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All Textbook Solutions for Physics for Scientists and Engineers with Modern Physics

Review. A rod of mass m and radius R rests on two parallel rails (Fig. P28.23) that are a distance d apart and have a length L. The rod carries a current I in the direction shown and rolls along the rails without slipping. A uniform magnetic field B is directed perpendicular to the rod and the rails. If it starts from rest, what is the speed of the rod as it leaves the rails?25P Consider the system pictured in Figure P28.26. A 15.0-cm horizontal wire of mass 15.0 g is placed between two thin, vertical conductors, and a uniform magnetic field acts perpendicular to the page. The wire is free to move vertically without friction on the two vertical conductors. When a 5.00-A current is directed as shown in the figure, the horizontal wire moves upward at constant velocity in the presence of gravity. (a) What forces act on the horizontal wire, and (b) under what condition is the wire able to move upward at constant velocity? (c) Find the magnitude and direction of the minimum magnetic Field required to move the wire at constant speed. (d) What happens if the magnetic field exceeds this minimum value? Figure P28.26 A strong magnet is placed under a horizontal conducting ring of radius r that carries current I as shown in Figure P28.27. If the magnetic field B makes an angle with the vertical at the rings location, what are (a) the magnitude and (b) the direction of the resultant magnetic force on the ring? Figure P28.27 In Figure P28.28, the cube is 40.0 cm on each edge. Four straight segments of wireab, bc, cd, and daform a closed loop that carries a current I = 5.00 A in the direction shown. A uniform magnetic field of magnitude B = 0.020 0 T is in the positive y direction. Determine the magnetic force vector on (a) ab, (b) bc, (c) cd, and (d) da. (c) Explain how you could find the force exerted on the fourth of these segments from the forces on the other three, without further calculation involving the magnetic field. Figure P28.28 29P A 50.0-turn circular coil of radius 5.00 cm can be oriented in any direction in a uniform magnetic field having a magnitude of 0.500 T. If the coil carries a current of 25.0 mA, find the magnitude of the maximum possible torque exerted on the coil.You are in charge of planning a physics magic show for an open house on your campus. You come up with the following plan for one trick. You will place a sphere on a rough inclined plane of angle , as shown in Figure P28.31, and it will not roll down the incline. Here is the secret that only you know: The sphere is nonconducting, has a mass of 80.0 g, and a radius 20.0 cm. A flat, compact coil of wire with five turns is wrapped tightly around it, with each turn concentric with the sphere. The sphere is placed on the incline so that the coil is parallel to the plane. You establish a uniform magnetic field of 0.350 T vertically upward in the region of the sphere. (a) What current in the coil do you need to make this trick work? (b) You explain the trick to a friend in confidence and he suggests lowering the angle of the plane to make the required current lower. How do you respond? Figure P28.31 32P A rectangular coil consists of N = 100 closely wrapped turns and has dimensions a = 0.400 m and b = 0.300 m. The coil is hinged along the y axis, and its plane makes an angle = 30.0 with the x axis (Fig. P28.33). (a) What is the magnitude of the torque exerted on the coil by a uniform magnetic field B = 0.800 T directed in the positive x direction when the current is I = 1.20 A in the direction shown? (b) What is the expected direction of rotation of the coil?A rectangular loop of wire has dimensions 0.500 m by 0.300 m. The loop is pivoted at the x axis and lies in the xy plane as shown in Figure P28.34. A uniform magnetic field of magnitude 1.50 T is directed at an angle of 40.0 with respect to the y axis with field lines parallel to the yz plane. The loop carries a current of 0.900 A in the direction shown. (Ignore gravitation.) We wish to evaluate the torque on the current loop. (a) What is the direction of the magnetic force exerted on wire segment ab? (b) What is the direction of the torque associated with this force about an axis through the origin? (c) What is the direction of the magnetic force exerted on segment cd? (d) What is the direction of the torque associated with this force about an axis through the origin? (e) Can the forces examined in parts (a) and (c) combine to cause the Loop to rotate around the x axis? (f) Can they affect the motion of the loop in any way? Explain. (g) What is the direction of the magnetic force exerted on segment bc? (h) What is the direction of the torque associated with this force about an axis through the origin? (i) What is the torque on segment ad about an axis through the origin? (j) From the point of view of Figure P28.34, once the loop is released from rest at the position shown, will it rotate clockwise or counterclockwise around the x axis? (k) Compute the magnitude of the magnetic moment of the loop. (l) What is the angle between the magnetic moment sector and the magnetic field? (m) Compute the torque on the loop using the results to parts (k) and (l). Figure P28.34 35P A Hall-effect probe operates with a 120-mA current. When the probe is placed in a uniform magnetic field of magnitude 0.080 0 T, it produces a Hall voltage of 0.700 V. (a) When it is used to measure an unknown magnetic field, the Hall voltage is 0.330 V. What is the magnitude of the unknown held? (b) The thickness of the probe in the direction of B is 2.00 mm. Find the density of the charge carriers, each of which has charge of magnitude e.37AP 38AP 39AP 40AP 41AP 42AP A proton having an initial velocity of 20.0iMm/s enters a uniform magnetic field of magnitude 0.300 T with a direction perpendicular to the protons velocity. It leaves the field-filled region with velocity 20.0jMm/s. Determine (a) the direction of the magnetic field. (b) the radius of curvature of the protons path while in the field, (c) the distance the proton traveled in the field, and (d) the time interval during which the proton is in the field.44AP 45AP Why is the following situation impossible? Figure P28.46 shows an experimental technique for altering the direction of travel for a charged particle. A particle of charge q = 1.00 C and mass m = 2.00 1015 kg enters the bottom of the region of uniform magnetic field at speed = 2.00 105 