speed of v 5 8.00 m/s. Find (a) the currents in both resistors, (b) the total power delivered to the resistance of the circuit, and (c) the magnitude of the applied force that is needed to move the rod with this constant velocity

Review. A particle with a mass of 2.00 3 10216 kg and a charge of 30.0 nC starts from rest, is accelerated through a potential difference DV, and is fired from a small source in a region containing a uniform, constant magnetic field of magnitude 0.600 T. The particles velocity is perpendicular to the magnetic field lines. The circular orbit of the particle as it returns to the location of the source encloses a magnetic flux of 15.0 mWb. (a) Calculate the particles speed. (b) Calculate the potential difference through which the particle was accelerated inside the source.

The plane of a square loop of wire with edge length a 5 0.200 m is oriented vertically and along an east west axis. The Earths magnetic field at this point is of magnitude B 5 35.0 mT and is directed northward at 35.08 below the horizontal. The total resistance of the loop and the wires connecting it to a sensitive ammeter is 0.500 V. If the loop is suddenly collapsed by horizontal forces as shown in Figure P31.65, what total charge enters one terminal of the ammeter?

In Figure P31.66, the rolling axle, 1.50 m long, is pushed along horizontal rails at a constant speed v 5 3.00 m/s. A resistor R 5 0.400 V is connected to the rails at points a and b, directly opposite each other. The wheels make good electrical contact with the rails, so the axle, rails, and R form a closed-loop circuit. The only significant resistance in the circuit is R. A uniform magnetic field B 5 0.080 0 T is vertically downward. (a) Find the induced current I in the resistor. (b) What horizontal force F is required to keep the axle rolling at constant speed? (c) Which end of the resistor, a or b, is at the higher electric potential? (d)What If? After the axle rolls past the resistor, does the current in R reverse direction? Explain your answer.

Figure P31.67 shows a stationary conductor whose shape is similar to the letter e. The radius of its circular portion is a 5 50.0 cm. It is placed in a constant magnetic field of 0.500 T directed out of the page. A straight conducting rod, 50.0 cm long, is pivoted about point O and rotates with a constant angular speed of 2.00 rad/s. (a) Determine the induced emf in the loop POQ. Note that the area of the loop is ua2/2. (b) If all the conducting material has a resistance per length of 5.00 V/m, what is the induced current in the loop POQ at the instant 0.250 s after point P passes point Q?

A conducting rod moves with a constant velocity in a direction perpendicular to a long, straight wire carrying a current I as shown in Figure P31.68. Show that the magnitude of the emf generated between the ends of the rod is 0e0 5 m0vI, 2pr In this case, note that the emf decreases with increasing r as you might expect.

A small, circular washer of radius a 5 0.500 cm is held directly below a long, straight wire carrying a current of I 5 10.0 A. The washer is located h 5 0.500 m above the top of a table (Fig. P31.69). Assume the magnetic field is nearly constant over the area of the washer and equal to the magnetic field at the center of the washer. (a) If the washer is dropped from rest, what is the magnitude of the average induced emf in the washer over the time interval between its release and the moment it hits the tabletop? (b) What is the direction of the induced current in the washer?

Figure P31.70 shows a compact, circular coil with 220 turns and radius 12.0 cm immersed in a uniform magnetic field parallel to the axis of the coil. The rate of change of the field has the constant magnitude 20.0 mT/s. (a) What additional information is necessary to determine whether the coil is carrying clockwise or counterclockwise current? (b) The coil overheats if more than 160 W of power is delivered to it. What resistance would the coil have at this critical point? (c) To run cooler, should it have lower resistance or higher resistance?

A rectangular coil of 60 turns, dimensions 0.100 m by 0.200 m, and total resistance 10.0 V rotates with angular speed 30.0 rad/s about the y axis in a region where a 1.00-T magnetic field is directed along the x axis. The time t 5 0 is chosen to be at an instant when the plane of the coil is perpendicular to the direction of B S. Calculate (a) the maximum induced emf in the coil, (b) the maximum rate of change of magnetic flux through the coil, (c) the induced emf at t 5 0.050 0 s, and (d) the torque exerted by the magnetic field on the coil at the instant when the emf is a maximum.

Review. In Figure P31.72, a uniform magnetic field decreases at a constant rate dB/dt 5 2K, where K is a positive constant. A circular loop of wire of radius a containing a resistance R and a capacitance C is placed with its plane normal to the field. (a)Find the charge Q on the capacitor when it is fully charged. (b) Which plate, upper or lower, is at the higher potential? (c) Discuss the force that causes the separation of charges.

