1 A horseshoe magnet is held vertically with the north pole on the left and south pole on the right. A wire passing between the poles, equidistant from them, carries a current directly away from you. In what direction is the force on the wire? 7 The circuit in the figure consists of wires at the top and bottom and identical metal springs in the left and right sides. The upper portion of the circuit is fixed. The wire at the bottom has a mass of 10.0 g and is 5.00 cm long. The springs stretch 0.500 cm under the weight of the wire and the circuit has a total resistance of 12.0 Ω. When a magnetic field is turned on, directed out of the page, the springs stretch an additional 0.300 cm. What is the magnitude of the magnetic field? 2 How much current is flowing in a wire 4.80 m long if the maximum force on it is 0.750 N when placed in a uniform 0.0800 T field? 1.95 A 3 The force on a wire carrying 8.75 A is a maximum of 1.28 N when placed between the pole faces of a magnet. If the pole faces are 55.5 cm in diameter, what is the approximate strength of the magnetic field? 0.588 T 4 0.264 T Suppose a straight 1.00 mm diameter copper wire ( density = 8.9 x 10 3 kg/m 3 ) could just float horizontally in air because of the force due to the Earth s magnetic field B, which is horizontal, perpendicular to the wire, and of magnitude 5.0x 10-5 T. (a) What current would the wire carry? (b) Does the answer seem feasible? Explain briefly. 8 A 13.0 g wire of length L = 62.0 cm is suspended by a pair of flexible leads in a uniform magnetic field of magnitude 0.440 T (see figure). (a) What is the magnitude of the current required to remove the tension in the supporting leads? (b) What is the direction (left or right) of the current required to remove the tension in the supporting leads? 1400 A 5 Calculate the force on an airplane which has acquired a net charge of 1550 μc and moves with a speed of 120 m/s perpendicular to the Earth s magnetic field of 5.0 x 10-5 T. (a) 467 ma; (b) right 6 9.3 x 10-6 N A wire carries a steady current of 2.40 A. A straight section of the wire is 0.750 m long and lies along the x axis within a uniform magnetic field, B = 1.60 T. If the current is in the +x direction, what is the magnetic force on the section of wire? 9 In the figure, charged particles move in the vicinity of a current carrying wire. For each charged particle, the arrow indicates the direction of motion of the particle, and the + or indicates the sign of the charge. For each of the particles, indicate the direction of the magnetic force due to the magnetic field produced by the wire. (-2.88 j) N
10 the following questions: (a) What is the magnitude of the force per meter of length on a straight wire carrying an 8.40 A current when perpendicular to a 0.90 T uniform magnetic field? (b) What if the angle between the wire and field is 45.0? 15 The power cable for an electric trolley shown in the figure carries a horizontal current of 330 A toward the east. The Earth s magnetic field has a strength 5.0 x 10-5 T and makes an angle of dip of 22 at this location. Calculate the magnitude and direction of the magnetic force on a 15 m length of this cable. (a) 7.6 N/m (b) 5.3 N/m 11 Calculate the magnitude of the magnetic force on a 160 m length of straight wire stretched between two towers carrying a 150 A current. The Earth s magnetic field of 5 x 10-5 T makes an angle of 65 with the wire. 1.1 N 12 A 1.5 m length of wire carrying 4.5 A of current is oriented horizontally. At that point on the Earth s surface, the dip angle of the Earth s magnetic field makes an angle of 38 to the wire. Estimate the magnitude of the magnetic force on the wire due to the Earth s magnetic field of 5.5 x 10-5 T at this point. 16 0.25 N northerly and 68 0 above the horizontal A wire having a mass per unit length of 0.500 g/cm carries a 2.00 A current horizontally to the south. What are the direction and magnitude of the minimum magnetic field needed to lift this wire vertically upward? 2.3 x 10-4 N 13 14 The magnetic force per meter on a wire is measured to be only 35% of its maximum possible value. Sketch the relationship of the wire and the field if the force had been a maximum, and sketch the relationship as it actually is, calculating the angle between the wire and the magnetic field. 20 0 The force on a wire is a maximum of 6.50 x 10-2 N when placed between the pole faces of a magnet. The current flows horizontally to the right and the magnetic field is vertical. The wire is observed to jump toward the observer when the current is turned on. (a) What type of magnetic pole is the top pole face? (b) If the pole faces have a diameter of 10.0 cm, estimate the current in the wire if the field is 0.16 T. (c) If the wire is tipped so that it makes an angle of 10.0 with the horizontal, what force will it now feel? (a) South Pole (b) 4.063 A (c) 6.40 x 10-2 N 17 18 B = 0.245 Tesla eastward A wire 2.80 m in length carries a current of 5.00 A in a region where a uniform magnetic field has a magnitude of 0.390 T. (a) Calculate the magnitude of the magnetic force on the wire assuming the angle between the magnetic field and the current is 60.0. (b) Calculate the magnitude of the magnetic force on the wire assuming the angle between the magnetic field and the current is 90.0. (c) Calculate the magnitude of the magnetic force on the wire assuming the angle between the magnetic field and the current is 120. (a) 4.73 N (b) 5.46 N (c) 4.73 N Assume that in Atlanta, Georgia, the Earth s magnetic field is 52.0 µt northward at 60.0 below the horizontal. A tube in a neon sign carries current 35.0 ma, between two diagonally opposite corners of a shop window, which lies in a north south vertical plane. The current enters the tube at the bottom south corner of the window. It exits at the opposite corner, which is 1.40 m farther north and 0.850 m higher up. Between these two points, the glowing tube spells out DONUTS. Use the theorem proved as Case 1 in the text to determine the total vector magnetic force on the tube. 2.98 µn west