m/s, with a velocity vector perpendicular to the field lines. The magnetic force on the particle causes its direction of travel to change so that it leaves the region of the magnetic field at the top traveling at an angle from its original direction. The magnetic field has magnitude B = 0.400 T and is directed out of the page. The length h of the magnetic field region is 0.110 m. An experimenter performs the technique and measures the angle at which the particles exit the top of the field. She finds that the angles of deviation are exactly as predicted. Figure P28.46 A heart surgeon monitors the flow rate of blood through an artery using an electromagnetic flowmeter (Fig. P28.47). Electrodes A and B make contact with the outer surface of the blood vessel, which has a diameter of 3.00 mm. (a) For a magnetic field magnitude of 0.040 0 T, an emf of 160 V appears between the electrodes. Calculate the speed of the blood. (b) Explain why electrode A has to be positive as shown. (c) Does the sign of the emf depend on whether the mobile ions in the blood are predominantly positively or negatively charged? Explain.48AP 49CP Protons having a kinetic energy of 5.00 MeV (1 eV = 1.60 1019 J) are moving in the positive x direction and enter a magnetic field B=0.0500k T directed out of the plane of the page and extending from x = 0 to x = 1.00 m as shown in Figure P28.50. (a) Ignoring relativistic effects, find the angle between the initial velocity vector of the proton beam and the velocity vector after the beam emerges from the field. (b) Calculate the y component of the protons momenta as they leave the magnetic field. Figure P28.50 Review. A wire having a linear mass density of 1.00 g/cm is placed on a horizontal surface that has a coefficient of kinetic friction of 0.200. The wire carries a current of 1.50 A toward the east and slides horizontally to the north at constant velocity. What are (a) the magnitude and (b) the direction of the smallest magnetic field that enables the wire to move in this fashion?Consider the magnetic field due to the current in the wire shown in Figure 29.2. Rank the points A, B, and C in terms of magnitude of the magnetic field that is due to the current in just the length element ds shown from greatest to least. Figure 29.2 (Quick Quiz 29.1) Where is the magnetic field due to the current element the greatest?29.2QQ 29.3QQ 29.4QQ Consider a solenoid that is very long compared with its radius. Of the following choices, what is the most effective way to increase the magnetic field in the interior of the solenoid? (a) double its length, keeping the number of turns per unit length constant (b) reduce its radius by half, keeping the number of turns per unit length constant (c) overwrap the entire solenoid with an additional layer of current-carrying wire Calculate the magnitude of the magnetic field at a point 25.0 cm from a long, thin conductor carrying a current of 2.00 A.2P In Niels Bohrs 1913 model of the hydrogen atom, an electron circles the proton at a distance of 5.29 1011 m with a speed of 2.19 106 m/s. Compute the magnitude of the magnetic field this motion produces at the location of the proton.4P 5P Consider a flat, circular current loop of radius R carrying a current I. Choose the x axis to be along the axis of the loop, with the origin at the loops center. Plot a graph of the ratio of the magnitude of the magnetic field at coordinate x to that at the origin for x = 0 to x = 5R. It may be helpful to use a programmable calculator or a computer to solve this problem.7P One long wire carries current 30.0 A to the left along the x axis. A second long wire carries current 50.0 A to the right along the line (y = 0.280 m, z = 0). (a) Where in the plane of the two wires is the total magnetic field equal to zero? (b) A particle with a charge of 2.00 C is moving with a velocity of 150iMm/s along the line (y = 0.100 m, z = 0). Calculate the vector magnetic force acting on the particle. (c) What If? A uniform electric field is applied to allow this particle to pass through this region undetected. Calculate the required vector electric field.Determine the magnetic field (in terms of I, a, and d) at the origin due to the current loop in Figure P29.9. The loop extends to infinity above the figure. Figure P29.9 10P Two long, parallel wires carry currents of I1 = 3.00 A and I2 = 5.00 A in the directions indicated in Figure P29.11 (page 792). (a) Find the magnitude and direction of the magnetic field at a point midway between the wires. (b) Find the magnitude and direction of the magnetic field at point P, located d = 20.0 cm above the wire carrying the 5.00-A current. Figure P29.11 12P 13P 14P You are part of a team working in a machine parts mechanics shop. An important customer has asked your company to provide springs with a very precise force constant k. To measure the spring constant, you fasten two of the springs between the ends of two very long wires of length L, separated by the unstretched length of the springs as shown in Figure P29.15. The specific attachment method that you use insulates the springs from the wires so that no current passes through the springs. You lay the apparatus flat on a table and then pass a current of magnitude I through the wires, in opposite directions. As a result the springs stretch by a distance d and come to equilibrium. You determine an expression for the spring constant in terms of L, I, , and d. Figure P29.15 Why is the following situation impossible? Two parallel copper conductors each have length = 0.500 m and radius r = 250 m. They carry currents I = 10.0 A in opposite directions and repel each other with a magnetic force FB = 1.00 N.17P 18P The magnetic coils of a tokamak fusion reactor are in the shape of a toroid having an inner radius of 0.700 m and an outer radius of 1.30 m. The toroid has 900 turns of large-diameter wire, each of which carries a current of 14.0 kA. Find the magnitude of the magnetic field inside the toroid along (a) the inner radius and (b) the outer radius.A packed bundle of 100 long, straight, insulated wires forms a cylinder of radius R = 0.500 cm. If each wire carries 2.00 A, what are (a) the magnitude and (b) the direction of the magnetic force per unit length acting on a wire located 0.200 cm from the center of the bundle? (c) What If? Would a wire on the outer edge of the bundle experience a force greater or smaller than the value calculated in parts (a) and (b)? Give a qualitative argument for your answer.21P 22P A long solenoid that has 1 