Unit 12B: Ampere's Law. Not Assigned. Ampere's Law

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1 Unit 12B: Ampere's Law Ampere's Law 1 A A current is charging a capacitor that has circular plates 10.0 cm in radius. The plate separation is 4.00 mm. (a) What is the time rate of increase of electric field between the plates? (b) What is the magnetic field between the plates 5.00 cm from the center? (a) 7.19 x V/m.s (b) 2.00 x 10-7 T 4 Show that a uniform magnetic field B cannot drop abruptly to zero (as is suggested by the lack of field lines to the right of point a in the figure) as one moves perpendicular to B, say along the horizonta) arrow in the figure. (Hint: Apply Ampere's law to the rectangular path shown by the dashed lines.) In actual magnets, "fringing" of the magnetic field lines always occurs, which means that B approaches zero in a gradual manner. Modify the field lines in the figure to indicate a more realistic situation 2 The figure here shows three equal currents i (two parallel and one antiparallel) and four Amperian loops, Rank the loops according to the magnitude of B ds along each, greatest first. 3 d, then a and c tie, then b The figure shows two closed paths wrapped around two conducting loops carrying currents i l = 5.0 A and i 2 = 10 A. (a) What is the value of the integral B ds for path 1? (b) What is the value of the integral B ds for path 2? 5 The force on a particle in a magnetic field is the idea behind electromagnetic pumping. It is used to pump metallic fluids (such as sodium) and to pump blood in artificial heart machines. The basic design is shown in the figure. An electric field is applied perpendicular to a blood vessel and to a magnetic field. Explain how ions are caused to move. Do positive and negative ions feel a force in the same direction? (a) -2.5 µt.m; (b) -16 µt.m Answer 6 Answer the following questions. (a) Is Ampere's law valid for all closed paths surrounding a conductor? (b) Why is it not useful for calculating B for all such paths? 7 Compare Ampere's law with the Biot-Savart law. Which is more generally useful for calculating B for a current carrying conductor?

2 Unit 12B: Ampere's Law Ampere's Law Describe the similarities between Ampere's law in magnetism and Gauss's law in electrostatics. A hollow copper tube carries a current along its length. (a) Why is B = 0 inside the tube? (b) Is B nonzero outside the tube? The magnetic field 40.0 cm away from a long straight wire carrying current 2.00 A is 1.00 µt. (a) At what distance is it µt? (b) What If? At one instant, the two conductors in a long household extension cord carry equal 2.00-A currents in opposite directions. The two wires are 3.00 mm apart. Find the magnetic field 40.0 cm away from the middle of the straight cord, in the plane of the two wires. (c) At what distance is it one tenth as large? (d) The center wire in a coaxial cable carries current 2.00 A in one direction and the sheath around it carries current 2.00 A in the opposite direction. What magnetic field does the cable create at points outside? Consider a column of electric current passing through plasma (ionized gas). Filaments of current within the column are magnetically attracted to one another. They can crowd together to yield a very great current density and a very strong magnetic field in a small region. Sometimes the current can be cut off momentarily by this pinch effect. (In a metallic wire a pinch effect is not important, because the current-carrying electrons repel one another with electric forces.) The pinch effect can be demonstrated by making an empty aluminum can carry a large current parallel to its axis. Let R represent the radius of the can and I the upward current, uniformly distributed over its curved wall. (a) Determine the magnetic field just inside the wall. (b) Determine the magnetic field just outside. (c) Determine the pressure on the wall. (a) zero (b) B = µ 0 I/(2πR) (c) P = µ 0 I 2 /(2πR) 2 Niobium metal becomes a superconductor when cooled below 9 K. Its superconductivity is destroyed when the surface magnetic field exceeds T. Determine the maximum current a mm-diameter niobium wire can carry and remain super-conducting, in the absence of any external magnetic field. 11 (a) 400 cm (b) 7.50 nt (c) 1.26 m (d) zero A packed bundle of 100 long, straight, insulated wires forms a cylinder of radius R = cm. (a) If each wire carries 2.00 A, what are the magnitude and direction of the magnetic force per unit length acting on a wire located cm from the center of the bundle? (b) What If? Would a wire on the outer edge of the bundle experience a force greater or smaller than the value calculated in part (a)? A A long cylindrical conductor of radius R carries a current I as shown in the figure. The current density J, however, is not uniform over the cross section of the conductor but is a function of the radius according to J = br, where b is a constant. (a) Find an expression for the magnetic field B at a distance r 1 < R at a distance r 1 < R measured from the axis. (b) Find an expression for the magnetic field B at a distance r 2 > R measured from the axis. (a) 6.34 x 10-3 N/m inward (b) greatest at the outer surface (a) (b)

3 Unit 12B: Ampere's Law Ampere's Law 15 In the figure, both currents in the infinitely long wires are in the negative x direction. (a) Sketch the magnetic field pattern in the yz plane. (b) At what distance d along the z axis is the magnetic field a maximum? 18 The figure shows a cross section across a diameter of a long cylindrical conductor of radius a = 2.00 cm carrying uniform current 170 A. (a) What is the magnitude of the current's magnetic field at radial distance 0? (b) What is the magnitude of the current's magnetic field at radial distance 1.00 cm? (c) What is the magnitude of the current's magnetic field at radial distance 2.00 cm (wire's surface)? (d) What is the magnitude of the current's magnetic field at radial distance 4.00 cm? 16 (a) (b) d = a Eight wires cut the page perpendicularly at the points shown in the figure. A wire labeled with the integer k (k = 1, 2,..., 8) carries the current ki, where i = 4.50 ma. For those wires with odd k, the current is out of the page; for those with even k, it is into the page. Evaluate B ds along the closed path in the direction shown. 19 (a) 0; (b) mt; (c) 1.70 mt; (d) mt In the figure, a long circular pipe with outside radius R = 2.6 cm carries a (uniformly distributed) current i = 8.00 ma into the page. A wire runs parallel to the pipe at a distance of 3.00R from center to center. (a) Find the magnitude of the current in the wire such that the net magnetic field at point P has the same magnitude as the net magnetic field at the center of the pipe but is in the opposite direction. (b) Find the direction (into or out of the page) of the current in the wire such that the net magnetic field at point P has the same magnitude as the net magnetic field at the center of the pipe but is in the opposite direction nt. m 17 Each of the eight conductors in the figure carries 2.0 A of current into or out of the page. Two paths are indicated for the line integral B ds. (a) What is the value of the integral for path 1? (b) What is the value of the integral for path 2? (a) 3.00 ma; (b) into (a) -2.5 µt.m; (b) 0

4 Unit 12B: Ampere's Law Ampere's Law The current density J inside a long, solid, cylindrical wire of radius a = 3.1 mm is in the direction of the central axis, and its magnitude varies linearly with radial distance r from the axis according to J = J 0 r/a, where J 0 = 310 A/m 2. (a) Find the magnitude of the magnetic field at r = 0. (b) Find the magnitude of the magnetic field at r = a/2. (c) Find the magnitude of the magnetic field at r = a. (a) 0; (b) 0.10 µt; (c) 0.40 µt The figure shows a cross section of a long cylindrical conductor of radius a = 4.00 cm containing a long cylindrical hole of radius b = 1.50 cm. The central axes of the cylinder and hole are parallel and are distance d = 2.00 cm apart; current i = 5.25 A is uniformly distributed over the tinted area. (a) What is the magnitude of the magnetic field at the center of the hole? (b) Discuss the two special cases b = 0 and d = 0. (c) Use Ampere's law to show that the magnetic field in the hole is uniform. 22 Two long straight thin wires with current lie against an equally long plastic cylinder, at radius R = 20.0 cm from the cylinder's central axis. Figure a shows, in cross section, the cylinder and wire 1 but not wire 2. With wire 2 fixed in place, wire 1 is moved around the cylinder, from angle θ 1 = 0 0 to angle θ 1 = 180 0, through the first and second quadrants of the xy coordinate system. The net magnetic field B at the center of the cylinder is measured as a function of θ 1. Figure b gives the x component B x of that field and figure c gives the y component B y both as a function of θ 1. (a) At what angle θ 2 is wire 2 located? (b) What is the size of the current in wire 1? (c) What is the direction (into or out of the page) of the current in wire 1? (d) What is the size of the current in wire 2? (e) What is the direction of the current in wire 2? (a) 15.3 µt (a) ; (b) 4.0A; (c) out; (d) 2.0A; (e) into 23 A long, hollow, cylindrical conductor (inner radius 2.0 turn, outer radius 4.0 mm) carries a current of 24 A distributed uniformly across its cross section. A long thin wire that is coaxial with the cylinder carries a current of 24 A in the opposite direction. (a) What is the magnitude of the magnetic field 1.0 from the central axis of the wire and cylinder?, (b) What is the magnitude of the magnetic field 3.0 mm, from the central axis of the wire and cylinder? (c) What is the magnitude of the magnetic field 5.0 mm from the central axis of the wire and cylinder? (a) 4.8 mt; (b) 0.93mT; (c) 0

