Solved: In Fig. 29-71, an arrangement known as Helmholtz

Figure 29-23 shows three circuits, each consisting of two radial lengths and two concentric circular arcs, one of radius r and the other of radius R > r. The circuits have the same current through them and the same angle between the two radial lengths. Rank the circuits according to the magnitude of the net magnetic field at the center, greatest first.

Figure 29-24 represents a snapshot of the velocity vectors of four electrons near a wire carrying current i. The four velocities have the same magnitude; velocity V2 is directed into the page. Electrons 1 and 2 are at the same distance from the wire, as are electrons 3 and 4. Rank the electrons according to the magnitudes of the magnetic forces on them due to current i, greatest first.

Figure 29-25 shows four arrangements in which long parallel wires cany equal currents directly into or out of the page at the corners of identical squares. Rank the arrangements according to the magnitude of the net magnetic field at the center of the square, greatest first.

Figure 29-26 shows cross sections ---O-- of two long straight wires; the left- P i] i2 hand wire carries current ij directly Fig. 29-26 Question 4. out of the page. If the net magnetic field due to the two currents is to be zero at point P, (a) should the direction of current i2 in the right-hand wire be directly into or out of the page and (b) should i2 be greater than, less than, or equal to ij 7

Figure 29-27 shows three circuits consisting of straight radial lengths and concentric circular arcs (either half- or quarter-circles of radii 1', 21', and 31'). The circuits carry the same current. Rank them according to the magnitude of the magnetic field produced at the center of curvature (the dot), greatest first.

Figure 29-28 gives, as a function of radial distance 1', the magnitude B of the magnetic field inside and outside four wires (a, b, c, and d), each of which carries a current that is uniformly distributed across the wire's cross section. Overlapping portions of the plots are indicated by double labels. Rank the wires according to (a) radius, (b) the magnitude of the magnetic field on the surface, and (c) the value of the current, greatest first. (d) Is the magnitude of the current density in wire a greater than, less than, or equal to that in wire c7

Figure 29-29 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 the magnitude of g; Ii df around each, greatest first.

Figure 29-30 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.

Figure 29-31 shows four circular Amperian loops (a, b, c, d) and, in cross section, four long circular con- a ---hH-lr-hl ductors (the shaded regions), all of which are concentric. Three of the b d conductors are hollow cylinders; the central conductor is a solid cylinder. The currents in the conductors are, from smallest radius to largest radius, Fig. 29-31 Question 9. 4 A out of the page, 9 A into the page, 5 A out of the page, and 3 A into the page. Rank the Amperian loops according to the magnitude of g; B df around each, greatest firs

Figure 29-32 shows four identical currents i and five Amperian paths (a through e) encircling them. Rank the paths according to the value of g; Ii df taken in the directions shown, most positive first.

Figure 29-33 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 457

In Fig. 29-42, two long straight wires at separation d = 16.0 cm carry currents il = 3.61 rnA and i2 = 3.00ij out of the page. (a) Where on the x axis is the net magnetic field equal to zero? (b) If the two currents are dou- Fig.29-42 Problem 12. bled, is the zero-field point shifted toward wire 1, shifted toward wire 2, or unchanged?

In Fig. 29-43, point PI is at dis- Plf P2t tance R = 13.1 cm on the perpendic- I ular bisector of a straight wire of R R length L = 18.0 cm carrying current i! ! i = 58.2 rnA. (Note that the wire is - _ _ not long.) What is the magnitude of I ... '----L------...J.I the magnetic field at PI due to i?

Equation 29-4 gives the magnitude B of the magnetic field set up Fig. 29-43 Problems 13 and 17. by a current in an infinitely long straight wire, at a point P at perpendicular distance R from the wire. Suppose that point P is actually at perpendicular distance R from the midpoint of a wire with a finite length L. Using Eq. 29-4 to calculate B then results in a certain percentage error. What value must the ratio LlR exceed if the percentage error is to be less than 1.00%? That is, what LlR gives (B from Eq. 29-4) - (B actual) (100%) = 1.00%?

Figure 29-44 shows two current segments. The lower segment carries a current of il = 0.40 A and includes a semicircular arc with radius 5.0 cm, angle 180, and center point P. The upper segment carries current i2 = 2il and includes a circufJ lar arc with radius 4.0 cm, angle Fig. 29-44 Problem 15. 120, and the same center point P. What are the (a) magnitude and (b) direction of the net magnetic field IJ at P for the indicated current directions? What are the (c) magnitude and (d) direction of IJ if i1 is reversed?