000 turns uniformly distributed over a length of 0.400 m produces a magnetic field of magnitude 1.00 104 T at its center. What current is required in the windings for that to occur?24P 25P 26P 27P You are working for a company that creates special magnetic environments. Your new supervisor has come from the financial side of the organization rather than the technical side. He has promised a client that the company can provide a device that will create a magnetic field inside a cylindrical chamber that is directed along the cylinder axis at all points in the chamber and increases in the axial direction as the square of the value of y, where y is the in the axial direction and y = 0 is at the bottom end of the cylinder. Prepare a calculation to show that the field requested by your supervisor and promised to a client is impossible.A solenoid of radius r = 1.25 cm and length = 30.0 cm has 300 turns and carries 12.0 A. (a) Calculate the flux through the surface of a disk-shaped area of radius R = 5.00 cm that is positioned perpendicular to and centered on the axis of the solenoid as shown in Figure P29.20a. (b) Figure P29.29b shows an enlarged end view of the same solenoid. Calculate the flux through the tan area, which is an annulus with an inner radius of a = 0.400 cm and an outer radius of b = 0.800 cm.30P 31AP Why is the following situation impossible? The magnitude of the Earths magnetic field at either pole is approximately 7.0 105 T. Suppose the field fades away to zero before its next reversal. Several scientists propose plans for artificially generating a replacement magnetic field to assist with devices that depend on the presence of the field. The plan that is selected is to lay a copper wire around the equator and supply it with a current that would generate a magnetic field of magnitude 7.00 105 T at the poles. (Ignore magnetization of any materials inside the Earth.) The plan is implemented and is highly successful.33AP 34AP 35AP 36AP A very large parallel-plate capacitor has uniform charge per unit area + on the upper plate and on the lower plate. The plates are horizontal, and both move horizontally with speed v to the right. (a) What is the magnetic field between the plates? (b) What is the magnetic field just above or just below the plates? (c) What are the magnitude and direction of the magnetic force per unit area on the upper plate? (d) At what extrapolated speed v will the magnetic force on a plate balance the electric force on the plate? Suggestion: Use Amperes law and choose a path that closes between the plates of the capacitor.Two circular coils of radius R, each with N turns, are perpendicular to a common axis. The coil centers are a distance R apart. Each coil carries a steady current I in the same direction as shown in Figure P29.38. (a) Show that the magnetic field on the axis at a distance x from the center of one coil is B=N0IR22[1(R2+x2)3/2+1(2R2+x22Rx)3/2]39AP Two circular loops are parallel, coaxial, and almost in contact, with their centers 1.00 mm apart (Fig. P29.40). Each loop is 10.0 cm in radius. The top loop carries a clockwise current of I = 140 A. The bottom loop carries a counter-clockwise current of I = 140 A. (a) Calculate the magnetic force exerted by the bottom loop on the top loop. (b) Suppose a student thinks the first step in solving part (a) is to use Equation 29.7 to find the magnetic field created by one of the loops. How would you argue for or against this idea? (c) The upper loop has a mass of 0.021 0 kg. Calculate its acceleration, assuming the only forces acting on it are the force in part (a) and the gravitational force. Figure P29.40 41AP Review. Rail guns have been suggested for launching projectiles into space without chemical rockets. A tabletop model rail gun (Fig. P29.42) consists of two long, parallel, horizontal rails = 3.50 cm apart, bridged by a bar of mass m = 3.00 g that is free to slide without friction. The rails and bar have low electric resistance, and the current is limited to a constant I = 24.0 A by a power supply that is far to the left of the figure, so it has no magnetic effect on the bar. Figure P29.42 shows the bar at rest at the midpoint of the rails at the moment the current is established. We wish to find the speed with which the bar leaves the rails after being released from the midpoint of the rails. (a) Find the magnitude of the magnetic field at a distance of 1.75 cm from a single long wire carrying a current of 2.40 A. (b) For purposes of evaluating the magnetic field, model the rails as infinitely long. Using the result of part (a), find the magnitude and direction of the magnetic field at the midpoint of the bar. (c) Argue that this value of the field will be the same at all positions of the bar to the right of the midpoint of the rails. At other points along the bar, the field is in the same direction as at the midpoint, but is larger in magnitude. Assume the average effective magnetic field along the bar is five times larger than the field at the midpoint. With this assumption, find (d) the magnitude and (e) the direction of the force on the bar. (f) Is the bar properly modeled as a particle under constant acceleration? (g) Find the velocity of the bar after it has traveled a distance d = 130 cm to the end of the rails. Figure P29.42 43AP An infinitely long, straight wire carrying a current I1 is partially surrounded by a loop as shown in Figure P29.44. The loop has a length L and radius R, and it carries a current I2. The axis of the loop coincides with the wire. Calculate the magnetic force exerted on the loop. Figure P29.44 45CP 46CP A wire carrying a current I is bent into the shape of an exponential spiral, r = e, from = 0 to = 2 as suggested in Figure P29.47. To complete a loop, the ends of the spiral are connected by a straight wire along the x axis. (a) The angle between a radial line and its tangent line at any point on a curve r = f() is related to the function by tan=rdr/d Use this fact to show that = /4. (b) Find the magnetic field at the origin. Figure P29.47 48CP 49CP 50CP 51CP A circular loop of wire is held in a uniform magnetic field, with the plane of the loop perpendicular to the field lines. Which of the following will not cause a current to be induced in the loop? (a) crushing the loop (b) rotating the loop about an axis perpendicular to the field lines (c) keeping the orientation of the loop fixed and moving it along the field lines (d) pulling the loop out of the field QUICK QUIZ 30.2 In Figure 30.8a, a given