19 A wire 1.80 m long carries a current of 13.0 A and makes an angle of 35.0 0 with a uniform magnetic field of magnitude B = 1.50 T. Calculate the magnetic force on the wire. 24 A conductor suspended by two flexible wires as shown in the figure has a mass per unit length of 0.040 0 kg/m. (a) What current must exist in the conductor in order for the tension in the supporting wires to be zero when the magnetic field is 3.60 T into the page? (b) What is the required direction for the current? 20.1 N 20 A wire 50.0 cm long carries a 0.500 A current in the positive direction of an x axis through a magnetic field B = (3.00 mt)j + (10.0 mt)k. In unit-vector notation, what is the magnetic force on the wire? (-2.50 x 10-3 N) j + (0.750 x 10-3 N) k 21 A long, rigid conductor, lying along an x axis, carries a current of 5.0 A in the negative x direction. A magnetic field B is present, given by B = 3.0i + 8.0x 2 j, with x in meters and B in milliteslas. Find, in unit-vector notation, the force on the 2.0 m segment of the conductor that lies between x = 1.0 m and x = 3.0 m. (-0.35 N) k 25 0.109 A to the right A metal rod having a mass per unit length λ carries a current I. The rod hangs from two vertical wires in a uniform vertical magnetic field as shown in the figure. The wires make an angle θ with the vertical when in equilibrium. Determine the magnitude of the magnetic field. 22 A wire lying along a y axis from y = 0 to y = 0.250 m carries a current of 2.00 ma in the negative direction of the axis. The wire fully lies in a nonuniform magnetic field given by B = (0.300 T/m)yi + (0.400 T/m)yj. In unit-vector notation, what is the magnetic force on the wire? (18.8 µn) k 23 A wire lying along an x axis from x = 0 to x = 1.00 m carries a current of 3.00 A in the positive x direction. The wire is immersed in a nonuniform magnetic field given by B = (4.00 T/m 2 )x 2 i - (0.600 T/m 2 )x 2 j. In unit-vector notation what is the magnetic force on the wire? (-0.600 N) k 26 A vertical straight wire carrying an upward 24 A current exerts an attractive force per unit length of 8.8 x 10 4 N/m on a second parallel wire 7.0 cm away. What current (magnitude and direction) flows in the second wire? 13 A upward 27 Determine the magnitude and direction of the force between two parallel wires 35 m long and 6.0 cm apart, each carrying 25 A in the same direction. 7.3 x 10-2 N Attractive
28 A long horizontal wire carries a current of 48 A. A second wire, made of 2.5 mm diameter copper wire and parallel to the first, is kept in suspension magnetically 15 cm below as shown in the figure. (a) Determine the magnitude and direction of the current in the lower wire. (b) Is the lower wire in stable equilibrium? (c) Repeat parts (a) and (b) if the second wire is suspended 15 cm above the first due to the latter s field. 31 32 Explain why two parallel wires carrying currents in opposite directions repel each other. Parallel current-carrying wires exert magnetic forces on each other. (a) What about perpendicular wires? (b) Imagine two such wires oriented perpendicular to each other, and almost touching. Does a magnetic force exist between the wires? (a) 6700 A, Right (b) not in stable equilibrium (c) 6700 A, Left (d) is in stable equilbrium 33 One very long wire carries current 30.0 A to the left along the x axis. A second very 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 µc is moving with a velocity of 150 î Mm/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 undeflected. Calculate the required vector electric field. 29 Two long straight parallel wires are 15 cm apart. Wire A carries 2.0 A current. Wire B s current is 4.0 A in the same direction. (a) Determine the magnetic field magnitude due to wire A at the position of wire B. (b) Determine the magnetic field due to wire B at the position of wire A. (c) Are these two magnetic fields equal and opposite? Why or why not? (d) Determine the force on wire A due to wire B, and the force on wire B due to wire A. Are these two forces equal and opposite? Why or why not? (a) 2.66 x 10-6 T (b) 5.333 x 10-6 T (c) the two fields are not equal and opposite (d) 1.067 x 10-5 N/m 34 (a) y = -0.420 m (b) 3.47 x 10-2 N (-j) (c) -1.73 x 10 4 j N/C Two long, parallel conductors, separated by 10.0 cm, carry currents in the same direction. The first wire carries current I 1 = 5.00 A and the second carries I 2 = 8.00 A. (a) What is the magnitude of the magnetic field created by I 1 at the location of I 2? (b) What is the force per unit length exerted by I 1 on I 2? (c) What is the magnitude of the magnetic field created by I 2 at the location of I 1? (d) What is the force per length exerted by I 2 on I 1? 30 Two long straight aluminum wires, each of diameter 0.50 mm, carry the same current but in opposite directions. They are suspended by 0.50 m long strings as shown in the figure. If the suspension strings make an angle of 3.0º with the vertical, what is the current in the wires? (a) B = 1.00 x10-5 T out of the page (b) F B = 8.00 x 10-5 N toward the first wire (c) B = 1.60 x 10-5 T into the page (d) F B 8.00 x 10-5 N toward the second Wire 8.4 A