In Fig. 29-45, two concencurrent tric circular in the loops same of direction wire carrying lie in r' ~ ~l ______ ) 2 the same plane. Loop 1 has radius '-~ 1.50 cm and carries 4.00 rnA. Loop 2 has radius 2.50 cm and carries 6.00 Fig. 29-45 Problem 16. rnA. Loop 2 is to be rotated about a diameter while the net magnetic field IJ set up by the two loops at their common center is measured. Through what angle must loop 2 be rotated so that the magnitude of that net field is 100 nT?

In Fig. 29-43, point P2 is at perpendicular distance R = 25.1 cm from one end of a straight wire of length L = 13.6 cm carrying current i = 0.693 A. (Note that the wire is not long.) What is the magnitude of the magnetic field at P2?

A current is set up in a wire loop consisting of a semicircle of radius 4.00 cm, a smaller concentric semicircle, and two radial straight lengths, all in the same plane. Figure (a) (b) 29-46a shows the arrangement but is Fig. 29-46 Problem 18. not drawn to scale. The magnitude of the magnetic field produced at the center of curvature is 47.25 pT. The smaller semicircle is then flipped over (rotated) until the loop is again entirely in the same plane (Fig. 29-46b). The magnetic field produced at the (same) center of curvature now has magnitude 15.75/LT, and its direction is reversed. What is the radius of the smaller semicircle?

One long wire lies along an x axis and carries a current of 30 A in the positive x direction. A second long wire is perpendicular to the xy plane, passes through the point (0,4.0 m, 0), and carries a current of 40 A in the positive z direction. What is the magnitude of the resulting magnetic field at the point (0,2.0 m, O)?

In Fig. 29-47, part of a long insulated wire carrying current i = 5.78 rnA is bent into a circular section of radius R = 1.89 cm. In unit-vector notation, what is the magnetic field at the center of curvature C if the circular section (a) lies in the plane of the page as shown and (b) is perpendicular to the plane of the page after being rotated 90 counterclockwise as indicated?

Figure 29-48 shows two very long straight wires (in cross section) that each carry a current of 4.00 A directly out of the page. Distance d1 = 6.00 m and distance d2 = 4.00 m. What is the magnitude of the net magnetic field at point P, which lies on a perpendicular bisector to the wires?

Figure 29-49a shows, in cross section, two long, parallel wires carrying current and separated by distance L. The ratio il/i2 of their currents is 4.00; the directions of the currents are not indicated. Figure 29-49b shows the y component By of their net magnetic field along the x axis to the right of wire 2. The vertical scale is set by Bys = 4.0 nT, and the horizontal scale is set by Xs = 20.0 cm. (a) At what value of x > 0 is By maximum? (b) If i2 = 3 rnA, what is the value of that maximum? What is the direction (irJto or out of the page) of (c) i1 and (d) i2

Figure 29-50 shows a snapshot of a proton moving at velocity j1 = (-200 m/s)] toward a long straight wire with current i = 350 rnA. At the instant shown, the proton's distance from the wire is d = 2.89 cm. In unit-vector notay IT lv ~=======.~.~=--x t Fig. 29-50 Problem 23. tion, what is the magnetic force on the proton due to the current?

Figure 29-51 shows, in cross section, four thin wires that are parallel, straight, and very long. They carry identical currents in the directions indicated. Initially all four wires are at distance d = 15.0 cm from the origin of the coordinate system, where they cre-ate a net magnetic field B. (a) To what value of x must you move wire 1 along the x axis in order to rotate B counterclockwise by 300? (b) With wire 1 in that new position, to what value of x must you move wire 3 along the x axis to rotate If by 300 back to its initial orientation?

A wire with current i = 3.00 A is shown in Fig. 29-52. Two semi-infinite straight sections, both tangent to the same circle, are connected by a circular arc that has a central angle 8 and runs along the circumference of the circle. The arc and the two straight sections all lie in the same plane. If B = 0 at the circle's center, what is 8?

In Fig. 29-53a, wire 1 consists of a circular arc and two radial lengths; it carries current il = 0.50 A in the direc)' __ ~. __ d~4-~d~~ ___ X 1 3 2 Fig. 29-51 Problem 24. Fig. 29-52 Problem 25. tion indicated. Wire 2, shown in cross section, is long, straight, and perpendicular to the plane of the figure. Its distance from the center of the arc is equal to the radius R of the arc, and it carries a current i2 that can be varied. The two currents set up a net magnetic field B at the center of the arc. Figure 29-53b gives the square of the field's magnitude B2 plotted versus the square of the current i~. The vertical scale is set by m = 10.0 X 1O-lO T2. What angle is sub tended by the arc?