applied force of magnitude F results in a constant speed v and a power input P. Imagine that the force is increased so that the constant speed of the bar is doubled to 2v. Under these conditions, what are the new force and the new power input? (a) 2F and 2P (b) 4F and 2P (c) 2F and 4P (d) 4F and 4P Figure 30.12 (Quick Quiz 30.3) QUICK QUIZ 30.3 Figure 30.12 shows a circular loop of wire falling toward a wire carrying a current to the left. What is the direction of the induced current in the loop of wire? (a) clockwise (b) counterclockwise (c) zero (d) impossible to determine 30.4QQ A circular loop of wire of radius 12.0 cm is placed in a magnetic field directed perpendicular to the plane of the loop as in Figure P30.1. If the field decreases at the rate of 0.050 0 T/s in some time interval, find the magnitude of the emf induced in the loop during this interval. Figure P30.1 An instrument based on induced emf has been used to measure projectile speeds up to 6 km/s. A small magnet is imbedded in the projectile as shown in Figure P30.2. The projectile passes through two coils separated by a distance d. As the projectile passes through each coil, a pulse of emf is induced in the coil. The time interval between pulses can be measured accurately with an oscilloscope, and thus the speed can be determined. (a) Sketch a graph of V versus t for the arrangement shown. Consider a current that flows counterclockwise as viewed from the starting point of the projectile as positive. On your graph, indicate which pulse is from coil 1 and which is from coil 2. (b) If the pulse separation is 2.40 ms and d = 1.50 m, what is the projectile speed? Figure P30.2 Scientific work is currently under way to determine whether weak oscillating magnetic fields can affect human health. For example, one study found that drivers of trains had a higher incidence of blood cancer than other railway workers, possibly due to long exposure to mechanical devices in the train engine cab. Consider a magnetic field of magnitude 1.00 103 T, oscillating sinusoidally at 60.0 Hz. If the diameter of a red blood cell is 8.00 m, determine the maximum emf that can be generated around the perimeter of a cell in this field.A long solenoid has n = 400 turns per meter and carries a current given by I = 30.0(1 e1.60t), 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 = 6.00 cm and consists of a total of N = 250 turns of fine wire (Fig. P30.4). What emf is induced in the coil by the changing current? Figure P30.4 An aluminum ring of radius r1 = 5.00 cm and resistance 3.00 104 is placed around one end of a long air-core solenoid with 1 000 turns per meter and radius r2 = 3.00 cm as shown in Figure P30.5. Assume the axial component of the field produced by the solenoid is one-half as strong over the area of the end of the solenoid 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 270 A/s. (a) What is the induced current in the ring? At the center of the ring, what are (b) the magnitude and (c) the direction of the magnetic field produced by the induced current in the ring? Figure P30.5 Problems 5 and 6.6P 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 30.0 with the direction of the field. When the magnetic field is increased uniformly from 200 T to 600 T 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?8P A toroid having a rectangular cross section (a = 2.00 cm by b = 3.00 cm) and inner radius R = 4.00 cm consists of N = 500 turns of wire that carry a sinusoidal current I = Imax sin t, with Imax = 50.0 A and a frequency f = /2 = 60.0 Hz. A coil that consists of N = 20 turns of wire is wrapped around one section of the toroid as shown in Figure P30.9. Determine the emf induced in the coil as a function of time. Figure P30.9 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 T 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 cast? (d) Can this emf be used to power a lightbulb in the passenger compartment? Explain your answer.A helicopter (Fig. P30.11) 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 T, what is the emf induced between the blade tip and the center hub? Figure P30.11 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 T and is directed northward and 53.0 below the horizontal. (a) Calculate the magnitude of the induced emf between the ends of the wire and (b) determine which end is positive.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 P30.13. 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. Figure P30.13 14P 15P 16P You are working for a company that manufactures motors and generators. At the end of your first day of work, your supervisor explains to you that you will be assigned to a team that is designing a new homopolar generator. You have no idea what that is, but agree wholeheartedly to the assignment. At home that evening, you go online to learn about the homopolar generator and find the following. 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 P30.17. A uniform magnetic field is applied perpendicular to the plane of the disk. 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. At work the next morning, your supervisor tells you that the homopolar generator under consideration will have a magnetic field of magnitude B = 0.900 T and the radius of the disk is r = 0.400 m. The desired emf to be generated with the device is E=25.0V. Your supervisor asks you to determine the required angular speed of the disk to achieve this result. Figure P30.17 You are working in a laboratory that uses motional emf to make magnetic measurements. You have found that it is difficult to create a uniform magnetic field across the entire sliding-bar apparatus shown in Figure 30.8a, with a resistance R connected between the rails. You decide to investigate creating the magnetic field with a long, straight, current-carrying conductor lying next to and parallel to one of the rails, as show in Figure P30.18. This will create a non-uniform field across the plane of the bar and rails. You set up the apparatus in this way, with the current-carrying wire a distance a from the upper rail. You wish to find an expression for the force necessary to slide the bar at a constant speed of v to the right in Figure P30.18 if the wire carries a current I. (Hint: Two separate integrations will be required.) Figure P30.18 You are working in a factory that produces long bars of copper with a square cross section. In one section of the production process, the bars must slide down a plane inclined at an angle = 21.0 to the horizontal. It has been found