5 Unit 12B: Ampere's Law Ampere's Law 24 A long vertical wire carries an unknown current. Coaxial with the wire is a long, thin, cylindrical conducting surface that carries a current of 30 ma upward. The cylindrical surface has a radius of 3.0 mm. The magnitude of the magnetic field at a point 5.0 mm from the wire is 1.0 µt. (a) What is the size f the current in the wire? (b) What is the direction of the current in the wire? 28 The figure shows four circular Amperian loops (a, b, c, d) and, in cross section, four long circular conductors (the shaded regions), all of which are concentric. Three of the conductors are hollow cylinders; the central conductor is a solid cylinder. The currents in the conductors are, from smallest radius to largest radius, 4A out of the page, 9A into the page, 5A out of the page, and 3A into the page. Rank the Amperian loops according to the magnitude of B ds around each, greatest first. (a) 5.0 ma; (b) downward 25 A cylindrical cable of radius 8.00 mm carries a current of 25.0 A, uniformly spread over its cross-sectional area. At what distance from the center of the wire is there a point within the wire where the magnetic field magnitude is mt? 1.28 mm b, a, d, c (zero) 26 The figure shows four circular Amperian loops (a, b, c, d) concentric with a wire whose current is directed out of the page. The current is uniform across the wire's circular cross section (the shaded region). Rank the loops according to B ds around each, greatest first. 29 Figure a shows the cross section of a long conducting cylinder with inner radius a = 2.0 cm and outer radius b = 4.0 cm. The cylinder carries a current out of the page, and the magnitude of the current density in the cross section is given by J = cr 2, with c = 3.0 x 10 6 A/m 4 and r in meters. What is the magnetic field B at a point that is 3.0 cm from the central axis of the cylinder? c and d tie, then b, a 27 The figure shows four identical currents i and five Amperian paths (a through e) encircling them. Rank the paths according to the value of B ds taken in the directions shown, most positive first. d, then a and e tie, then b, c

6 Unit 12B: Ampere's Law Ampere's Law 30 Show that a uniform magnetic field B cannot drop abruptly to zero (as is suggested by the lack of field lines to the right of point a in the figure) as one moves perpendicular to B, say along the horizonta) arrow in the figure. (Hint: Apply Ampere's law to the rectangular path shown by the dashed lines.) In actual magnets, "fringing" of the magnetic field lines always occurs, which means that B approaches zero in a gradual manner. Modify the field lines in the figure to indicate a more realistic situation 32 The figure shows a closed surface. Along the flat top face, which has a radius of 2.0 cm, a perpendicular magnetic field B of magnitude 0.30 T is directed outward. Along the flat bottom face, a magnetic flux of 0.70 mwb is directed outward. (a) What is the magnitude of the magnetic flux through the curved part of the surface? (b) What is the direction (inward or outward) of the magnetic flux through the curved part of the surface? (a) 1.1 mwb; (b) inward 31 The figure here shows four closed surfaces with flat top and bottom faces and curved sides. The table gives the areas A of the faces and the magnitudes B of the uniform and perpendicular magnetic fields through those faces; the units of A and B are arbitrary but consistent. Rank the surfaces according to the magnitudes of the magnetic flux through their curved sides, greatest first. 33 The magnetic flux through each of five faces of a die (singular of "dice") is given by Φ B = ± N Wb, where N (= 1 to 5) is the number of spots on the face. The flux is positive (outward) for N even and negative (inward) for N odd. What is the flux through the sixth face of the die? +3 Wb 34 A Gaussian surface in the shape of a right circular cylinder with end caps has a radius of 12.0 cm and a length of 80.0 cm. Through one end there is an inward magnetic flux of 25.0 µwb. At the other end there is a uniform magnetic field of 1.60 mt, normal to the surface and directed outward. (a) What is the magnitude of the net magnetic flux through the curved surface? (b) What is the direction (inward or outward) of the net magnetic flux through the curved surface? d, b, c, a (zero) (a) 47.4 µwb; (b) inward

7 Gauss' Law for Magnetic Fields 35 Two wires, parallel to a z axis and a distance 4r apart, carry equal currents i in opposite directions, as shown in the figure. A circular cylinder of radius r and length L has its axis on the z axis, midway between the wires. Use Gauss' law for magnetism to derive an expression for the net outward magnetic flux through the half of the cylindrical surface above the x axis. (Hint: Find the flux through the portion of the xz plane that lies within the cylinder.) 37 The figure is a cross-sectional view of a coaxial cable. The center conductor is surrounded by a rubber layer, which is surrounded by an outer conductor, which is surrounded by another rubber layer. In a particular application, the current in the inner conductor is 1.00 A out of the page and the current in the outer conductor is 3.00 A into the page. Determine the magnitude and direction of the magnetic field at points a and b. (µ 0 il//π) ln 3 36 Answer the following questions: (a) Use Ampère s law to show that the magnetic field between the conductors of a coaxial cable shown in the figure is B = (μ 0 I)/(2πr) if r is greater than the radius of the inner wire and less than the radius of the outer cylindrical braid. (b) Show that B = 0 outside the coaxial cab Answer µT toward top of page 133 µt toward bottom of page We have seen that a long solenoid produces a uniform magnetic field directed along the axis of a cylindrical region. However, to produce a uniform magnetic field directed parallel to a diameter of a cylindrical region, one can use the saddle coils illustrated in the figure. The loops are wrapped over a somewhat flattened tube. Assume the straight sections of wire are very long. The end view of the tube shows how the windings are applied. The overall current distribution is the superposition of two overlapping circular cylinders of uniformly distributed current, one toward you and one away from you. The current density J is the same for each cylinder. The position of the axis of one cylinder is described by a position vector a relative to the other cylinder. Prove that the magnetic field inside the hollow tube is µ 0 Ja/2 downward. Suggestion: The use of vector methods simplifies the calculation. (a) B = µ 0 I/2πr (b) B = µ 0 I enc /2πr = 0

8 Unit 12B: Ampere's Law Magnetic Fields and Cylinders/Cables 39 A long cylindrical conductor of radius a has two cylindrical cavities of diameter a through its entire length, as shown in the figure. A current I is directed out of the page and is uniform through a cross section of the conductor. (a) Find the magnitude and direction of the magnetic field in terms of µ 0, I, r, and a at point P 1. (b) Find the magnitude and direction of the magnetic field in terms of µ 0, I, r, and a at point P The figure shows a cross section of a long conducting coaxial cable and gives its radii (a, b, c). Equal but opposite currents i are uniformly distributed in the two conductors. (a) Derive expressions for B(r) with radial distance r in the ranges r < c. (b) Derive expressions for B(r) with radial distance r in the ranges c < r < b. (c) Derive expressions for B(r) with radial distance r in the ranges b < r < a. (d) Derive expressions for B(r) with radial distance r in the ranges r > a. (e) Test these expressions for all the special cases that occur to you. (f) Assume that a = 2.0 cm, b = 1.8 cm, c = 0.40 cm, and i = 120 A and plot the function B(r) over the range 0 < r < 3 cm. 40 The figure shows a cross section of a hollow cylindrical conductor of radii a and b, carrying a uniformly distributed current i. (a) Show that the magnetic field magnitude B(r) for the radial distance r in the range b < r < a is given by the equation below the figure. (b) Show that when r = a, this equation gives the magnetic field magnitude B at the surface'of a long straight wire carrying current i; when r = b, it gives zero magnetic field; and when b = 0, it gives the magnetic field inside a solid conductor of radius a carrying cuirent i. (c) Assume that a = 2.0 cm, b = 1.8 cm, and i = 100 A, and then plot B(r) for the range 0 < r < 6 cm. 42 (a) µ 0 ir/2πc 2 ; (b) µ 0 i/2πr; (c) µ 0 i(a 2 - r 2 )/2π(a 2 - b 2 )r; (d) 0 The figure at the bottom shows graphs of the electric field magnitude E versus time t for four uniform electric fields, all contained within identical circular regions as in the figure at the top. Rank the fields according to the magnitudes of the magnetic fields they induce at the edge of the region, greatest first. (a) 2; (b) 1 43 The induced magnetic field at radial distance 6.0 mm from the central axis of a circular parallel-plate capacitor is 2.0 x 10-7 T. The plates have radius 3.0 mm. At what rate de/dt is the electric field between the plates changing? 2.4 x V/m.s