In Fig. 29-54, two long straight wires (shown in cross section) carry currents il = 30.0 rnA and i2 = 40.0 rnA directly out of the page. They are equal distances from the origin, where they set up a magnetic field B. To what value must current il be changed in order to rotate B 20.00 clockwise?

Figure 29-55a shows two )' ---{\i.)-----+---------x i2 Fig.29-54 Problem 27. wires, each carrying a current. Wire 1 consists of a circular arc of radius R and two radial lengths; it carries current il = 2.0 A in the direction indicated. Wire 2 is long and straight; it carries a current i2 that can be varied; and it is at distance R/2 from the center of the arc. The net magnetic field B due to the two currents is measured at the center of curvature of the arc. Figure 29-55b is a plot of the component of B in the direction perpendicular to the figure as a function of current i2 The horizontal scale is set by i2s = 1.00 A. What is the angle sub tended by the arc?

In Fig. 29-56, four long straight wires are perpendicular to the page, and their cross sections form a square of edge length a = 20 cm. The currents are out of the page in wires 1 and 4 and into the page in wires 2 and 3, and each wire carries 20 A. In unit-vector notation, what is the net magnetic field at the square's center?

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. Fig. 29-56 Problems 29, 37, and 40. Figure 29-57 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 81 = 00 to angle 81 = 1800 , 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 8). Figure 29-57b gives the x component B, of that field as a function of 8) (the vertical scale is set by Bn = 6.0 fLT), and Fig. 29-57c gives the y component By (the vertical scale is set by Bys = 4.0 fLT). (a) At what angle 82 is wire 2 located? What are the (b) size and (c) direction (into or out of the page) of the current in wire 1 and the (d) size and (e) direction of the current in wire 2?

In Fig. 29-58, length a is 4.7 cm (short) and current i is 13 A. What are the (a) magnitude and (b) direction (into or out of the page) of the magnetic field at point P?

The current- carrying wire loop in Fig. 29-59a lies all in one plane and consists of a semicircle of radius 10.0 cm, a smaller semicircle with the same center, and two radial lengths. The smaller semicircle is rotated out of that plane by angle 8, until it is perpendicular to the plane (Fig. 29-59b). Figure 29-59c gives the magnitude of the net magnetic field at the center of curvature versus angle 8. The vertical scale is set by Ba = 10.0 fLT andBb = 12.0 fLT.Whatis the radius of the smaller semicircle?

Figure 29-60 shows a cross section of a long thin ribbon of width w = 4.91 cm that is carrying a uniformly distributed total current i = 4.61 fLA into the page. In unit-vector notation, what is the magnetic field B at a point P in the plane of the ribbon at a distance d = l-d~~lV~1 Fig. 29-60 Problem 33. 2.16 cm from its edge? (Hint: Imagine the ribbon as being constructed from many long, thin, parallel wires.)

Figure 29-61 shows, in cross y Wire 2 section, two long straight wires held -r'/---I'---'---I-X against a plastic cylinder of radius Wire 1 20.0 cm. Wire 1 carries current i) = 60.0 rnA out of the page and is fixed in place at the left side of the cylinder. Wire 2 carries current i2 = 40.0 F rnA out of the page and can be moved around the cylinder. At what (positive) angle 82 should wire 2 be positioned such that, at the origin, the net magnetic field due to the two currents has magnitude 80.0 nT?

Figure 29-62 shows wire 1 in cross section; the wire is long and straight, carries a current of 4.00 rnA out of the page, and is at distance d) = 2.40 cm from a surface. Wire 2, which is parallel to wire 1 and also long, is at horizontal distance d2 = 5.00 cm from Fig. 29-62 Problem 35. wire 1 and carries a current of 6.80 rnA into the page. What is the x component of the magnetic force per unit length on wire 2 due to wire 1 ?

In Fig. 29-63, five long par'allel z wires in an xy plane are separated by 1_ _ _ . _ distance d = 8.00 cm, have lengths ~ y of 10.0 m, and carry identical cur- I- d + d + d -..1...- d-l rents of 3.00 A out of the page. Each wire experiences a magnetic force Fig. 29-63 Problems 36 due to the other wires. In unit-vector and 39. notation, what is the net magnetic force on (a) wire 1, (b) wire 2, (c) wire 3, (d) wire 4, and (e) wire 5?

In Fig. 29-56, four long straight wires are perpendicular to the page, and their cross sections form a square of edge length a = 13.5 cm. Each wire carries 7.50 A, and the currents are out of the page in wires 1 and 4 and into the page in wires 2 and 3. In unitvector notation, what is the net magnetic force per meter of wire length on wire 4?