that the bars travel with too high a speed and become dented or bent when they arrive at the bottom of the plane and must be discarded. In order to prevent this waste, you devise a way to deliver the bars at the bottom of the plane at a lower speed. You replace the inclined plane with a pair of parallel metal rails, shown in Figure P30.19, separated by a distance = 2.00 m. The smooth bars of mass m = 1.00 kg will slide down the smooth rails, with the length of the bar always perpendicular to the rails. The rails are immersed in a magnetic field of magnitude B, and a resistor of resistance R = 1.00 is connected between the upper ends of the rails. Determine the magnetic field necessary in your device so that the bars will arrive at the bottom of the plane with a maximum speed v = 1.00 m/s. Figure P30.19 Problems 19 and 20.20P Within the green dashed circle show in Figure P30.21, the magnetic field changes with time according to the expression B = 2.00t3 4.00t2 + 0.800, where B is in teslas, t is in seconds, and R = 2.50 cm. When t = 2.00 s, calculate (a) the magnitude and (b) the direction of the force exerted on an electron located at point P, which is at a distance r = 5.00 cm from the center of the circular field region. (c) At what instant is this force equal to zero? Figure P30.21 22P 23P Figure P30.24 (page 820) is a graph of the induced emf versus time for a coil of N turns rotating with angular speed 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. Figure P30.24 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 magnetic 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 , (d) the power delivered to the loop, and (e) the torque that must be exerted to rotate the loop.In Figure P30.26, a semicircular conductor of radius R = 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 P30.26 and (e) when the field is extended as described in part (c). Figure P30.26 27P 28AP 29AP 30AP A circular coil enclosing an area of 100 cm2 is made of 200 turns of copper wire (Figure P30.31). The wire making up the coil has no resistance; the ends of the wire are connected across a 5.00- 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 interval? Figure P30.31 32AP A guitars steel string vibrates (see Fig. 30.5). The component of magnetic field perpendicular to the area of a pickup coil nearby is given by B=50.0+3.20sin1046t 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.34AP A conducting rod of length = 35.0 cm is free to slide on two parallel conducting bars as shown in Figure P30.35. Two resistors R1 = 2.00 and R2 = 5.00 are connected across the ends of the bars to form a loop. A constant magnetic field B = 2.50 T is directed perpendicularly into the page. An external agent pulls the rod to the left with a constant speed of v = 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. Figure P30.35 36AP 37AP In Figure P30.38, the rolling axle, 1.50 m long, is pushed along horizontal rails at a constant speed v = 3.00 m/s. A resistor R = 0.400 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 = 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 P30.38 Figure P30.39 shows a stationary conductor whose shape is similar to the letter e. The radius of its circular portion is a = 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 a2/2. (b) If all the conducting material has a resistance per length of 5.00 /m, what is the induced current in the loop POQ at the instant 0.250 s after point P passes point Q? Figure P30.39 40AP Figure P30.41 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? Figure P30.41 Review. In Figure P30.42, a uniform magnetic field decreases at a constant rate dB/dt = K, 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. Figure P30.42 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 P30.43. At t = 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 = 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 = 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? Figure P30.43 A conducting rod of length moves with velocity v 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. P30.44). Show that the magnitude of the emf induced in the rod is E=0Iv2ln(1+lr) Figure P30.44 A long, straight wire carries a current given by I = Imax sin (t + ). The wire lies in the plane of a rectangular coil of N turns of wire as shown in Figure P30.45. The quantities Imax, , and are all constants. Assume Imax = 50.0 A, = 200 s1, N = 100, h = = 5.00 cm, and L = 20.0 cm. Determine the emf induced in the coil by the magnetic field created by the current in the straight wire. Figure P30.45 A rectangular loop of dimensions and w moves with a constant velocity v away from a long wire that carries a current I in the plane of the loop (Fig. P.10.46). 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. Figure P30.46 A thin wire = 30.0 cm long is held parallel to and d = 80.0 cm above a long, thin wire carrying I = 200 A and fixed in position (Fig. P30.47). The 30.0-cm wire is released at the instant t = 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? Figure P30.47 48CP 49CP 50CP Review. The bar of mass m in Figure P30.51 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 = 0, derive an expression that gives the bars horizontal speed as a function of time. Figure P30.51 A coil with zero resistance has its ends labeled a and b. The potential at a is higher than at b. Which of the following could be consistent with this situation? (a) The current is constant and is directed from a to b. (b) The current is constant and is directed from b to a. (c) The current is increasing and is directed from a to b. (d) The current is decreasing and is directed from a to b. (e) The current is increasing and is directed from b to a. (f) The current is decreasing and is directed from b to a.31.2QQ 31.3QQ 31.4QQ (i) At an instant of time during the oscillations of an LC circuit, the current is at its maximum value. At this instant, what happens to the voltage across the capacitor? (a) It is different from that across the inductor. (b) It is zero. (c) It has its maximum value. (d) It is impossible to determine. (ii) Now consider an instant when the current is momentarily zero. From the same choices, describe the magnitude of the voltage across the capacitor at this instant.1P 2P 3P 4P 5P A toroid has a major radius R and a minor radius r and is tightly wound with N turns of wire on a hollow cardboard torus. Figure P31.6 shows half of this toroid, allowing us to see its cross section. If R r, the magnetic field in the