35 Two long, parallel conductors carry currents in the same direction as shown in the figure. Conductor A carries a current of 150 A and is held firmly in position. Conductor B carries a current I B and is allowed to slide freely up and down (parallel to A) between a set of nonconducting guides. If the mass per unit length of conductor B is 0.100 g/cm, what value of current I B will result in equilibrium when the distance between the two conductors is 2.50 cm? 38 A third wire is placed in the plane of the two wires shown in the figure, parallel and just to the right. If it carries 25.0 A upward, what force per meter of length does it exert on each of the other two wires? Assume it is 2.8 mm from the nearest wire, center to center. 4.5 x 10-2 N/m, attract 2.2 x 10-2 N/m, repel 36 81.7 A The figure shows wire 1 in cross section; the wire is long and straight, carries a current of 4.00 ma out of the page, and is at distance d 1 = 2.40 cm from a surface. Wire 2, which is parallel to wire 1 and also long, is at horizontal distance d 2 = 5.00 cm from wire I and carries a current of 6.80 ma into the page. What is the x component of the magnetic force per unit length on wire 2 due to wire 1? 39 Three long parallel wires are 3.8 cm from one another. (Looking along them, they are at three corners of an equilateral triangle.) The current in each wire is 8.00 A, but its direction in wire M is opposite to that in wires N and P as shown in the figure. Determine the magnetic force per unit length on each wire due to the other two. 88.4 pn/m 37 A singly charged ion completes five revolutions in a uniform magnetic field of magnitude 5.00 10 2 T in 1.50 ms. Calculate the mass of the ion in kilograms. 5.8 x10-4 N/m, 90 0 3.4 x 10-4 N/m, 300 0 3.4 x 10-4 N/m, 240 0 3/82 x 10-25 kg 40 In the figure the top wire is 1.00 mm diameter copper wire and is suspended in air due to the two magnetic forces from the bottom two wires. The current flow through the two bottom wires is 95 A in each. Calculate the required current flow in the suspended wire. 79 A
41 Three long wires (wire 1, wire 2, and wire 3) hang vertically. The distance between wire 1 and wire 2 is 20.0 cm. On the left, wire 1 carries an upward current of 1.50 A. To the right, wire 2 carries a downward current of 4.00 A. Wire 3 is located such that when it carries a certain current, each wire experiences no net force. (a) Find the position of wire 3. (b) Find the magnitude and direction of the current in wire 3. 45 The figure shows three arrangements of three long straight wires carrying equal currents directly into or out of the page. (a) Rank the arrangements according to the magnitude of the net force on wire A due to the currents in the other wires, greatest first. (b) In arrangement 3, is the angle between the net force on wire A and the dashed line equal to, less than, or more than 45 0? (a) 12.0 cm to the left of wire l (b) 2.40 A down 42 The figure here shows three long, straight, parallel, equally spaced wires with identical currents either into or out of the page. Rank the wires according to the magnitude of the force on each due to the currents in the other two wires, greatest first. b, c, a (a) 1, 3, 2; (b) less 43 Figure a shows, in cross section, three current carrying wires that are long, straight, and parallel to one another. Wires 1 and 2 are fixed in place on an x axis, with separation d. Wire 1 has a current of 0.750 A, but the direction of the current is not given. Wire 3, with a current of 0.250 A out of the page, can be moved along the x axis to the right of wire 2. As wire 3 is moved, the magnitude of the net magnetic force P, on wire 2 due to the currents in wires 1 and 3 changes. The y component of that force is F 2y and the value per unit length of wire 2 is F 2Y /L 2. Figure b gives F 2y /L 2 versus the position x of wire 3. The plot hits an asymptote F 2y /L 2 = -0.627 µn/m as x --> infinity. (a) What is the size of the current in wire 2? (b) What is the direction (into or out of the page) of the current in wire 2? 46 In the figure, the cube is 40.0 cm on each edge. Four straight segments of wire ab, bc, cd, and da form 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 magnitude and direction of the magnetic force on each segment. ab = 0 bc = (40.0 mn) (-i) cd = (40.0 mn) (-k) da = (40.0 mn) (k + i) 44 (a) 0.50 A; (b) out Three long wires all lie in an xy plane parallel to the x axis. They are spaced equally, 10 cm apart. The two outer wires each carry a current of 5.0 A in the positive x direction. (a) What is the magnitude of the force on a 3.0 m section of either of the outer wires if the current in the center wire is 3.2 A in the positive x direction? (b) What is the magnitude of the force on a 3.0 m section of either of the outer wires if the current in the center wire is 3.2 A in the negative x direction? 47 In the figure, five long parallel wires in an xy plane are separated by distance d = 50.0 cm. The currents into the page are i l = 2.00 A, i 3 = 0.250 A, i 4 = 4.00 A, and i 5 = 2.00 A; the current out of the page is i 2 = 4.00 A. What is the magnitude of the net force per unit length acting on wire 3 due to the currents in the other wires? 800 nn/m (a) 0.17 mn; (b) 0.021 mn
48 In the figure, five long parallel wires in an xy plane are separated by distance d = 8.00 cm, have lengths of 10.0 m and carry identical currents of 3.00 A out of the page. Each wire experiences a magnetic force due to the other wires. (a) In unit-vector notation, what is the net magnetic force on wire 1? (b) In unit-vector notation, what is the net magnetic force on wire 2? (c) In unit-vector notation, what is the net magnetic force on wire 3? (d) In unit-vector notation, what is the net magnetic force on wire 4? (e) In unit-vector notation, what is the net magnetic force on wire 5? 51 Two stiff parallel wires a distance l apart in a horizontal plane act as rails to support a light metal rod of mass m (perpendicular to each rail) as shown in the figure. A magnetic field B, directed vertically upward (outward in the diagram), acts throughout. At t = 0, wires connected to the rails are connected to a constant current source and a current I begins to flow through the system. (a) Determine the speed of the rod, which starts from rest at t = 0, as a function of time assuming no friction between the rod and the rails. (b) Determine the speed of the rod, which starts from rest at t = 0, as a function of time if the coefficient of friction is μ k. (c) Does the rod move east or west if the current through it heads north? (a) (469 µn) j; (b) (188 µn) j; (c) 0; (d) (-188 µn) j; (e) (-469 µn) j (a) (ILB)/(m) t (b) ((ILB)/(m)- µ k g)t (c) east 49 In the figure, four long straight wires are perpendicular to the page, and their cross sections form a square of edge length a = 8.50 cm. Each wire carries 15.0 A, and all the currents are out of the page. In unit-vector notation, what is the net magnetic force per meter of wire length on wire 1? 