Figure 29-64a 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 Fz on wire 2 due to the currents in wires 1 and 3 changes. The x component of that force is Fl , and the value per unit length of wire 2 is FlJ L2. Figure 29-64b gives FlJ L2 versus the position x of wire 3. The plot has an asymptote Fl/L2 = -0.627 fLN/m as x -> 00. The horizontal scale is set by Xs = 12.0 cm. What are the (a) size and (b) direction (into or out of the page) of the current in wire 2?

In Fig. 29-63, five long parallel wires in an xy plane are separated by distance d = 50.0 cm. The currents into the page are ij = 2.00 A, i3 = 0.250 A, i4 = 4.00 A, and is = 2.00 A; the current out of the page is i2 = 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?

In Fig. 29-56, 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 afwire length on wire 1?

In Fig. 29-65, a long Lx straight wire carries a current i j = 30.0 A and a rectangular loop carries current i2 = 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 ij ?

In a particular region there is a uniform current density of 15 A/m2 in the positive z direction. What is the value of ~ B d'S when that line integral is calculated along the three straight-line segments from (x, y, z) coordinates (4d,0, 0) to (4d, 3d, 0) to (0, 0, 0) to (4d, 0, 0), where d = 20 cm?

Figure 29-66 shows a cross section across a diameter of a long cylindrical conductor of radius a = 2.00 cm carrying uniform current 170 A. What is the magnitude of the current's magnetic field at radial distance (a) 0, (b) 1.00 cm, (c) 2.00 cm (wire's surface), and (d) 4.00 cm?

Figure 29-67 shows two closed paths wrapped around two conducting loops carrying currents ij = 5.0 A and i2 = 3.0 A. What is the value of the integral ~ B d'S for (a) path 1 and (b) path2?

Each of the eight conductors in Fig. 29-68 carries 2.0 A of current into or out of the page. Two paths are indicated for the line integral ~ B . d'S. What is the value of the integral for (a) path 1 and (b) path2?

Eight wires cut the page perpendicularly at the points shown in Fig. 29-69. A wire labeled with the integer k (k = 1, 2, ... , 8) carries the current ki, where i = 4.50 rnA. 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 d'S along the closed path in the direction shown.

The current density 7 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 I' from the axis according to J = Jor/a, where Jo = 310 Alm2 Find the magnitude of the magnetic field at (a) r = O,(b)r = a/2,and (c) I' = a.

In Fig. 29-70, a long circular pipe with outside radius R = 2.6 cm carries a (uniformly distributed) current i = 8.00 rnA into the page. A wire runs parallel to the pipe at a distance of 3.00R from center to center. Find the (a) magnitude and (b) 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.

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 0.800 A. (It is made up of a square solenoid-instead of a round one as in Fig. 29-16-bent into a doughnut shape.) What is the magnetic field inside the toroid at (a) the inner radius and (b) the outer radius?

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.

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.

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.

A long solenoid has 100 turns/cm and carries current i. 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 0.0460c (c = speed of light). Find the current i in the solenoid.

An electron is shot into one end of a solenoid. As it enters the uniform magnetic field within the solenoid, its speed is 800 mls and its velocity vector makes an angle of 30 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 solenoid'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.)

A long solenoid with 10.0 turns/cm and a radius of 7.00 cm carries a current of 20.0 rnA. 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 45.0 to the axial direction? (b) What is the magnitude of the magnetic field there?

Figure 29-71 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 distance s = R. The two coils carry equal currents i = 12.2 rnA in the same direction. Find the magnitude of the net magnetic field at P, midway between the coils.

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 ;l> d will the magnetic field have the magnitude 5.0 f-LT (approximately one-tenth that of Earth's magnetic field)?

Figure 29-72a shows a length of wire carrying a current i and bent into a circular coil of one turn. In Fig. 29-72b the same length of wire has been bent to give a coil of two turns, each of half the original radius. (a) If Ba and Bb are the magnitudes of the magnetic fields at the centers of the two coils, what is the ratio BblBa? (b) What is the ratio f-Lbl f-La of the dipole moment magnitudes of the coils?

What is the magnitude of the magnetic dipole moment Jl of the solenoid described in Problem 51?

In Fig. 29-73a, two circular loops, with different currents but the same radius of 4.0 cm, are centered on a y axis. They are initially separated by distance L = 3.0 cm, with loop 2 positioned at the origin of the axis. The currents in the two loops produce a net magnetic field at the origin, with y component B)" That component is to be measured as loop 2 is gradually moved in the positive direction of the y axis. Figure 29-73b gives By as a function of the position y of loop 2. The curve approaches an asymptote of By = 7.20 f-LT as y ---4 00. The horizontal scale is set by Ys = 10.0 cm. What are (a) current i1 in loop 1 and (b) current i2 in loop 2?