9 Induced Magnetic Fields 44 Suppose that a parallel-plate capacitor has circular plates with radius R = 30 mm and a plate separation of 5.0 mm. Suppose also that a sinusoidal potential difference with a maximum value of 150 V and a frequency of 60 Hz is applied across the plates; that is, V = (150 V) sin[2π(60 Hz) t]. (a) Find B max (R), the maximum value of the induced magnetic field that occurs at r = R. (b) Plot B max (r) for 0 < r < 10 cm. 47 In the figure, a uniform electric field is directed out of the page within a circular region of radius R = 3.00 cm. The magnitude of the electric field is given by E = (4.50 x 10-3 V/m.s)t where t is in seconds. (a) What is the magnitude of the induced magnetic field at radial distances 2.00 cm? (b) What is the magnitude of the induced magnetic field at radial distances 5.00 cm? (a) 1.9 pt 45 The figure shows a circular region of radius R = 3.00 cm in which a uniform electric flux is directed out of the page. The total electric flux through the region is given by Φ E = (3.00mV.m/s)t, where t is in seconds. (a) What is the magnitude of the magnetic field that is induced at a radial distance of 2.00 cm? (b) What is the magnitude of the magnetic field that is induced at a radial distance of 5.00 cm? 48 (a) 5.01 x T; (b) 4.51 x T In the figure, an electric field is directed out of the page within a circular region of radius R = 3.00 cm. The magnitude of the electric field is given by E = (0.500 V/m. s)(1 - r/r)t, where t is in seconds and r is the radial distance (r R). (a) What is the magnitude of the induced magnetic field at radial distance 2.00 cm? (b) What is the magnitude of the induced magnetic field at radial distance 5.00 cm? (a) 1.18 x T; (b) 1.06 x T 46 The figure shows a circular region of radius R = 3.00 cm in which an electric flux is directed out of the page. The flux encircled by a concentric circle of radius r is given by Φ E.emc = (0.600 V.m/s)(r/R)t, where r R and t is in seconds. (a) What is the magnitude of the induced magnetic field at radial distances 2.00 cm? (b) What is the magnitude of the induced magnetic field at radial distances 5.00 cm? (a) 3.09 x T; (b) 1.67 x T (a) 3.54 x T; (b) 2.13 x T

10 Induced Magnetic Fields 49 A parallel-plate capacitor with circular plates of radius R is being charged as in the figure. (a) Derive an expression for the magnetic field at radius r for the case r R. (b) Evaluate the field magnitude B for r = R/5 = 11.0 mm. and de/dt = 1.50 x V/m.s. (c) Derive an expression for the induced magnetic field for the case r R. 53 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? Answer (a) 4.01 x 10-5 N.m (b) 54 A thin 12 cm long solenoid has a total of 420 turns of wire and carries a current of 2.0 A. Calculate the field inside near the center. Answer 8.8 x 10-3 T 55 A 30.0 cm long solenoid 1.25 cm in diameter is to produce a field of T at its center. How much current should the solenoid carry if it has 975 turns of the wire? 50 Measurements of the magnetic field of a large tornado were made at the Geophysical Observatory in Tulsa, Oklahoma, in The tornado s field was measured to be B = T pointing north when the tornado was 9.00 km east of the observatory. What current was carried up or down the funnel of the tornado, modeled as a long straight wire? 675 A positive current is downward or negative charge flows upward 56 Answer 94.3 A A 550 turn solenoid is 15 cm long. The current in it is 33 A. A 3.0 cm long straight wire cuts through the center of the solenoid, along a diameter. This wire carries a 22 A current downward (and is connected by other wires that don t concern us). What is the force on this wire assuming the solenoid s field points due east? Answer 51 The figure shows an arrangement known as a Helmholtz coil. It consists of two circular coaxial coils, each of 200 turns and radius R = 25.0 cm, separated by a distances = R. The two coils carry equal currents i = 12.2 ma in the same direction. Find the magnitude of the net magnetic field at P, midway between the coils N You have 1.0 kg of copper and want to make a practical solenoid that produces the greatest possible magnetic field for a given voltage. Should you make your copper wire long and thin, short and fat, or something else? Consider other variables, such as solenoid diameter, length, and so on. Answer short and fat 8.78 µt 58 A long thin solenoid has 430 loops of wire per meter, and a 25 A current flows through the wire. If the permeability of the iron is 3000 μ 0 what is the total field B inside the solenoid? 41 T 52 Many loops of wire are wrapped around a nail and the ends of the wire are connected to a battery. Identify the source of M, of H, and of B.

11 Unit 12B: Ampere's Law Magnetic Fields and Solenoids 59 An iron core solenoid is 38 cm long and 1.8 cm in diameter, and has 640 turns of wire. The magnetic field inside the solenoid is 2.2 T when 48 A flows in the wire. What is the permeability μ at this high field strength? Answer 2.7 x 10-5 Tm/A approximately 22µ o 65 A solenoid 2.50 cm in diameter and 30.0 cm long has 300 turns and carries 12.0 A. (a) Calculate the flux through the surface of a disk of radius 5.00 cm that is positioned perpendicular to and centered on the axis of the solenoid, as shown in Figure a. (b) Figure b shows an enlarged end view of the same solenoid. Calculate the flux through the blue area, which is defined by an annulus that has an inner radius of cm and outer radius of cm. 60 A 32 cm long solenoid, 1.8 cm in diameter, is to produce a 0.30 T magnetic field at its center. If the maximum current is 5.7 A, how many turns must the solenoid have? Answer 1.3 x 10 4 turns 61 A current in a solenoid having air in the interior creates a magnetic field B = µ 0 H. (a) Describe qualitatively what happens to the magnitude of B as aluminum is placed in the interior. (b) Describe qualitatively what happens to the magnitude of B as copper, is placed in the interior. (c) Describe qualitatively what happens to the magnitude of B as iron is placed in the interior. (a) 7.40 µwb (b) 2.27 µwb 62 (a) B increases slightly (b) B decreases slightly (c) B increases significantly What current is required in the windings of a long solenoid that has 1000 turns uniformly distributed over a length of m, to produce at the center of the solenoid a magnetic field of magnitude T? 66 For a research project, a student needs a solenoid that produces an interior magnetic field of T. She decides to use a current of 1.00 A and a wire mm in diameter. She winds the solenoid in layers on an insulating form 1.00 cm in diameter and 10.0 cm long. Determine the number of layers of wire needed and the total length of the wire ma 12 layers 120 m 63 Consider a solenoid of length l and radius R, containing N closely spaced turns and carrying a steady current I. (a) In terms of these parameters, find the magnetic field at a point along the axis as a function of distance a from the end of the solenoid. (b) Show that as l becomes very long, B approaches µ 0 NI/2l at each end of the solenoid. 67 A solenoid that is 95.0 cm long has a radius of 2.00 cm and a winding of 1200 turns; it carries a current of 3.60 A. Calculate the magnitude of the magnetic field inside the solenoid mt (a) (b) 68 A 200-turn solenoid having a length of 25 cm and a diameter of 10 cm carries a current of 0.29 A. Calculate the magnitude of the magnetic field B inside the solenoid. 64 A single-turn square loop of wire, 2.00 cm on each edge, carries a clockwise current of A. The loop is inside a solenoid, with the plane of the loop perpendicular to the magnetic field of the solenoid. The solenoid has 30 turns/cm and carries a clockwise current of 15.0 A. Find the force on each side of the loop and the torque acting on the loop mt 226 µn directed away from the center