52 A rod of mass 0.720 kg and radius 6.00 cm rests on two parallel rails shown in the figure that are d = 12.0 cm apart and L = 45.0 cm long. The rod carries a current of I = 48.0 A (in the direction shown) and rolls along the rails without slipping. A uniform magnetic field of magnitude 0.240 T 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? (0.794 mn/m) i - (0.794 mn/m) j 50 The figure shows four arrangements in which long, parallel, equally spaced wires carry equal currents directly into or out of the page. Rank the arrangements according to the magnitude of the net force on the central wire due to the currents in the other wires, greatest first. 1.07 m/s 53 A 0.200 kg metal rod carrying a current of 10.0 A glides on two horizontal rails 0.500 m apart. What vertical magnetic field is required to keep the rod moving at a constant speed if the coefficient of kinetic friction between the rod and rails is 0.100? 39.2 mt b, d, c, a (zero)
54 In the figure, a metal wire of mass m = 24.1 mg can slide with negligible friction on two horizontal parallel rails separated by distance d = 2.56 cm. The track lies in a vertical uniform magnetic field of magnitude 56.3 mt. At time t = 0, device G is connected to the rails, producing a constant current i = 9.13 ma in the wire and rails (even as the wire moves). (a) At t = 61.1 ms, what is the wire's speed? (b) At t = 61.1 ms, what is the wire's direction of motion (left or right)? (a) 3.34 cm/s; (b) left 56 Rail guns have been suggested for launching projectiles into space without chemical rockets, and for ground-to-air antimissile weapons of war. A tabletop model rail gun shown in the figure, consists of two long parallel horizontal rails 3.50 cm apart, bridged by a bar BD of mass 3.00 g. The bar is originally at rest at the midpoint of the rails and is free to slide without friction. When the switch is closed, electric current is quickly established in the circuit ABCDEA. The rails and bar have low electric resistance, and the current is limited to a constant 24.0 A by the power supply. (a) Find the magnitude of the magnetic field 1.75 cm from a single very long straight wire carrying current 24.0 A. (b) Find the magnitude and direction of the magnetic field at point C in the diagram, the midpoint of the bar, immediately after the switch is closed. Suggestion: Consider what conclusions you can draw from the Biot Savart law. (c) At other points along the bar BD, the field is in the same direction as at point C, but larger in magnitude. Assume that the average effective magnetic field along BD is five times larger than the field at C. With this assumption, find the magnitude and direction of the force on the bar. (d) Find the acceleration of the bar when it is in motion. (e) Does the bar move with constant acceleration? (f) Find the velocity of the bar after it has traveled 130 cm to the end of the rails. 55 The figure is an idealized schematic drawing of a rail gun. Projectile P sits between two wide rails of circular cross section; a source of current sends current through the rails and through the (conducting) projectile (a fuse is not used). (a) Let w be the distancebetween the rails, R the radius of each rail, and i the current. Show that the force on the projectile is directed to the right along the rails and is given approximately by the equation above the figure. (b) If the projectile starts from the left end of the rails at rest, find the speed v at which it is expelled at the right. Assume that i = 450 ka, w = 12 mm, R = 6.7 cm, L = 4.0 m, and the projectile mass is 10 g. (a) 2.74 x 10-4 T (b) 2.74 x 10-4 T (-j) (c) (1.15 x 10-3 N) i (d) (0.384 m/s 2 ) i (e) acceleration is constant (f) (0.999 m/s) i 57 A thin copper bar of length l = 10.0 cm is supported horizontally by two (nonmagnetic) contacts. The bar carries current I 1 = 100 A in the x direction, as shown in the figure. At a distance h = 0.500 cm below one end of the bar, a long straight wire carries a current I 2 = 200 A in the z direction. Determine the magnetic force exerted on the bar. (b) 2.3 km/s 1.20 x 10-2 N (-k) 58 A small bar magnet is suspended in a uniform 0.250 T magnetic field. The maximum torque experienced by the bar magnet is 4.60 10 3 N m. Calculate the magnetic moment of the bar magnet. 18.4 ma.m 2
59 The magnetic moment of the Earth is approximately 8.00 10 22 A m 2. (a) If this were caused by the complete magnetization of a huge iron deposit, how many unpaired electrons would this correspond to? (b) At two unpaired electrons per iron atom, how many kilograms of iron would this correspond to? (Iron has a density of 7900 kg/m 3, and approximately 8.50 10 28 iron atoms/m 3. 62 A nonuniform magnetic field exerts a net force on a magnetic dipole. A strong magnet is placed under a horizontal conducting ring of radius r that carries current I as shown in the figure. If the magnetic field B makes an angle θ with the vertical at the ring s location, what are the magnitude and direction of the resultant force on the ring? (a) 8.63 x 10 45 (b) 4.01 x 10 20 kg 60 A magnetic dipole with a dipole moment of magnitude 0.020 J/T is released from rest in a uniform magnetic field of magnitude 52 mt. The rotation of the dipole due to the magnetic force on it is unimpeded. When the dipole rotates through the orientation where its dipole moment is aligned with the magnetic field, its kinetic energy is 0.80 mj. (a) What is the initial angle between the dipole moment and the magnetic field? (b) What is the angle when the dipole is next (momentarily) at rest? I2πrBsinθ up (a) 77 0 ; (b) 77 0 63 A current loop with magnetic dipole moment µ is placed in a uniform magnetic field B, with its moment making angle θ with the field. With the arbitrary choice of U = 0 for θ = 90, prove that the potential energy of the dipole field system is U = µ B. 61 The figure gives the potential energy U of a magnetic dipole in an external magnetic field B, as a function of angle φ between the directions of B and the,dipole moment. The dipole can be rotated about an axle with negligible friction so as to change φ. Counterclockwise rotation from φ = 0 yields positive values of and clockwise rotations yield negative values.the dipole is to be released at angle φ = 0 with a rotational kinetic energy of 6.7 x 10-4 J, so that it rotates counterclock-wise. To what maximum value of φ will it rotate? (In the language of energy, what value φ is the turning point in the potential well of the figure?) 