region enclosed by the wire is essentially the same as the magnetic field of a solenoid that has been bent into a large circle of radius R. Modeling the field as the uniform field of a long solenoid, show that the inductance of such a toroid is approximately L=120N2r2R Figure P31.6 7P 8P 9P 10P 11P 12P 13P You are working as a demonstration assistant for a physics professor. He shows you the circuit in Figure P31.14, which he wants you to build for an upcoming class. The lightbulb is a household incandescent bulb that receives energy at the rate of 40.0 W when operating at 120 V. It has a resistance R1, which, for simplicity, we will assume is constant at all operating voltages. The battery in the circuit has an emf of 12.0 V. When the switch has been closed for a long time, the bulb glows dimly, since it is powered by only 12.0 V. When the switch is opened, however, the bulb flashes brightly and then gradually dims to darkness. Your professor wants you to determine two values: (a) the resistance R2 that is necessary for the bulb to initially flash, when the switch is opened, at the same brightness it would have if plugged into a 120-V socket; (b) the inductance L necessary to keep the current in the lightbulb above 50.0% of its value when the switch is opened, for a time interval of 2.00 s after it is opened. Assume a resistance-free inductor and that the resistance of the lightbulb does not vary with temperature. Figure P31.14 15P 16P 17P 18P 19P 20P 21P 22P 23P 24P 25P 26P 27P 28P In the circuit of Figure P31.29, the battery emf is 50.0 V, the resistance is 250 , and the capacitance is 0.500 F. The switch S is closed for a long time interval, and zero potential difference is measured across the capacitor. After the switch is opened, the potential difference across the capacitor reaches a maximum value of 150 V. What Ls the value of the inductance? Figure P31.29 30P 31P 32P In Figure 31.15, let R = 7.60 , L = 2.20 mH, and C = 1.80 F. (a) Calculate the frequency of the damped oscillation of the circuit when the switch is thrown to position b. (b) What is the critical resistance for damped oscillations?34P Electrical oscillations are initiated in a series circuit containing a capacitance C, inductance L, and resistance R. (a) If R4L/C (weak damping), what time interval elapses before the amplitude of the current oscillation falls to 50.0% of its initial value? (b) Over what time interval does the energy decrease to 50.0% of its initial value?36AP A capacitor in a series LC circuit has an initial charge Q and is being discharged. When the charge on the capacitor is Q/2, find the flux through each of the N turns in the coil of the inductor in terms of Q, N, L, and C.38AP 39AP At the moment t = 0, a 24.0-V battery is connected to a 5.00-mH coil and a 6.00- resistor. (a) Immediately thereafter, how does the potential difference across the resistor compare to the emf across the coil? (b) Answer the same question about the circuit several seconds later. (c) Is there an instant at which these two voltages are equal in magnitude? If so, when? Is there more than one such instant? (d) After a 4.00-A current is established in the resistor and coil, the battery is suddenly replaced by a short circuit. Answer parts (a), (b), and (c) again with reference to this new circuit.41AP 42AP 43AP 44AP 45AP At t = 0, the open switch in Figure P31.46 is thrown closed. We wish to find a symbolic expression for the current in the inductor for time t 0. Let this current be called i and choose it to be downward in the inductor in Figure P31.46. Identify i1 as the current to the right through R1 and i2 as the current downward through R2. (a) Use Kirchhoffs junction rule to find a relation among the three currents. (b) Use Kirchhoffs loop rule around the left loop to find another relationship. (c) Use Kirchhoffs loop rule around the outer loop to find a third relationship. (d) Eliminate i1 and i2 among the three equations to find an equation involving only the current i. (e) Compare the equation in part (d) with Equation 31.6 in the text. Use this comparison to rewrite Equation 31.7 in the text for the situation in this problem and show that i(t)=R1[1e(R/L)t] where R = R1R2/(R1 + R2). Figure P31.46 47AP 48AP 49AP 50CP 51CP 52CP 53CP Consider the voltage phasor in Figure 32.4, shown at three instants of time. (i) Choose the part of the figure, (a), (b), or (c), that represents the instant of time at which the instantaneous value of the voltage has the largest magnitude. (ii) Choose the part of the figure that represents the instant of time at which the instantaneous value of the voltage has the smallest magnitude.Consider the AC circuit in Figure 32.8. The frequency of the AC source is adjusted while its voltage amplitude is held constant. When does the lightbulb glow the brightest? (a) It glows brightest at high frequencies. (b) It glows brightest at low frequencies. (c) The brightness is the same at all frequencies.Consider the AC circuit in Figure 32.11. The frequency of the AC source is adjusted while its voltage amplitude is held constant. When does the lightbulb glow the brightest? (a) It glows brightest at high frequencies. (b) It glows brightest at low frequencies. (c) The brightness is the same at all frequencies.Consider the AC circuit in Figure 32.12. The frequency of the AC source is adjusted while its voltage amplitude is held constant. When does the lightbulb glow the brightest? (a) It glows brightest at high frequencies. (b) It glows brightest at low frequencies. (c) The brightness is the same at all frequencies.Label each part of Figure 32.16, (a), (b), and (c), as representing XL, XC, XL = XC, or XL, XC. Figure 32.16 (Quick Quiz 32.5) Match the phasor diagrams to the relationships between the reactances.32.6QQ 32.7QQ (a) What is the resistance of a lightbulb that uses an average power of 75.0 W when connected to a 60.0-Hz power source having a maximum voltage of 170 V? (b) What If? What is the resistance of a 100-W lightbulb?A certain lightbulb is rated at 60.0 W when operating at an rms voltage of 120 V. (a) What is the peak voltage applied across the bulb? (b) What is the resistance of the bulb? (c) Does a 100-W bulb have greater or less resistance than a 60.0-W bulb? Explain.The current in the circuit shown in Figure P32.3 equals 60.0% of the peak current at t = 7.00 ms. What is the lowest source frequency that gives this current? Figure P32.3 Problems 3 and 5.Figure P32.4 shows three lightbulbs connected to a 120-V AC (rms) household