12 Unit 12B: Ampere's Law Magnetic Fields and Solenoids 69 A solenoid 1.30 m long and 2.60 cm in diameter carries a current of 18.0 A. The magnetic field inside the solenoid is 23.0 mt. Find the length of the wire forming the solenoid. 108 m 74 Answer the following questions: (a) What would be the effect on B inside a long solenoid if the diameter of all the loops was doubled? (b) What would be the effect on B inside a long solenoid if the spacing between loops was doubled? (c) What would be the effect on B inside a long solenoid if the solenoid s length was doubled along with a doubling in the total number of loops? 70 A long solenoid has 100 turns/cm and carries current L,. An electron moves within the solenoid in a circle of radius 2.30 cm perpendicular to the solenoid axis. The speed of the electron is c (c = speed of light). Find the current i in the solenoid. Answer (a) (b) (c) A A long solenoid with 10.0 turns/cm and a radius of 7.00 cm carries a current of 20.0 ma. A current of 6.00 A exists in a straight conductor located along the central axis of the solenoid. (a) At what radial distance from the axis will the direction of the resulting magnetic field be at to the axial direction? (b) What is the magnitude of the magnetic field there? 75 A type of magnetic switch similar to a solenoid is a relay. A relay is an electromagnet (the iron rod inside the coil does not move) which, when activated, attracts a piece of iron on a pivot. Design a relay to close an electrical switch. A relay is used when you need to switch on a circuit carrying a very large current but you do not want that large current flowing through the main switch. For example, the starter switch of a car is connected to a relay so that the large current needed for the starter doesn t pass to the dashboard switch. Answer 72 (a) 4.77 cm; (b) 35.5 µt An electron is shot into one end of a solenoid. As it enters the uniform magnetic field within the solenoid, its speed is 800 m/s and its velocity vector makes an angle of 30 0 with the central axis of the solenoid. The solenoid carries 4.0 A and has 8000 turns along its length. How many revolutions does the electron make along its helical path within the solenoid by the time it emerges from the solinoid's opposite end? (In a real solenoid, where the field is not uniform at the two ends, the number of revolutions would be slightly less than the answer here.) 76 A solenoid 2.50 cm in diameter and 30.0 cm long has 300 turns and carries 12.0 A. (a) Calculate the flux through the surface of a disk of radius 5.00 cm that is positioned perpendicular to and centered on the axis of the solenoid, as shown in Figure a. (b) Figure b shows an enlarged end view of the same solenoid. Calculate the flux through the blue area, which is defined by an annulus that has an inner radius of cm and outer radius of cm. 1.6 x 10 6 rev 73 A solenoid has length L = 1.23 m and inner diameter d = 3.55 cm, and it carries a current i = 5.57 A. It consists of five close-packed layers, each with 850 turns along length L. What is B at its center? 77 (a) 7.40 µwb (b) 2.27 µwb A solenoid that is 95.0 cm long has a radius of 2.00 cm and a winding of 1200 turns; it carries a current of 3.60 A. Calculate the magnitude of the magnetic field inside the solenoid mt

13 Unit 12B: Ampere's Law Magnetic Fields and Dipoles 78 The force on a magnetic dipole µ aligned with a nonuniform magnetic field in the x direction is given by F x = µ db/dx. Suppose that two flat loops of wire each have radius R and carry current I. (a) The loops are arranged coaxially and separated by a variable distance x, large compared to R. Show that the magnetic force between them varies as 1/x 4. (b) Evaluate the magnitude of this force if I = 10.0 A, R = cm, and x = 5.00 cm. 82 Assume the average value of the vertical component of Earth's magnetic field is 43 µt (downward) for all of Arizona, which has an area of 2.95 x 10 5 km 2. (a) What is the magnitude of the net magnetic flux through the rest of Earth's surface (the entire surface excluding Arizona)? (b) What is the direction (inward or outward) of the net magnetic flux through the rest of Earth's surface (the entire surface excluding Arizona)? (a) (b) 5.92 x 10-8 N (a) 13 MWb; (b) outward 79 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. (a) 12.7 A; (b) N m 83 A toroid is a solenoid in the shape of a circle as shown in the figure. (a) Use Ampère s law along the circular path, shown dashed in figure a, to determine that the magnetic field inside the toroid is B = (μ 0 NI)/(2πr), where N is the total number of turns, and (b) Use Ampère s law along the circular path, shown dashed in figure a, to determine that the magnetic field outside the toroid is B = 0. (c) Is the field inside a toroid uniform like a solenoid s? If not, how does it vary? 80 A student makes a short electromagnet by winding 300 turns of wire around a wooden cylinder of diameter d = 5.0 cm. The coil is connected to a battery producing a current of 4.0 A in the wire. (a) What is the magnitude of the magnetic dipole moment of this device? (b) At what axial distance z >> d will the magnetic field have the magnitude 5.0 µt (approximately one-tenth that of Earth's magnetic field)? (a) 2.4 A.m 2 ; (b) 46 cm Answer 81 In New Hampshire the average horizontal component of Earth's magnetic field in 1912 was 16 µt, and the average inclination or "dip" was What was the corresponding magnitude of Earth's magnetic field? 55 µt 84 (a) µ 0 NI/2πR (b) B = 0 (c) B proportional 1/R Is the magnetic field inside a toroid uniform? Explain. 85 The magnetic coils of a tokamak fusion reactor are in the shape of a toroid having an inner radius of m and an outer radius of 1.30 m. The toroid has 900 turns of largediameter wire, each of which carries a current of 14.0 ka. (a) Find the magnitude of the magnetic field inside the toroid along the inner radius. (b) Find the magnitude of the magnetic field inside the toroid along the outer radius.

14 Unit 12B: Ampere's Law Magnetic Fields and Torrids 86 A toroid with a mean radius of 20.0 cm and 630 turns shown in the figure is filled with powdered steel whose magnetic susceptibility Χ is 100. The current in the windings is 3.00 A. Find B (assumed uniform) inside the toroid. 89 A sphere of radius R has a uniform volume charge density ρ. Determine the magnetic dipole moment of the sphere when it rotates as a rigid body with angular velocity ω about an axis through its center. 87 B = T A toroid having a square cross section, 5.00 cm on a side, and an inner radius of 15.0 cm has 500 turns and carries a current of A. (It is made up of a square solenoid instead of a round one as in the figure bent into a doughnut shape.) (a) What is the magnetic field inside the toroid at the inner radius? (b) What is the magnetic field inside the toroid at the outer radius? 90 A sphere of radius R has a uniform volume charge density ρ. Determine the magnetic field at the center of the sphere when it rotates as a rigid object with angular speed ω about an axis through its center as shown in the figure. (a) 533 µt; (b) 400 µt 88 Show that if the thickness of a toroid is much smaller than its radius of curvature (a very skinny toroid), then Eq for the field inside a toroid reduces to Eq for the field inside a solenoid. Explain why this result is to be expected. 91 The figure shows two diamagnetic spheres located near the south pole of a bar magnet. (a) Is the magnetic forces on the spheres directed toward or away from the bar magnet? (b) Is the magnetic dipole moments of the spheres directed toward or away from the bar magnet? (c) Is the magnetic force on sphere 1 greater than, less than, or equal to that on sphere 2? a, c, b, d (zero)

15 Unit 12B: Ampere's Law Torques and Current Loops 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? 95 A parallel-plate capacitor with circular plates of radius 40 mm is being discharged by a current of 6.0 A. (a) At what radius inside the capacitor gap is the magnitude of the induced magnetic field equal to 75% of its maximum value? (b) At what radius outside the capacitor gap is the magnitude of the induced magnetic field equal to 75% of its maximum value? (c) What is that maximum value? (a) 30 mm; (b) 53mm; (c) 3.0 x 10-5 T (-4.3 x 10-3 N.m) j 96 At what rate must the potential difference between the plates of a parallel-plate capacitor with a 2.0 µf capacitance be changed to produce a displacement current of 1.5 A? 7.5 x 10 5 V/s 93 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? 97 Prove that the displacement current in a parallel-plate capacitor of capacitance C can be written as i d = C(dV/dt), where V is the potential difference between the plates. (a) 79 µt; (b) 1.1 x 10-6 N.m 98 A parallel-plate capacitor with circular plates of radius 0.10 m is being discharged. A circular loop of radius 0.20 m is concentric with the capacitor and half way between the plates. The displacement current through the loop is 2.0 A. At what rate is the electric field between the plates changing? 94 The figure is a view of one plate of a parallel-plate capacitor from within the capacitor. The dashed lines show four integration paths (path b follows the edge of the plate). Rank the paths according to the magnitude of B ds along the paths during the discharging of the capacitor, greatest first x V/m.s A parallel-plate capacitor with circular plates of radius R is being charged as in the figure. Show that the magnitude of the current density of the displacement current is J d = ε 0 (de/dt) for r R. (a) away; (b) away; (c) less