An N-turn square coil with side , and resistance R is pulled to the right at constant speed v in the presence of a uniform magnetic field B acting perpendicular to the coil as shown in Figure P31.73. At t 5 0, the right side of the coil has just departed the right edge of the field. At time t, the left side of the coil enters the region where B 5 0. In terms of the quantities N, B, ,, v, and R, find symbolic expressions for (a) the magnitude of the induced emf in the loop during the time interval from t 5 0 to t, (b) the magnitude of the induced current in the coil, (c) the power delivered to the coil, and (d) the force required to remove the coil from the field. (e) What is the direction of the induced current in the loop? (f) What is the direction of the magnetic force on the loop while it is being pulled out of the field?

A conducting rod of length , moves with velocity v S parallel to a long wire carrying a steady current I. The axis of the rod is maintained perpendicular to the wire with the near end a distance r away (Fig. P31.74). Show that the magnitude of the emf induced in the rod is

The magnetic flux through a metal ring varies with time t according to FB 5 at 3 2 bt 2, where FB is in webers, a 5 6.00 s23, b 5 18.0 s22, and t is in seconds. The resistance of the ring is 3.00 V. For the interval from t 5 0 to t 5 2.00 s, determine the maximum current induced in the ring.

A rectangular loop of dimensions , and w moves with a constant velocity v S away from a long wire that carries a current I in the plane of the loop (Fig. P31.76). The total resistance of the loop is R. Derive an expression that gives the current in the loop at the instant the near side is a distance r from the wire.

A long, straight wire carries a current given by I 5 I max sin (vt 1 f). The wire lies in the plane of a rectangular coil of N turns of wire as shown in Figure P31.77. The quantities Imax, v, and f are all constants. Assume I max 5 50.0 A, v 5 200p s21, N 5 100, h 5 w 5 5.00 cm, and L 5 20.0 cm. Determine the emf induced in the coil by the magnetic field created by the current in the straight wire.

A thin wire , 5 30.0 cm long is held parallel to and d 5 80.0 cm above a long, thin wire carrying I 5 200 A and fixed in position (Fig. P31.78). The 30.0-cm wire is released at the instant t 5 0 and falls, remaining parallel to the current-carrying wire as it falls. Assume the falling wire accelerates at 9.80 m/s2. (a) Derive an equation for the emf induced in it as a function of time. (b) What is the minimum value of the emf? (c) What is the maximum value? (d) What is the induced emf 0.300 s after the wire is released?

Two infinitely long solenoids (seen in cross section) pass through a circuit as shown in Figure P31.79. The magnitude of B S inside each is the same and is increasing at the rate of 100 T/s. What is the current in each resistor?

An induction furnace uses electromagnetic induction to produce eddy currents in a conductor, thereby raising the conductors temperature. Commercial units operate at frequencies ranging from 60 Hz to about 1 MHz and deliver powers from a few watts to several megawatts. Induction heating can be used for warming a metal pan on a kitchen stove. It can be used to avoid oxidation and contamination of the metal when welding in a vacuum enclosure. To explore induction heating, consider a flat conducting disk of radius R, thickness b, and resistivity r. A sinusoidal magnetic field Bmax cos vt is applied perpendicular to the disk. Assume the eddy currents occur in circles concentric with the disk. (a) Calculate the average power delivered to the disk. (b) What If? By what factor does the power change when the amplitude of the field doubles? (c) When the frequency doubles? (d) When the radius of the disk doubles?

A bar of mass m and resistance R slides without friction in a horizontal plane, moving on parallel rails as shown in Figure P31.81. The rails are separated by a distance d. A battery that maintains a constant emf e is connected between the rails, and a constant magnetic field B S is directed perpendicularly out of the page. Assuming the bar starts from rest at time t 5 0, show that at time t it moves with a speed v 5 e Bd 11 2 e 2B 2 d 2 t /mR 2

A betatron is a device that accelerates electrons to energies in the MeV range by means of electromagnetic induction. Electrons in a vacuum chamber are held in a circular orbit by a magnetic field perpendicular to the orbital plane. The magnetic field is gradually increased to induce an electric field around the orbit. (a) Show that the electric field is in the correct direction to make the electrons speed up. (b) Assume the radius of the orbit remains constant. Show that the average magnetic field over the area enclosed by the orbit must be twice as large as the magnetic field at the circles circumference.

Review. The bar of mass m in Figure P31.83 is pulled horizontally across parallel, frictionless rails by a massless string that passes over a light, frictionless pulley and is attached to a suspended object of mass M. The uniform upward magnetic field has a magnitude B, and the distance between the rails is ,. The only significant electrical resistance is the load resistor R shown connecting the rails at one end. Assuming the suspended object is released with the bar at rest at t 5 0, derive an expression that gives the bars horizontal speed as a function of time.