supply voltage. Bulbs 1 and 2 have a power rating of 150 W, and bulb 3 has a 100-W rating. Find (a) the rms current in each bulb and (b) the resistance of each bulb. (c) What is the total resistance of the combination of the three lightbulbs? Figure P32.4 5P 6P 7P 8P An AC source has an output rms voltage of 78.0 V at a frequency of 80.0 Hz. If the source is connected across a 25.0-mH inductor, what are (a) the inductive reactance of the circuit, (b) the rms current in the circuit, and (c) the maximum current in the circuit?10P 11P An AC source with an output rms voltage of 86.0 V at a frequency of 60.0 Hz is connected across a 12.0-F capacitor. Find (a) the capacitive reactance, (b) the rms current, and (c) the maximum current in the circuit. (d) Does the capacitor have its maximum charge when the current has its maximum value? Explain.What is the maximum current in a 2.20-F capacitor when it is connected across (a) a North American electrical outlet having Vrms = 120 V and f = 60.0 Hz and (b) a European electrical outlet having Vrms = 240 V and f = 50.0 Hz?14P In addition to phasor diagrams showing voltages such as in Figure 32.15, we can draw phasor diagrams with resistance and reactances. The resultant of adding the phasors is the impedance. Draw to scale a phasor diagram showing Z, XL, XC, and for an AC series circuit for which R = 300 , C = 11.0 F, L = 0.200 H, and f = 500/ Hz.An AC source with Vmax = 150 V and f = 50.0 Hz is connected between points a and d in Figure P32.16. Calculate the maximum voltages between (a) points a and b, (b) points b and c, (c) points c and d, and (d) points b and d. Figure P32.16 Problems 16 and 51.You are working in a factory and have been tasked with determining the electrical needs for a new motor that will be installed on an assembly line. The motor has been tested under load conditions and found to have a resistance of 35. and an inductive reactance of 50.0 . We can model the motor as a series RL circuit. The motor will have its own dedicated circuit with an rms voltage of 480 V. You need to determine the peak current drawn by the motor to determine the size of the circuit breaker needed to protect the circuit.18P 19P A 60.0-ft resistor is connected in series with a 30.0-F capacitor and a source whose maximum voltage is 120 V, operating at 60.0 Hz. Find (a) the capacitive reactance of the circuit, (b) the impedance of the circuit, and (c) the maximum current in the circuit. (d) Does the voltage lead or lag the current? (e) How will adding an inductor in series with the existing resistor and capacitor affect the current? Explain.A series RLC circuit has a resistance of 45.0 and an impedance of 75.0 . What average power is delivered to this circuit when Vrms = 210 V?22P 23P An AC voltage of the form v = 90.0 sin 350t, where v is in volts and t is in seconds, is applied to a series RLC circuit. If R = 50.0 . C = 25.0 F, and L = 0.200 H, find (a) the impedance of the circuit, (b) the rms current in the circuit, and (c) the average power delivered to the circuit.25P A series RLC circuit has components with the following values: L = 20.0 mH, C = 100 nF, R = 20.0 , and Vmax = 100 V, with v = Vmax sin t. Find (a) the resonant frequency of the circuit, (b) the amplitude of the current at the resonant frequency, (c) the Q of the circuit, and (d) the amplitude of the voltage across the inductor at resonance.You wish to build a series RLC circuit for a project you are working on. Looking in your electronics parts box, you are disappointed to find that you have only two resistors, each of resistance 47.0 , two capacitors, each of capacitance 5.0 nF, and one inductor of inductance 5.00 mH. You need to determine the lowest possible angular frequency at resonance that you can obtain from all five components by connecting the inductor in series with a combination of the two resistors and a combination of the two capacitors.A 10.0- resistor, 10.0-mH inductor, and 100-F capacitor are connected in series to a 50.0-V (rms) source having variable frequency. If the operating frequency is twice the resonance frequency, find the energy delivered to the circuit during one period.29P The primary coil of a transformer has N1 = 350 turns, and the secondary coil has N2 = 2 000 turns. If the input voltage across the primary coil is v = 170 cos t, where v is in volts and t is in seconds, what rms voltage is developed across the secondary coil?31P A transmission line that has a resistance per unit length of 4.50 104 /m is to be used to transmit 5.00 MW across 400 mi (6.44 105 m). The output voltage of the source is 4.50 kV. (a) What is the line loss if a transformer is used to step up the voltage to 500 kV? (b) What fraction of the input power is lost to the line under these circumstances? (c) What If? What difficulties would be encountered in attempting to transmit the 5.00 MW at the source voltage of 4.50 kV?33AP 34AP 35AP 36AP 37AP 38AP 39AP 40AP 41AP (a) Sketch a graph of the phase angle for an RLC series circuit as a function of angular frequency from zero to a frequency much higher than the resonance frequency. (b) Identify the value of at the resonance angular frequency 0. (c) Prove that the slope of the graph of versus at the resonance point is 2Q/0.A series RLC circuit contains the following components: R = 150 , L = 0.250 H, C = 2.00 F, and a source with Vmax = 210 V operating at 50.0 Hz. Our goal is to find the phase angle, the power factor, and the power input for this circuit. (a) Find the inductive reactance in the circuit. (b) Find the capacitive reactance in the circuit. (c) Find the impedance in the circuit. (d) Calculate the maximum current in the circuit. (e) Determine the phase angle between the current and source voltage. (f) Find the power factor for the circuit. (g) Find the power input to the circuit.Review. In the circuit shown in Figure P32.44, assume all parameters except C are given. Find (a) the current in the circuit as a function of time and (b) the power delivered to the circuit. (c) Find the current as a function of time after only switch 1 is opened. (d) After switch 2 is also opened, the current and voltage are in phase. Find the capacitance C. Find (e) the impedance of the circuit when both switches are open, (f) the maximum energy stored in the capacitor during oscillations, and (g) the maximum energy stored in the inductor during oscillations. (h) Now the frequency of the voltage source is doubled. Find the phase difference between the current and the voltage. (i) Find the