16 Displacement Current 100 The magnitude of the electric field between the two circular parallel plates in the figure is E = (4.0 x 10 5 ) - (6.0 x 10 4 t), with E in volts per meter and t in seconds. At t = 0, E is upward. The plate area is 4.0 x 10-2 m 2. (a) For t 0, what is the magnitude of the displacement current between the plates? (b) For t 0, what is the direction (up or down) of the displacement current between the plates? (c) Is the direction of the induced magnetic field clockwise or counterclockwise in the figure? 103 In the figure, a uniform electric field E collapses. Calculate the magnitude of the displacement current through a 1.6 m 2 area perpendicular to the field during each of the time intervals a, b, and c shown on the graph. (Ignore the behavior at the ends of the intervals.) (a) 2.1 x 10-8 A; (b) downward; (c) clockwise As a parallel-plate capacitor with circular plates 20 cm in diameter is being charged, the current density of the displacement current in the region between the plates is uniform and has a magnitude of 20 A/m 2. (a) Calculate the magnitude B of the magnetic field at a distance r = 50 mm from the axis of symmetry of this region. (b) Calculate de/dt in this region. (a) 0.63 µt; (b) 2.3 x V/m.s A capacitor with parallel circular plates of radius R = 1.20 cm is discharging via a current of 12.0 A. Consider a loop of radius R/3 that is centered on the central axis between the plates. (a) How much displacement current is encircled by the loop? The maximum induced magnetic field has a magnitude of 12.0 mt. (b) At what radius inside the capacitor gap is the magnitude of the induced magnetic field 3.00 mt? (c) At what radius outside the capacitor gap is the magnitude of the induced magnetic field 3.00 mt? 104 (a) 0.71 A; (b) 0; (c) 2.8A The circuit in the figure consists of switch S, a 12.0 V ideal battery, a 20.0 MΩ resistor, and an air-filled capacitor. The capacitor has parallel circular plates of radius 5.00 cm, separated by 3.00 mm. At time t = 0, switch S is closed to begin charging the capacitor. The electric field between the plates is uniform. At t = 250 µs, what is the magnitude of the magnetic field within the capacitor, at radial distance 3.00 cm? 8.40 x T (a) 1.33 A; (b) cm and 4.80 cm

17 Displacement Current 105 In the figure, a parallel-plate capacitor has square plates of edge length L = 1.0 m. A current of 2.0 A charges the capacitor, producing a uniform electric field E between the plates, with E perpendicular to the plates. (a) What is the displacement current i d through the region between the plates? (b) What is de/dt in this region? (c) What is the displacement current encircled by the square dashed path of edge length d = 0.50 m? (d) What is B ds' around this square dashed path? (a) 2.0A; (b) 2.3 x V/m.s; (c) 0.50 A; (d) 0.63 µt.m 106 Figure a shows the current i that is produced in a wire of resistivity 1.62 x 10-8 Ω.m. The magnitude of the current versus time t is shown in figure b. Point P is at radial distance 9.00 mm from the wire's center. Determine the magnitude of the magnetic field B i at point P due to the actual current i in the wire at (a) Determine the magnitude of the magnetic field B i at point P due to the actual current i in the wire at t = 20 ms. (b) Determine the magnitude of the magnetic field B i at point P due to the actual current i in the wire at t = 40 ms. (c) Determine the magnitude of the magnetic field B i at point P due to the actual current i in the wire at t = 60 ms. Next, assume that the electric field driving the current is confined to the wire. (d) Then determine the magnitude of the magnetic field B id at point P due to the displacement current id in the wire at t = 20 ms. (e) Then determine the magnitude of the magnetic field B id at point P due to the displacement current id in the wire at t = 40 ms. (f) Then determine the magnitude of the magnetic field B id at point P due to the displacement current id in the wire at t = 60 ms. (g) At point P at t = 20 s, what is the direction (into or out of the page) of B i? (h) At point P at t = 20 s, what is the direction (into or out of the page) of B id? (a) mt; (b) 0.18 mt; (c) 0.22 mt; (d) 6.4 x T; (e) 6.4 x T; (f) 0; (g) out; (h) out 107 The figure shows a circular region of radius R = 3.00 cm in which a displacement current is directed out of the page. The displacement current has a uniform density of magnitude J d = 6.00 A/m 2. (a) What is the magnitude of the magnetic field due to the displacement current at a radial distance of 2.00 cm? (b) What is the magnitude of the magnetic field due to the displacement current at a radial distance of 5.00 cm? (a) 75.4 nt; (b) 67.9 nt

18 Displacement Current 108 The figure shows a circular region of radius R = 3.00 cm in which a uniform displacement current i d = A is out of the page. (a) What is the magnitude of the magnetic field due to the displacement current at a radial distance of 2.00 cm? (b) What is the magnitude of the magnetic field due to the displacement current at a radial distance of 5.00 cm? 111 A parallel-plate capacitor with circular plates of radius R is being charged with a current i. (a) Between the plates, what is the magnitude of B ds in terms of µ 0 and i, at a radius r = R/5 from their center? (b) In terms of the maximum induced magnetic field, what is the magnitude of the magnetic field induced at r = R/5, inside the capacitor? (a) 2.22 µt; (b) 2.00 µt 109 The figure shows a circular region of radius R = 3.00 cm in which a displacement current is directed out of the page. The magnitude of the density of this displacement current is given by J d = (4.00 A/m 2 )(1 - r/r), where r is the radial distance r R. (a) What is the magnitude of the magnetic field due to the displacement current at r = 2.00 cm? (b) What is the magnitude of the magnetic field due to the displacement current at r = 5.00 cm? 110 (a) 27.9 nt; (b) 15.1 nt The figure shows a circular region of radius R = 3.00 cm in which a displacement current i d is directed out of the page. The magnitude of the displacement current is given by i d = (3.00 A)(r/R), where r is the radial distance r R. (a) What is the magnitude of the magnetic field due to i d at radial distances 2.00 cm? (b) What is the magnitude of the magnetic field due to i d at radial distances 5.00 cm? (a) 20.0 µt; (b) 12.0 µt 112 The figure here shows the spin orientations of two particles in an external magnetic field B ext. (a) If the particles are electrons, which spin orientation is at lower potential energy? (b) If, instead, the particles are protons, which spin orientation is at lower potential energy? (a) toward; (b) toward; (c) less 113 Answer the following questions. (a) What is the measured component of the orbital magnetic dipole moment of an electron with m l = 1? (b) What is the measured component of the orbital magnetic dipole moment of an electron withm l = -2? (a) -9.3 x J/T; (b) 1.9 x J/T

19 Magntism & Electrons What is the energy difference between parallel and antiparallel alignment of the z component of an electron's spin magnetic dipole moment with an external magnetic field of magnitude 0.25 T, directed parallel to the z axis? 4.6 x J An electron in an atom has an orbital angular momentum with m l = 0. (a) What is the component of L orb,z? (b) What is the component of µ orb,z? The atom is in an external magnetic field B that has magnitude 35 mt and is directed along the z axis. (c) What is the potential energy U orb associated with µ orb? (d) What is the potential energy U spin associated with µ s? If, instead, the electron has m l = -3. (e) What is L orb,z? (f) What is µ orb,z? (g) What is U orb? (h) What is U spin? (a) 0; (b) 0; (c) 0; (d) ±3.2 x J; (e) -3.2 x J.s, 2.8 x J/T, +9.7 x J, ±3.2 x J An electron is placed in a magnetic field B that is directed along a z axis. The energy difference between parallel and antiparallel alignments of the z component of the electron's spin magnetic moment with B is 6.00 x J. What is the magnitude of B? Figure a is a one-axis graph along which two of the allowed energy values (levels) of an atom are plotted. When the atom is placed in a magnetic field of T, the graph changes to that of figure b because of the energy associated with µ orb B. (We neglect µ s.) Level E l is unchanged, but level E 2 Splits into a (closely spaced) triplet of levels. What are the allowed values of m l associated with (a) energy level E l? (b) energy level E 2? (c) In joules, what amount of energy is represented by the spacing between the triplet levels? (a) 0; (b) -1, 0, 1; (c) 4.64 x J The figure shows a loop model (loop L) for a diamagnetic material. (a) Sketch the magnetic field lines within and about the material due to the bar magnet. (b) What is the direction of the loop's net magnetic dipole moment µ? (c) What is the direction of the conventional current i in the loop (clockwise or counterclockwise in the figure)? (d) What is the direction of the magnetic force on the loop? 32.3 mt (a) +x; (b) (c) clockwise; (d) +x 119 Assume that an electron of mass m and charge magnitude e moves in a circular orbit of radius r about a nucleus. A uniform magnetic field B is then established perpendicular to the plane of the orbit. Assuming also that the radius of the orbit does not change and that the change in the speed of the electron due to field B is small, find an expression for the change in the orbital magnetic dipole moment of the electron due to the field. e 2 r 2 B/4m