64 U = -µb In Bohr s 1913 model of the hydrogen atom, the electron is in a circular orbit of radius 5.29 10 11 m and its speed is 2.19 10 6 m/s. (a) What is the magnitude of the magnetic moment due to the electron s motion? (b) If the electron moves in a horizontal circle, counterclockwise as seen from above, what is the direction of this magnetic moment vector? (a) 9.27 x 10-24 A.m 2 (b) downward 65 A circular coil of 160 turns has a radius of 1.90 cm. (a) Calculate the current that results in a magnetic dipole moment of magnitude 2.30 A.m 2. (b) Find the maximum magnitude of the torque that the coil, carrying this current, can experience in a uniform 35.0 mt magnetic field. 110 0 (a) 12.7 A; (b) 0.0805N m
66 A circular wire loop of radius 15.0 cm carries a current of 2.60 A. It is placed so that the normal to its plane makes an angle of 41.0 0 with a uniform magnetic field of magnitude 12.0 T. (a) Calculate the magnitude of the magnetic dipole moment of the loop. (b) What is the magnitude of the torque acting on the loop? 71 A uniform magnetic field of magnitude 0.150 T is directed along the positive x axis. A positron moving at 5.00 10 6 m/s enters the field along a direction that makes an angle of 85.0 with the x axis in the figure. The motion of the particle is expected to be a helix. (a) Calculate the pitch p of the trajectory (b) Calculate the radius r of the trajectory. (a) 0.184 A.m 2 ; (b) l1.45 N.m 67 A current loop, carrying a current of 5.0 A, is in the shape of a right triangle with sides 30, 40, and 50 cm. The loop is in a uniform magnetic field of magnitude 80 mt whose direction is parallel to the current in the 50 cm side of the loop. (a) Find the magnitude of the magnetic dipole moment of loop. (b) Find the magnitude of the torque on the loop. (a) 0.30 J/T; (b) 0.024 N.m (a) 1.04 x 10-4 m (b) 1.89 x 10-4 m 68 In figure a, two concentric coils, lying in the same plane, carry currents in opposite directions. The current in the larger coil 1 is fixed. Current i 2 in coil 2 can be varied. Figure b gives the net magnetic moment of the two-coil system as a function of i 2. If the current in coil 2 is then reversed, what is the magnitude of the net magnetic moment of the two coil system when i 2 = 7.0 ma? 72 The figure shows a wire ring of radius a = 18 cm that is perpendicular to the general direction of a radially symmetric, diverging magnetic field. The magnetic field at the ring is everywhere of the same magnitude B = 3.4 mt, and its direction at the ring everywhere makes an angle θ = 20 0 with a normal to the plane of the ring. The twisted lead wires have no effect on the problem. Find the magnitude of the force the field exerts on the ring if the ring carries a current i = 4.6 ma. 0.60 µn 4.8 x 10-5 A.m 2 69 A 200-turn solenoid having a length of 25 cm and a diameter of 10 cm carries a current of 0.29 A. What is the magnitude of the magnetic dipole moment µ of the solenoid? 73 In the figure, a long straight wire carries a current i l = 30.0 A and a rectangular loop carries current i 2 = 20.0 A. Take a = 1.00 cm, b = 8.00 cm, and L = 30.0 cm. In unit-vector notation, what is the net force on the loop due to i l? 0.47A.m 2 70 A circular coil 16.0 cm in diameter and containing nine loops lies flat on the ground. The Earth s magnetic field at this location has magnitude 5.50 x 10 5 T and points into the Earth at an angle of 56.0 below a line pointing due north. (a) If a 7.20 A clockwise current passes through the coil determine the torque on the coil (b) If a 7.20 A clockwise current passes through the coil which edge of the coil rises up: north, east, south, or west? (3.20 mn) j (a) 4.01 x 10-5 N.m (b)
74 A rectangular loop of wire lies in the same plane as a straight wire, as shown in the figure. There is a current of 2.5 A in both wires. Determine the magnitude and direction of the net force on the loop. 76 Two circular loops are parallel, coaxial, and almost in contact, 1.00 mm apart as shown in the figure. Each loop is 10.0 cm in radius. The top loop carries a clockwise current of 140 A. The bottom loop carries a counterclockwise current of 140 A. (a) Calculate the magnetic force exerted by the bottom loop on the top loop. (b) The upper loop has a mass of 0.021 0 kg. Calculate its acceleration, assuming that the only forces acting on it are the force in part (a) and the gravitational force. Suggestion: Think about how one loop looks to a bug perched on the other loop. 2.6 x 10-6 N, towards twire 75 In the figure, the current in the long, straight wire is I 1 = 5.00 A and the wire lies in the plane of the rectangular loop, which carries the current I 2 = 10.0 A. The dimensions are c = 0.100 m, a = 0.150 m, and l = 0.450 m. Find the magnitude and direction of the net force exerted on the loop by the magnetic field created by the wire. 77 (a) 2.46 N (b) 107 m/s 2 An infinitely long straight wire carrying a current I 1 is partially surrounded by a loop as shown in the figure. The loop has a length L, radius R, and carries a current I 2. The axis of the loop coincides with the wire. Calculate the force exerted on the loop. F = 2.70 x 10-5 N toward the left 78 A single square loop of wire 22.0 cm on a side is placed with its face parallel to the magnetic field between the pole pieces of a large magnet. When 6.30 A flows in the coil, the torque on it is 0.325 m N. What is the magnetic field strength? 1.07 T 79 A galvanometer needle deflects full scale for a 53.0 μa current. What current will give full scale deflection if the magnetic field weakens to 0.860 of its original value? 61.6 µa