frequency that makes the inductive reactance one-half the capacitive reactance. Figure P32.44 You have decided to build your own speaker system for your home entertainment system. The system will consist of two loudspeakers: a large woofer, to which you want to send low audio frequencies (bass), and a small tweeter, which should receive high audio frequencies (treble). To separate the high and low frequencies of the audio signal, you build the crossover network shown in Figure P32.45. The input voltage is the audio output of the amplifier in your system, shown in the figure as an AC source. You have two outputs as shown: one across the resistor and one across the capacitor. (a) Across which element should you connect the woofer? (b) Across which element should you connect the tweeter? (c) To choose the appropriate values of R and C, you need to determine an expression for the ratio of the output voltage to the input voltage as a function of angular frequency for the resistor as an output. (d) You need to determine a similar expression for the ratio of the output voltage to the input voltage as a function of angular frequency for the capacitor as an output. Figure P32.45 46AP 47AP A series RLC circuit in which R = l.00 , L = 1.00 mH, and C = 1.00 nF is connected to an AC source delivering 1.0 V (rms). (a) Make a precise graph of the power delivered to the circuit as a function of the frequency and (b) verify that the full width of the resonance peak at half-maximum is R/2L.The resistor in Figure P32.49 represents the midrange speaker in a three-speaker system. Assume its resistance to be constant at 8.00 . The source represents an audio amplifier producing signals of uniform amplitude Vmax = 10.0 V at all audio frequencies. The inductor and capacitor are to function as a band-pass filter with Vout/Vin=12 at 200 Hz and at 4.00 103 Hz. Determine the required values of (a) L and (b) C. Find (c) the maximum value of the ratio Vout/Vin; (d) the frequency fo at which the ratio has its maximum value; (e) the phase shift between vin and vout at 200 Hz, at fo, and at 4.00 103 Hz; and (f) the average power transferred to the speaker at 200 Hz, at f0, and at 4.00 103 Hz. (g) Recognizing that the diagram represents an RLC circuit driven by an AC source, find its quality factor. Figure P32.49 50CP 51CP 33.1QQ What is the phase difference between the sinusoidal oscillations of the electric and magnetic fields in Figure 33.8? (a) 180 (b) 90 (c) 0 (d) impossible to determine 33.3QQ 33.4QQ 33.5QQ 33.6QQ 33.7QQ 1P 2P 3P 4P The distance to the North Star, Polaris, is approximately 6.44 1018 m. (a) If Polaris were to burn out today, how many years from now would we see it disappear? (b) What time interval is required for sunlight to reach the Earth? (c) What time interval is required for a microwave signal to travel from the Earth to the Moon and back?6P 7P 8P 9P 10P 11P 12P If the intensity of sunlight at the Earths surface under a fairly clear sky is 1 000 W/m2, how much electromagnetic energy per cubic meter is contained in sunlight?14P 15P Review. Model the electromagnetic wave in a microwave oven as a plane traveling wave moving to the left, with an intensity of 25.0 kW/m2. An oven contains two cubical containers of small mass, each full of water. One has an edge length of 6.00 cm, and the other, 12.0 cm. Energy falls perpendicularly on one face of each container. The water in the smaller container absorbs 70.0% of the energy that falls on it. The water in the larger container absorbs 91.0%. That is, the fraction 0.300 of the incoming microwave energy passes through a 6.00-cm thickness of water, and the fraction (0.300)(0.300) = 0.090 passes through a 12.0-cm thickness. Assume a negligible amount of energy leaves either container by heat. Find the temperature change of the water in each container over a time interval of 480 s.17P 18P 19P 20P 21P The intensity of sunlight at the Earths distance from the Sun is 1 370 W/m2. Assume the Earth absorbs all the sunlight incident upon it. (a) Find the total force the Sun exerts on the Earth due to radiation pressure. (b) Explain how this force compares with the Suns gravitational attraction.23P 24P 25P 26P Extremely low-frequency (ELF) waves that can penetrate the oceans are the only practical means of communicating with distant submarines. (a) Calculate the length of a quarter-wavelength antenna for a transmitter generating ELF waves of frequency 75.0 Hz into air. (b) How practical is this means of communication?A large, flat sheet carries a uniformly distributed electric current with current per unit width Js. This current creates a magnetic field on both sides of the sheet, parallel to the sheet and perpendicular to the current, with magnitude B=120Js. If the current is in the y direction and oscillates in time according to Jmax(cost)j=Jmax[cos(t)]j the sheet radiates an electromagnetic wave. Figure P33.28 shows such a wave emitted from one point on the sheet chosen to be the origin. Such electromagnetic waves arc emitted from all points on the sheet. The magnetic field of the wave to the right of the sheet is described by the wave function B=120Jmax[cos(kxt)]k (a) Find the wave function for the electric field of the wave to the right of the sheet. (b) Find the Poynting vector as a function of x and t. (c) Find the intensity of the wave. (d) What If? If the sheet is to emit radiation in each direction (normal to the plane of the sheet) with intensity 570 W/m2, what maximum value of sinusoidal current density is required? Figure P33.28 29P 30P 31P 32P 33AP 34AP 35AP 36AP 37AP One goal of the Russian space program is to illuminate dark northern cities with sunlight reflected to the Earth from a 200-m diameter mirrored surface in orbit. Several smaller prototypes have already been constructed and put into orbit. (a) Assume that sunlight with intensity 1 370 W/m2 falls on the mirror nearly perpendicularly and that the atmosphere of the Earth allows 74.6% of the energy of sunlight to pass though it in clear weather. What is the power received by a city when the space mirror is reflecting light to it? (b) The plan is for the reflected sunlight to cover a circle of diameter 8.00 km. What is the intensity of light (the average magnitude of the Poynting vector) received by the city? (c) This intensity is what percentage of the vertical component of sunlight at St. Petersburg in January, when the sun reaches an angle of 7.00 above the horizon at noon?

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