20 Paramagnetism 120 The figure here shows two paramagnetic spheres located near the south pole of a bar magnet. (a)is the magnetic forces on the spheres directed toward or away from the bar magnet? (b) Is the magnetic dipole moments of the spheres directed toward or away from the bar magnet? (c) Is the magnetic force on sphere Igreater than, less than, or equal to that on sphere 2? 124 A sample of the paramagnetic salt to which the magnetization curve of the figure applies is to be tested to see whether it obeys Curie's law. The sample is placed in a uniform 0.50 T magnetic field that remains constant throughout the experiment. The magnetization M is then measured at temperatures ranging from 10 to 300 K. Will it be found that Curie's law is valid under these conditions? tie of b, c, and d, then a A 0.50 T magnetic field is applied to a paramagnetic gas whose atoms have an intrinsic magnetic dipole moment of 1.0 x J/T. At what temperature will the mean kinetic energy of translation of the atoms equal the energy required to reverse such a dipole end for end in this magnetic field? 0.48 K A magnet in the form of a cylindrical rod has a length of 5.00 cm and a diameter of 1.00 cm. It has a uniform magnetization of 5.30 x 10 3 A/m. What is its magnetic dipole moment? 125 yes The figure gives the magnetization curve for a paramagnetic material. Let µ sam be the measured net magnetic moment of a sample of the material and µ max, be the maximum possible net magnetic moment of that sample. According to Curie's law, what would be the ratio µ sam /µ max were the sample placed in a uniform magnetic field of magnitude T, at a temperatureof 2.00 K? 20.8 mj/t 123 A sample of the paramagnetic salt to which the magnetization curve of the figure applies is held at room temperature (300 K). (a) At what applied magnetic field will the degree of magnetic saturation of the sample be 50%? (b) At what applied magnetic field will the degree of magnetic saturation of the sample be 90%? (c) Are these fields attainable in the laboratory? 0.30 (a) 1.5 x 10 2 T; (b) 6.0 x 10 2 T; (c) no

21 Paramagnetism 126 An electron with kinetic energy K, travels in a circular path that is perpendicular to a uniform magnetic field, which is in the positive direction of a z axis. The electron's motion is subject only to the force due to the field. (a) Show that the magnetic dipole moment of the electron due to its orbital motion has magnitude µ = K e /B and that it is in the direction opposite that of B. (b) What is the magnitude of the magnetic dipole moment of a positive ion with kinetic energy K i under the same circumstances? (c) What is the direction of the magnetic dipole moment of a positive ion with kinetic energy K i under the same circumstances? (d) An ionized gas consists of 5.3 x electrons/m 3 and the same number density of ions. Take the average electron kinetic energy to be 6.2 x J and the average ion kinetic energy to be 7.6 x J. Calculate the magnetization of the gas when it is in a magnetic field of 1.2 T. 131 Consider the hemispherical closed surface shown in the fiugre. The hemisphere is in a uniform magnetic field that makes an angle θ with the vertical. (a) Calculate the magnetic flux through the flat surface S 1. (b) Calculate the magnetic flux through the hemispherical surface S K i /B; (b) -z; (c) 0.31 ka/m A paramagnetic gas at room temperature (T = 300 K) is placed in an external uniform magnetic field of magnitude B = 1.5 T; the atoms of the gas have magnetic dipole moment µ = 1.0µ B. Calculate the mean translational kinetic energy K of an atom of the gas and the energy difference U B between parallel alignment and antiparallel alignment of the atom's magnetic dipole moment with the external field. 132 (a) (Φ B ) flat = -BπR 2 cos θ (b) (Φ B ) curved = -BπR 2 cos θ A cube of edge length l = 2.50 cm is positioned as shown in the figure. A uniform magnetic field given by B = (5i + 4j + 3k )T exists throughout the region. (a) Calculate the flux through the shaded face. (b) What is the total flux through the six faces? 128 A compass needle made of pure iron (density 7900 kg/m 3 ) has a length L of 3.0 cm, a width of 1.0 mm, and a thickness of 0.50 mm. The magnitude of the magnetic dipole moment of an iron atom is µ Fe = 2.1 x J/T. (a) If the magnetization of the needle is equivalent to the alignment of 10% of the atoms in the needle, what is the magnitude of the needle's magnetic dipole moment µ? (b) If the compass needle is jarred slightly from its (horizontal) north-south equilibrium position, it oscillates about that position. If the period of oscillation is 2.2 s, what is the horizontal component of the local magnetic field? (a) 3.12 mwb (b) What new concept did Maxwell's generalized form of Ampere's law include? 130 What new concept did Maxwell's generalized form of Ampere's law include?

22 Unit 13: Gauss s Law for Magnetism Odd Problem A very large parallel-plate capacitor carries charge with uniform charge per unit area +σ on the upper plate and σ 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 close to the plates but outside of the capacitor? (c) What is 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? Calculate this speed numerically. (a) (b) (c) (d) 3.00 x 10 8 m/s Measurements in mines and boreholes indicate that Earth's interior temperature increases with depth at the average rate of 30 C 0 /km. Assuming a surface temperature of 10 0 C, at what depth does iron cease to be ferromagnetic? (The Curie temperature of iron varies very little with pressure.) 25 km The exchange coupling as being responsible for ferromagnetism is not the mutual magnetic interaction between two elementary magnetic dipoles. (a) To show this, calculate the magnitude of the magnetic field a distance of 10 nm away, along the dipole axis, from an atom with magnetic dipole moment 1.5 x J/T (cobalt). (b) To show this, calculate the minimum energy required to turn a second identical dipole end for end in this field. (c) By comparing the latter with the results of Sample Problem 32-3, what can you conclude? (a) 3.0 µt; (b) l5.6 x ev The magnitude of the dipole moment associated with an atom of iron in an iron bar is 2.1 x J/T. Assume that all the atoms in the bar, which is 5.0 cm long and has a cross sectional area of 1.0 cm 2, have their dipole moments aligned. (a) What is the dipole moment of the bar? (b) What torque must be exerted to hold this magnet perpendicular to an ternal field of magnitude 1.5 T? (The density of iron is 7.9 g/cm 3.) The saturation magnetization M max of the ferromagnetic metal nickel is 4.70 x 10 5 A/m. Calculate the magnetic dipole moment of a single nickel atom. (The density of nickel is 8.90 g/cm 3, and its molar mass is g/mol.) 5.15 x A.m 2 A magnetic rod with length 6.00 cm, radius 3.00 mm, and (uniform) magnetization 2.70 x 10 3 A/m can turn about its center like a compass needle. It is placed in a uniform magnetic field B of magnitude 35.0 mt, such that the directions of its dipole moment and Li make an angle of (a) What is the magnitude of the torque on the rod due to B? (b) What is the change in the magnetic potential energy of the rod if the angle changes to ? (a) 1.49 x 10-4 N.m; (b) µj A Rowland ring is formed of ferromagnetic material. It is circular in cross section, with an inner radius of 5.0 cm and an outer radius of 6.0 cm, and is wound with 400 turns of wire. (a) What current must be set up in the windings to attain a toroidal field of magnitude B 0 = 0.20 mt? (b) A secondary coil wound around the toroid has 50 turns and resistance 8.0 Ω. If, for this value of B 0 we have B m = 800B 0, how much charge moves through the secondary coil when the current in the toroid windings is turned on? (a) 0.14 A; (b) 79 µc You place a magnetic compass on a horizontal surface, allow the needle to settle into its equilibrium position, and then give the compass a gentle wiggle to cause the needle to oscillate about that equilibrium position. The frequency oscillation is Hz. Earth's magnetic field at the location of the compass has a horizontal component of 18.0 µt. The needle has a magnetic moment of mj/t. What is the needle's rotational inertia about its (vertical) axis of rotation? 3.19 x 10-9 kg.m 2 (a) 8.9 A.m 2 ; (b) l13 N.m

23 Ferromagnetism 141 The magnitude of the magnetic dipole moment of Earth is 8.0 x J/T. (a) If the origin of this magnetism were a magnetized iron sphere at the center of Earth, what would be its radius? (b) What fraction of the volume of Earth would such a sphere occupy? Assume complete alignment of the dipoles. The density of Earth's inner core is 14 g/cm 3. The magnetic dipole moment of an iron atom is 2.1 x J/T. (Note: Earth's inner core is in fact thought to be in both liquid and solid forms and partly iron, but a permanent magnet as the source of Earth's magnetism has been ruled out by several considerations. For one, the temperature is certainly above the Curie point.) 143 A capacitor with square plates of edge length L is beingdischarged by a current of 0.75A. The figure is a head-on view of one of the plates from inside the capacitor. A dashed rectangular path is shown. If L = 12 cm, W = 4.0 cm, and H = 2.0 cm, what is the value of B ds around the dashed path? 142 (a) 1.8 x 10 2 km; (b) 2.3 x 10-5 Charge is sprayed onto a large non-conducting belt above the left-hand roller in the figure. The belt carries the charge with a uniform surface charge density σ as it moves with a speed v between the rollers as shown. The charge is removed by a wiper at the right-hand roller. Consider a point just above the surface of the moving belt. (a) Find an expression for the magnitude of the magnetic field B at this point. (b) If the belt is positively charged, what is the direction of B? (Note that the belt may be considered as an infinite sheet.) nt.m 51 Two plates (as in the figure) are being discharged by a constant current. Each plate has a radius of 4.00 cm. During the discharging, at a point between the plates at radial distance 2.00 cm from the central axis, the magnetic field has a magnitude of 12.5 nt. (a) What is the magnitude of the magnetic field at radial distance 6.00 cm? (b) What is the current in the wires attached to the plates? (a) 16.7 nt; (b) 5.00 ma (a) (b) out of the page parallel to the roller axes. 145 A charge q is distributed uniformly around a thin ring of radius r. The ring is rotating about an axis through its center and perpendicular to its plane, at an angular speed ω. (a) Show that the magnetic moment due to the rotating charge has magnitude µ = (1/2)qωr 2. (b) What is the direction of this magnetic moment if the charge is positive? (a) (b) in the direction of the angular momentum vector