80 If the restoring spring of a galvanometer weakens by 25% over the years, what current will give full scale deflection if it originally required 36 μa? 27 µa 86 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 θ = 30.0 with the x axis shown in the figure. (a) What is the magnitude of the torque exerted on the coil by a uniform magnetic field B = 0.800 T directed along the x axis when the current is I = 1.20 A in the direction shown? (b) What is the expected direction of rotation of the coil? 81 If the current to a motor drops by 12%, by what factor does the output torque change? 0.88 times the initial torque 82 Is it possible to orient a current loop in a uniform magnetic field such that the loop does not tend to rotate? Explain. 83 How can a current loop be used to determine the presence of a magnetic field in a given region of space? (a) 9.98 N.m (b) clockwise 84 A current of 17.0 ma is maintained in a single circular loop of 2.00 m circumference. A magnetic field of 0.800 T is directed parallel to the plane of the loop. (a) Calculate the magnetic moment of the loop. (b) What is the magnitude of the torque exerted by the magnetic field on the loop? (a) 5.41 ma.m 2 (b) 4.33 mn.m 87 A 40.0 cm length of wire carries a current of 20.0 A. It is bent into a loop and placed with its normal perpendicular to a magnetic field with a magnitude of 0.520 T. (a) What is the torque on the loop if it is bent into an equilateral triangle? (b) What is the torque if the loop is a square? (c) What is the torque if the loop is a circle? (d) Which torque is greatest? (a) 80.1 mn.m (b) 0.104 N.m (c) 0.132 N.m (d) the cvircular loop experiences the largest torque. 85 A long piece of wire with a mass of 0.100 kg and a total length of 4.00 m is used to make a square coil with a side of 0.100 m. The coil is hinged along a horizontal side, carries a 3.40 A current, and is placed in a vertical magnetic field with a magnitude of 0.0100 T. (a) Determine the angle that the plane of the coil makes with the vertical when the coil is in equilibrium. (b) Find the torque acting on the coil due to the magnetic force at equilibrium. 88 A wire is formed into a circle having a diameter of 10.0 cm and placed in a uniform magnetic field of 3.00 mt. The wire carries a current of 5.00 A. (a) Find the maximum torque on the wire (b) Find the range of potential energies of the wire field system for different orientations of the circle. (a) 118 µn.m (b) -118 µj U +118 µj (a) 3.97 0 (b) 3.39 mn.m
89 The rotor in a certain electric motor is a flat rectangular coil with 80 turns of wire and dimensions 2.50 cm by 4.00 cm. The rotor rotates in a uniform magnetic field of 0.800 T. When the plane of the rotor is perpendicular to the direction of the magnetic field, it carries a current of 10.0 ma. In this orientation, the magnetic moment of the rotor is directed opposite the magnetic field. The rotor then turns through onehalf revolution. This process is repeated to cause the rotor to turn steadily at 3600 rev/min. (a) Find the maximum torque acting on the rotor. (b) Find the peak power output of the motor. (c) Determine the amount of work performed by the magnetic field on the rotor in every full revolution. (d) What is the average power of the motor? 92 The figure shows a rectangular 20-turn coil of wire, of dimensions 10 cm by 5.0 cm. It carries a current of 0.10 A and is hinged along one long side. It is mounted in the xy plane, at angle θ = 30 0 to the direction of a uniform magnetic field of magnitude 0.50 T. In unit-vector notation, what is the torque acting on the coil about the hinge line? (a) 6.40 x 10-4 N.m (b) 0.241 W (c) 2.56 x 10-3 J (d) 0.154 W 90 A hand-held electric mixer contains an electric motor. Model the motor as a single flat compact circular coil carrying electric current in a region where a magnetic field is produced by an external permanent magnet. You need consider only one instant in the operation of the motor. The coil moves because the magnetic field exerts torque on the coil. Make order-ofmagnitude estimates of the magnetic field, the torque on the coil, the current in it, its area, and the number of turns in the coil, so that they are related according to the equation: τ = µ x B Note that the input power to the motor is electric, given by P = I V, and the useful output power is mechanical, P = τ ω. 93 (-4.3 x 10-3 N.m) j A single-turn current loop, carrying a current of 4.00 A, is in the shape of a right triangle with sides 50.0, 120, and 130 cm. The loop is in a uniform magnetic field of magnitude 75.0 mt whose direction is parallel to the current in the 130 cm side of the loop. (a) What is the magnitude of the magnetic force on the 130 cm side? (b) What is the magnitude of the magnetic force on the 50.0 cm side? (c) What is the magnitude of the magnetic force on the 120 cm side? (d) What is the magnitude of the net force on the loop? Power is about I = 1 amp τ =10-1 N.m Area = 10 3=-3 m 2 B = 10-1 N = 10 3 (a) 0; (b) 0.138 N; (c) 0.138 N; (d) 0 91 Consider an electron orbiting a proton and maintained in a fixed circular path of radius R = 5.29 10 11 m by the Coulomb force. Treating the orbiting charge as a current loop, calculate the resulting torque when the system is in a magnetic field of 0.400 T directed perpendicular to the magnetic moment of the electron. 3.70 x 10-24 N.m