24 Odd Problems 146 A parallel-plate capacitor with circular plates of radius R = 16 mm and gap width d = 5.0 mm has a uniform electric field between the plates. Starting at time t = 0, the potential difference between the plates is V = (100 V)e -τ /T, where the time constant τ = 12 ms. (a) At radial distance r = 0.80R from the central axis, what is the magnetic field magnitude as a function of time for t 0 (b) At radial distance r = 0.80R from the central axis, what is the magnetic field magnitude at time t = 3τ? A sample of the paramagnetic salt to which the magnetization curve of the figure applies is immersed in a uniform magnetic field of 2.0 T. (a) At what temperature will the degree of magnetic saturation of the sample be 50%? (b) At what temperature will the degree of magnetic saturation of the sample be 90%? (a) (1.2 x T)e -t/(0.012 s ); (b) 5.9 x T 147 The capacitor in Fig is being charged with a 2.50 A current. The wire radius is 1.50 mm, and the plate radius is 2.00 cm. Assume that the current i in the wire and the displacement current i d in the capacitor gap are both uniformly distributed. What is the magnitude of the magnetic field due to i at the following radial distances from the wire's center: (a) 1.00 mm (inside the wire) (b) 3.00 mm (outside the wire) (c) 2.20 cm (outside the wire) What is the magnitude of the magnetic field due to i d at the following radial distances from the central axis between the plates: (d) 1.00 mm (inside the gap) (e) 3.00 mm (inside the gap) (f) 2.20 cm (outside the gap) (g) Explain why the fields at the two smaller radii are so different for the wire and the gap but the fields at the largest radius are not. 149 (a) 4K; (b) 1K Suppose that ±4 are the limits to the values of m l for an electron in an atom. (a) How many different values of the z component µ orb,z of the electron's orbital magnetic dipole moment are possible? (b) What is the greatest magnitude of those possible values? Next, suppose that the atom is in a magnetic field of magnitude T, in the positive direction of the z axis. (c) What is the maximum potential energy associated with those possible values of µ orb,z? (d) What is the minimum potential energy associated with those possible values of µ orb,z? (a) 9; (b) 3.71 x J/T; (c) x J; (d) x J (a) 222 µt; (b) 167 µt; (c) 22.7 µt; (d) 1.25 µt; (e) 3.75 µt; (f) 22.7 µt 150 A silver wire has resistivity ρ = 1.62 x 10-8 Ω.m and a cross-sectional area of 5.00 mm 2. The current in the wire is uniform and changing at the rate of 2000 A/s when the current is 100 A. (a) What is the magnitude of the (uniform) electric field in the wire when the current in the wire is 100 A? (b) What is the displacement current in the wire at that time? (c) What is the ratio of the magnitude of the magnetic field due to the displacement current to that due to the current at a distance r from the wire? (a) V/m; (b) 2.87 x A; (c) 2.87 x 10-18

25 Odd Problems 151 Answer the following questions. (a) What is the measured component of the orbital magnetic dipole moment of an electron with the value of m l = 3? (b) What is the measured component of the orbital magnetic dipole moment of an electron with the value of m l = -4? (a) x J/T; (b) 3.71 x J/T 154 In the figure, a capacitor with circular plates of radius R = 18.0 cm is connected to asource of emf l = l m sin ωt, where l m, = 220 V and ω = 130 rad/s. The maximum value of the displacement current is i d = 7.60 µa. Neglect fringing of the electric field at the edges of the plates. (a) What is the maximum value of the current i in the circuit? (b) What is the maximum value of dφ E /dt, where Φ E is the electric flux through the region between the plates? (c) What is the separation d between the plates? (d) Find the maximum value ofthe magnitude of B between the plates at a distance r = 11.0 cm from the center. 152 A parallel-plate capacitor with circular plates of radius R is being discharged. The displacement current through a central circular area, parallel to the plates and with radius R/2, is 2.0 A. What is the discharging current? 8.0 A 153 A magnetic compass has its needle, of mass kg and length 4.0 cm, aligned with the horizontal component of Earth's magnetic field at a place where that component has the value B h = 16 µt. After the compass is given a momentary gentle shake, the needle oscillates with angular frequency ω = 45 rad/s. Assuming that the needle is a uniform thin rod mounted at its center, find the magnitude of its magnetic dipole moment kj/t 155 (a) 7.60 µa; (b) 859 kv.m/s; (c) 3.39 mm; (d) 5.16 pt A parallel-plate capacitor with circular plates is being charged. Consider a circular loop centered on the central axis and located between the plates. If the loop radius of 3.00 cm is greater than the plate radius, what is the displacement current between the plates when the magnetic field along the loop has magnitude 2.00 µt? 0.300A

26 Odd Problems Answer the following questions. (a) If an electron in an atom has orbital angular momentum with m l values limited by ±3, how many values of L orb,z can the electron have? (b) If an electron in an atom has orbital angular momentum with m l values limited by ±3, how many values of µ orb,z can the electron have? (c) In terms of h, m, and e, what is the greatest allowed magnitude for L orb,z? (d) In terms of h, m, and e, what is the greatest allowed magnitude for µ orb,z? (e) What is the greatest allowed magnitude for the z component of the electron's net angular momentum (orbital plus spin)? (f) How many values (signs included) are allowed for the z component of its net angular momentum? (a) 7; (b) 7; (c) 3h/2π; (d) 3eh/4πm; (e) 3.5h/2π; (f) 8 gives the variation of an electric field that is perpendicular to a circular area of 2.0 m 2. During the time period shown, what is the greatest displacement current through the area? In the lowest energy state of the hydrogen atom, the most probable distance of the single electron from the central proton (the nucleus) is 5.2 x m, (a) Compute the magnitude of the proton's electric field at that distance. The component µ s,z of the proton's spin magnetic dipole moment measured on a z axis is 1.4 x J/T. (b) Compute the magnitude of the proton's magnetic field at the distance 5.2 x m on the z axis. (c) What is the ratio of the spin magnetic dipole moment of the electron to that of the proton? (a) 5.3 x V/m; (b) 20 mt; (c) 6.6 x 10 2 Earth has a magnetic dipole moment of 8.0 x J/T. (a) What current would have to be produced in a single turn of wire extending around Earth at its geomagnetic equator if we wished to set up such a dipole? (b) Could such an arrangement be used to cancel out Earth's magnetism at points in space well above Earth's surface? (c) Could such an arrangement be used to cancel out Earth's magnetism on Earth's surface? (a) 6.3 x 10 8 A; (b) yes; (c) no 3.5 x 10-5 A 161 In the figure, a parallel-plate capacitor is being discharged by a current i = 5.0 A.The plates are square with edge length L = 8.0 mm. (a) What is the rate at which the electric field between the plates is changing? (b) What is the value of B ds around the dashed path, where H = 2.0 mm and W = 3.0 mm? 158 The figure shows a loop model (loop L) for a paramagnetic material. (a) Sketch the field lines through and about the material due to the magnet. (b) What is the direction of the loop's net magnetic dipole moment µ? (c) What is the direction of the conventional current i in the loop (clockwise or counterclockwise in the figure)? (d) What is the direction of the magnetic force acting on the loop? (a) -8.8 x V/m.s; (b) 5.9 x 10-7 T m (a) (b) -x; (c) counterclockwise; (d) -x

27 Odd Problems 162 Consider a solid containing N atoms per unit volume, each atom having a magnetic dipole moment µ. Suppose the direction of µ can be only parallel or antiparallel to an externally applied magnetic field B (this will be the case if g is due to the spin of a single electron). According to statistical mechanics, the probability of an atom being in a state with energy U is proportional to e -U/kT, where T is the temperature and k is Boltzmann's constant. Thus because energy U is -µ B, the fraction of atoms whose dipole moment is parallel to B is proportional to e U/kT and the fraction of atoms whose dipole moment is antiparallel to B is proportional to e -U/kT. (a) Show that the magnitude of the magnetization of this solid is M = Nµ tanh(µb/kt). Here tanh is the hyperbolic tangent function: tanh(x) = (e x - e -x )/(e x + e -x ). (b) Show that the result given in (a) reduces to M = Nµ 2 B/kT for µb << kt (c) Show that the result of (a) reduces to M = Nµ for µb >>kt (d) Show that both (b) and (c) agree qualitatively with the figure. 164 The magnetic field of Earth can be approximated as the magnetic field of a dipole. The horizontal and vertical components of this field at any distance r from Earth's center are given by the top equation shown below where λ m is the magnetic latitude (this type of latitude is measured from the geomagnetic equator toward the north or south geomagnetic pole). Assume that Earth's magnetic dipole moment has magnitude µ = 8.00 x A.m 2. (a) Show that the magnitude of Earth's field at latitude A. is given by the bottom equation. (b) Show that the inclination φ i of the magnetic field is related to the magnetic latitude λ m by tan φ i = 2 tan λ m 163 A magnetic flux of 7.0 mwb is directed outward through the flat bottom face of the closed surface shown in the figure. Along the flat top face (which has a radius of 4.2 cm) there is a 0.40 T magnetic field B directed perpendicular to the face. (a) What is the magnitude of the magnetic flux through the curved part of the surface? (b) What is the direction (inward or outward) of the magnetic flux through the curved part of the surface? 165 The magnetic field of Earth can be approximated as the magnetic field of a dipole. The horizontal and vertical components of this field at any distance r from Earth's center are given by the top equation shown below where λ m is the magnetic latitude (this type of latitude is measured from the geomagnetic equator toward the north or south geomagnetic pole). Assume that Earth's magnetic dipole moment has magnitude µ = 8.00 x A.m 2. (a) Use the equations to help predict the magnitude of Earth's magnetic field at the geomagnetic equator. (b) Use the equations to help predict the inclination of Earth's magnetic field at the geomagnetic equator. (c) Use the equations to help predict the magnitude at geomagnetic latitude (d) Use the equations to help predict the inclination at geomagnetic latitude (e) Use the equations to help predict the magnitude aat the north geomagnetic pole. (f) Use the equations to help predict the inclination at the north geomagnetic pole. (a) 9.2 mwb; (b) inward (a) 31.0 µt; (b) 0 0 ; (c) 55.9 µt; (d) ; (e) 62.0 µt; (f)

28 Odd Problems 166 The magnetic field of Earth can be approximated as the magnetic field of a dipole. The horizontal and vertical components of this field at any distance r from Earth's center are given by the top equation shown below where λ m is the magnetic latitude (this type of latitude is measured from the geomagnetic equator toward the north or south geomagnetic pole). Assume that Earth's magnetic dipole moment has magnitude µ = 8.00 x A.m 2. (a) Find the altitude above Earth's surface where the magnitude of its magnetic field is 50.0% of the surface value at the same latitude; (b) Find the maximum magnitude of the magnetic field at the core-mantle boundary, 2900 km below Earth's surface; and the (c) Find magnitude of Earth's magnetic field at the north geographic pole. (d) Findinclination of Earth's magnetic field at the north geographic pole. (e) Suggest why the values you calculated for (c) and (d) differ from measured values The figure shows, in two situations, an electric field vector E and an induced magnetic field line. In each, is the magnitude of E increasing or decreasing? a, decreasing; b, decreasing The figure shows a face-on view of one of the two square plates of a parallel-plate capacitor, as well as four loops that are located betweenthe plates. The capacitor is being discharged. (a) Neglecting fringing of the magnetic field,rank the loops according to the magnitude of B ds along them, greatest first. (b) Along which loop, if any, is the angle between the directions of B and ds constant (so that their dot product can easily be eval-uated)? (c) Along which loop, if any, is B constant (so that Bcan be brought in front of the integral sign in)? (a) 1.66 x 10 3 km; (b) 383 µt; (c) 61.1 µt; (d) A parallel-plate capacitor with circular plates of radius 55.0 mm is being charged. (a) At what radius inside he capacitor gap is the magnitude of the induced magnetic field equal to 50.0% of its maximum value? (b) At what radius outside the capacitor gap is the magnitude of the induced magnetic field equal to 50.0% of its maximum value? (a) 27.5mm; (b) 110mm In the figure, a bar magnet lies near a paper cylinder. (a) Sketch the magnetic field lines that pass through the surface of the cylinder. (b) What is the sign of B da for every area da on the surface? (c) Does this contradict Gauss' law for magnetism? Explain. 171 (a) a and b tie, then c, d; (b) none (because plate lacks circular symmetry, B not tangent to any circular loop); (c) none The figure shows a parallel-plate capacitor and the current in the connecting wires that is discharging the capacitor. (a) Is the direction of electric field E leftward or rightward between the plates? (b) Is the direction of displacement current i d leftward or rightward between the plates? (c) Is the magnetic field at point P into or out of the page? (a) (b) sign is minus; (c) no, because there is compensating positive flux through the open end nearest to magnet (a) rightward; (b) leftward; (c) into

29 Odd Problems 172 Figure a shows a capacitor, with circular plates that is being charged. Point a (near one of the connecting wires) and point b (inside the capacitor gap) are equidistant from the central axis, as are point c (not so near the wire) and point d (between the plates but outside the gap). In figure b, one curve gives the variation with distance r of the magnitude of the magnetic field inside and outside thewire. The other curve gives the variation with distance r of the magnitude of the magnetic field inside and outside the gap. The two curves partially overlap. Which of the three points on the curves correspond to which of the four points of figure a? 175 The figure shows three loop models of an electron orbiting counterclockwise within a magnetic field. The fields are nonuniform for models 1 and 2 and uniform for model 3. (a) For each model, is the magnetic dipole moment of the loop directed up, directed down, or zero? (b) For each model, is the magnetic force on the loop directed up, directed down, or zero? (a) all down; (b) 1 up, 2 down, 3 zero 173 1, a; 2, b; 3, c and d An electron in an external magnetic field B ext has its spin angular momentum S z antiparallel to B ext. If the electron undergoes a spin-flip so that S z is then parallel with B ext energy must be supplied to or lost by the electron? 176 The figure shows three loop models of an electron orbiting counterclockwise within a magnetic field. The fields are nonuniform for models 1 and 2 and uniform for model 3. Replace the current loops with diamagnetic spheres. (a) For each field, is the magnetic dipole moment of the sphere directed up, directed down, or zero? (b) For each field, is the magnetic force on the sphere directed up, directed down, or zero? 174 supplied Figure a shows a pair of opposite spin orientations for an electron in an external magnetic field B ext. Figure b gives three choices for the graph of the potential energies associated with those orientations as a function of the magnitude of B ext. Choices b and c consist of intersecting lines, choice a of parallel lines. Which is the correct choice? 177 (a) 1 up, 2 up, 3 down; (b) 1 down, 2 up, 3 zero The figure shows three loop models of an electron orbiting counterclockwise within a magnetic field. The fields are nonuniform for models 1 and 2 and uniform for model 3. Replace the current loops with paramagnetic spheres. (a) For each field, is the magnetic dipole moment of the sphere directed up, directed down, or zero? (b) For each field, is the magnetic force on the sphere directed up, directed down, or zero? b (a) 1 down, 2 down, 3 up; (b) 1 up, 2 down, 3 zero

30 Odd Problems 178 The figure represents three rectangular samples of a ferromagnetic material in which the magnetic dipoles of the domains have been directed out of the page (encircled dot) by a very strong applied field B 0. In each sample, an island domain still has its magnetic field directed into the page (encircled X). Sample 1 is one (pure) crystal. The other samples contain impurities collected along lines; domains cannot easily spread across such lines. The applied field is now to be reversed and its magnitude kept moderate. The change causes the island domain to grow. (a) Rank the three samples according to the success of that growth, greatest growth first. Ferromagnetic materials in which the magnetic dipoles are easily changed are said to be magnetically soft; when the changes are difficult, requiring strong applied fields, the materials are said to be magnetically hard. (b) Of the three samples, which is the most magnetically hard? (a) 1, 3, 2; (b) Answer the following question (a) Does the magnitude of the net force on the current loop of figures a and b increase, decrease, or remain the same if we increase the magnitude of B ext? (b) Does the magnitude of the net force on the current loop of figures a and b increase, decrease, or remain the same if we increase the divergence of B ext? (a) increase; (b) increase