94 The coil of a certain galvanometer has a resistance of 75.3 Ω; its needle shows a full scale deflection when a current of 1.62 ma passes through the coil. (a) Determine the value of the auxiliary resistance required to convert the galvanometer to a voltmeter that reads 1.00 V at full-scale deflection. (b) Should this resistance hconnected in series or in parallel with the galvanometer? (c) Determine the value of the auxiliary resistance required Lo convert the galvanometer to an ammeter that reads 50.0 ma at full-scale deflection. (d) Should this resistance be connected in series or in parallel? 96 A circular loop of wire having a radius of 8.0 cm carries a current of 0.20 A. A vector of unit length and parallel to the dipole moment μ of the loop is given by 0.60i - 0.80j. The loop is located in a uniform magnetic field given by B = (0.25 T)i + (0.30 T)k. (a) Find the torque on the loop (in unit vector notation). (b) Find the magnetic potential energy of the loop. (a) (-9.7 x 10-4 N.m) i - (7.2 x 10-4 N.m) j + (8.0 x 10-4 N.m) k; (b) -6.0 x 10-4 J 97 A wire of length 25.0 cm. carrying a current of 4.51 ma is to be formed into a circular coil and placed in a uniform magnitude field B of magnitude 5.71 mt. (a) If the torque on the coil from the field is maximized, what is the angle between B and the coil's magnetic dipole moment? (b) If the torque on the coil from the field is maximized, what is the number of turns in the coil? (c) What is the magnitude of that maximum torque? 95 (a) 542 Ω; (b) series; (c) 2.52 Ω; (d) parallel The figure shows a wood cylinder of mass m = 0.250 kg and length L = 0.100 m, with N = 10.0 turns of wire wrapped around it longitudinally, so that the plane of the wire coil contains the long central axis of the cylinder. The cylinder is released on a plane inclined at an angle θ to the horizontal, with the plane of the coil parallel to the incline plane. If there is a vertical uniform magnetic field of magnitude 0.500 T, what is the least current i through the coil that keeps the cylinder from rolling down the plane? 98 (a) 90 0 ; (b) 1; (c) 1.28 x 10-7 N.m The coil in the figure carries current i = 2.00 A in the direction indicated, is parallel to an xz plane, has 3.00 turns and an area of 4.00 x 10-3 m 2, and lies in a uniform magnetic field B = (2.00i - 3.00j - 4.00k) mt. (a) What is the magnetic potential energy of the coil-magnetic field system? (b) What is the magnetic torque (in unit-vector notation) on the coil? (a) -72.0 µj; (b) (96.0 µn.m) i + (48.0 µn.m) ˆk 2.45 A 99 A circular loop of radius 12 cm carries a current of 15 A. A flat coil of radius 0.82 cm, having 50 turns and a current of 1.3 A, is concentric with the loop. The plane of the loop is perpendicular to the plane of the coil. Assume the loop's magnetic field is uniform across the coil. (a) What is the magnitude of the magnetic field produced by the loop at its center? (b) What is the magnitude of the torque on the coil due to the loop? (a) 79 µt; (b) 1.1 x 10-6 N.m
100 The figure shows a wood cylinder of mass m = 0.250 kg and length L = 0.100 m, with N = 10.0 turns of wire wrapped around it longitudinally, so that the plane of the wire coil contains the long central axis of the cylinder. The cylinder is released on a plane inclined at an angle θ to the horizontal, with the plane of the coil parallel to the incline plane. If there is a vertical uniform magnetic field of magnitude 0.500 T, what is the least current i through the coil that keeps the cylinder from rolling down the plane? 103 A flat ribbon of silver having a thickness t = 0.200 mm is used in a Hall effect measurement of a uniform magnetic field perpendicular to the ribbon, as shown in the figure. The Hall coefficient for silver is RH = 0.840 10 10 m 3 /C. (a) What is the density of charge carriers in silver? (b) If a current I = 20.0 A produces a Hall voltage V H = 15.0 µv, what is the magnitude of the applied magnetic field? 101 2.45 A The needle of a magnetic compass has magnetic moment 9.70 ma m 2. At its location, the Earth s magnetic field is 55.0 µt north at 48.0 below the horizontal. (a) Identify the orientations of the compass needle that represent minimum potential energy and maximum potential energy of the needle field system. (b) How much work must be done on the needle to move it from the former to the latter orientation? 104 (a) 7.44 x 10 28 m -3 (b) 1.79 T A flat copper ribbon 0.330 mm thick carries a steady current of 50.0 A and is located in a uniform 1.30 T magnetic field directed perpendicular to the plane of the ribbon. If a Hall voltage of 9.60 µv is measured across the ribbon, what is the charge density of the free electrons? What effective number of free electrons per atom does this result indicate? 1.28 x 10 29 m -3 1.52 conduction electrons per atom 102 (a) point north at 48.0 0 below the horizontal south at 48.0 0 above the horizontal (b) 1.07 µj A non-conducting sphere has mass 80.0 g and radius 20.0 cm. A flat compact coil of wire with 5 turns is wrapped tightly around it, with each turn concentric with the sphere. As shown in the figure, the sphere is placed on an inclined plane that slopes downward to the left, making an angle θ with the horizontal, so that the coil is parallel to the inclined plane. A uniform magnetic field of 0.350 T vertically upward exists in the region of the sphere. What current in the coil will enable the sphere to rest in equilibrium on the inclined plane? Show that the result does not depend on the value of θ. 105 106 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 µv. (a) When it is measuring an unknown magnetic field, the Hall voltage is 0.330 µv. What is the magnitude of the unknown field? (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. (a) 37.7 mt (b).29 x 10 25 m -3 In an experiment that is designed to measure the Earth s magnetic field using the Hall effect, a copper bar 0.500 cm thick is positioned along an east west direction. If a current of 8.00 A in the conductor results in a Hall voltage of 5.10 10 12 V, what is the magnitude of the Earth s magnetic field? (Assume that n = 8.49 10 28 electrons/m 3 and that the plane of the bar is rotated to be perpendicular to the direction of B.) 43.3 µt 0.713 A
107 A very long, thin strip of metal of width w carries a current I along its length as shown in the figure. Find the magnetic field at the point P in the diagram. The point P is in the plane of the strip at distance b away from it. 110 In a Hall-effect experiment, a current of 3.0 A sent lengthwise through a conductor 1.0 cm. wide, 4.0 cm long, and 10 μm thick produces a transverse (across the width) Hall potential difference of 10 μv when a magnetic field of 1.5 T is passed perpendicularly through the thickness of the conductor. (a) From these data, find the drift velocity of the charge carriers. (b) From these data, find the number density of charge carriers. (c) Show on a diagram the polarity of the Hall potential difference with assumed current and magnetic field directions, assuming also that the charge carriers are electrons. (a) 0.67 mm/s; (b) 2.8 x 10 29 m -3 108 A metal strip 6.50 cm long, 0.850 cm wide, and 0.760 mm thick moves with constant velocity V through a uniform magnetic field B = 1.20 mt directed perpendicular to the strip, as shown in the figure. A potential difference of 3.90 μv is measured between points x and y across the strip. Calculate the speed v. 111 The figure shows the Hall effect. (a) Show that the ratio of the Hall electric field magnitude E to the magnitude E C of the electric field responsible for moving charge (the current) along the length of the strip is E/E c = B/neρ where ρ is the resistivity of the material and n is the number density of the charge carriers. (b) A strip of copper 150 μm thick and 4.5 mm wide is placed in a uniform magnetic field B of magnitude 0.65 T, which is perpendicular to the strip. A current i = 23 A is then sent through the strip such that a Hall potential difference V appears across the width of the strip. (The number of charge carriers per unit volume for copper is 8.47 x 10 8 electrons/m 3.) Compute this ratio numerically from the data above. 38.2 cm/s 109 A strip of copper 150 μm thick and 4.5 mm wide is placed in a uniform magnetic field B of magnitude 0.65 T, perpendicular to the strip. A current i = 23 A is then sent through the strip such that a Hall potential difference V appears across the width of the strip. Calculate V. (The number of charge carriers per unit volume for copper is 8.47 x 10 28 electrons/m 3.) 112 (a) 2.84 x 10-3 What type of magnetic field is required to exert a resultant force on a magnetic dipole? What is the direction of the resultant force? 7.4 µv
113 Sodium melts at 99 C. Liquid sodium, an excellent thermal conductor, is used in some nuclear reactors to cool the reactor core. The liquid sodium is moved through pipes by pumps that exploit the force on a moving charge in a magnetic field. The principle is as follows. Assume the liquid metal to be in an electrically insulating pipe having a rectangular cross section of width w and height h. A uniform magnetic field perpendicular to the pipe affects a section of length L shown in the figure. An electric current directed perpendicular to the pipe and to the magnetic field produces a current density J in the liquid sodium. (a) Explain why this arrangement produces on the liquid a force that is directed along the length of the pipe. (b) Show that the section of liquid in the magnetic field experiences a pressure increase JLB. 116 Given the following background:what objects experience a force in an electric field? We found that any electric charge, stationary or moving, other than the charge that created the field. What creates an electric field? Any electric charge, stationary or moving. What objects experience a force in a magnetic field? An electric current or a moving electric charge, other than the current or charge that created the field,. What creates a magnetic field? An electric current or a moving electric charge. (a) To display how a moving charge creates a magnetic field, consider a charge q moving with velocity v. Define the vector r = r r to lead from the charge to some location. Show that the magnetic field at that location is given by the shown equation. (b) Find the magnitude of the magnetic field 1.00 mm to the side of a proton moving at 2.00 10 7 m/s. (c) Find the magnetic force on a second proton at this point, moving with the same speed in the opposite direction. (d) Find the electric force on the second proton. (a) electric current experiences a magnetic force (b) JLB (a) (b) 3.20 x 10-13 T (c) 1.02 x 10-24 N directed away from the first proton (d) 2.30 x 10-22 N directed away from the first proton 114 The table shows measurements of a Hall voltage and corresponding magnetic field for a probe used to measure magnetic fields. (a) Plot these data, and deduce a relationship between the two variables. (b) If the measurements were taken with a current of 0.200 A and the sample is made from a material having a charge-carrier density of 1.00 10 26 / m 3, what is the thickness of the sample? 117 A beam of electrons is directed toward a horizontal wire carrying a current from left to right (see figure). In what direction is the beam deflected? 118 A long straight wire lies on a horizontal table and carries a current of 1.20 µa. In a vacuum, a proton moves parallel to the wire (opposite the current) with a constant speed of 2.30 10 4 m/s at a distance d above the wire. Determine the value of d. You may ignore the magnetic field due to the Earth but not its gravitational force. (a) V H =(1.00 x 10-4 V/T)B (b) 0.125 mm 5.40 cm 115 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. What are the magnitude and direction of the smallest magnetic field that enables the wire to move in this fashion? B min = 0.128 T point north at an angle o 78.7 0 below the horizontal