A circular loop of radius 12 cm carries a current of 15 A. A fiat 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. What is the magnitude of (a) the magnetic field produced by the loop at its center and (b) the torque on the coil due to the loop?

In Fig. 29-74, current i = 56.2 rnA is set up in a loop having two radial lengths and two semicircles of radii a = 5.72 cm and b = 9.36 cm with a common center P. What are the (a) magnitude and (b) direction (into or out of the page) of the magnetic field at P and the (c) magnitude and (d) direction of the loop's magnetic di- Fig.29-74 Problem 62. pole moment?

In Fig. 29-75, a conductor car- y ries 6.0 A along the closed path abedefgha running along 8 of the 12 edges of a cube of edge length 10 cm. (a) Taking the path to be a combination of three square current loops (befgb, abgha, and edefe), find the net magnetic moment of the path in unit-vector notation. (b) What is the magnitude of the net magnetic field at the xyz coordi- d nates of (0,5.0 m,O)?

In Fig. 29-76, a closed loop carries current i = 200 rnA. The loop consists of two radial straight wires and two concentric circular arcs of radii 2.00 m and 4.00 ill. The angle () is 1T14 rad. What are the (a) magnitude and (b) direction (into or out of the page) of the net magnetic field at the center of curvature P?

A cylindrical cable of radius Fig.29-76 Problem 64. 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 0.100 mT?

Two long wires lie in an xy plane, and each carries a current in the positive direction of the x axis. Wire 1 is at y = 10.0 cm and carries 6.00 A; wire 2 is at y = 5.00 cm and carries 10.0 A. (a) In unitvector notation, what is the net magnetic field B at the origin? (b) At what value of y does B = O? (c) If the current in wire 1 is reversed, at what value of y does B = O?

Tho wires, both of length L, are formed into a circle and a square, and each carries current i. Show that the square produces a greater magnetic field at its center than the circle produces at its center.

A long straight wire carries a current of 50 A. An electron, traveling at 1.0 X 107 mis, is 5.0 cm from the wire. What is the magnitude of the magnetic force on the electron if the electron velocity is directed (a) toward the wire, (b) parallel to the wire in the direction of the current, and (c) perpendicular to the two directions defined by (a) and (b)?

Three long wires are parallel to a z axis, and each carries a current of 10 A in the positive z direction. Their points of intersection with the xy plane form an equilateral triangle with sides of 50 cm, as shown in Fig. 29-77. A fourth wire (wire b) passes through the midpoint of the base of the triangle and is parallel to the other three wires. If the net magnetic force on wire a is zero, what are the (a) size and (b) direction (+z or -z) of the current in wire b?

Figure 29-78 shows a closed loop with current i = 2.00 A. The loop consists of a half-circle of radius 4.00 m, two quarter-circles each of radius 2.00 m, and three radial straight wires. What is the magnitude of the @{G I \ I \ I \ I \ I \ I \ I \ I \ I \)' '-~---~----~' Lx b Fig. 29-77 Problem 69. net magnetic field at the common Fig. 29-78 Problem 70. center of the circular sections?

A 10-gauge bare copper wire (2.6 mm in diameter) can carry a current of 50 A without overheating. For this current, what is the magnitude of the magnetic field at the surface of the wire?

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 rnA upward. The cylindrical surface has a radius of 3.0 mm. If the magnitude of the magnetic field at a point 5.0 mm from the wire is 1.0 /LT, what are the (a) size and (b) direction of the current in the wire?

Figure 29-79 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 = O.

The magnitude of the magnetic field 88.0 cm from the axis of a long straight wire is 7.30 /LT. What is the current in the wire?

Figure 29-80 shows a wire segment of length /).s = 3.0 cm, centered at the origin, carrying current i = 2.0 A in the positive y direction (as part of some complete circuit). To calculate the magnitude of the magnetic field lJ produced by the segment at a point several meters from the origin, we can use B = (/Lo/41T)i /).s (sin 8)/ras the Biot-Savart law. This is because rand 8 are essentially conz Fig. 29-80 Problem 75. stant over the segment. Calculate lJ (in unit-vector notation) at the (x, y, z) coordinates (a) (0,0,5.0 m), (b) (0,6.0 m, 0), (c) (7.0 m, 7.0 m, 0), and (d) ( - 3.0 m, -4.0 m,O).

Figure 29-81 shows, in cross section, two long parallel wires spaced by distance d = 10.0 cm; each carries 100 A, out of the page in wire 1. Point P is on a perpendicular bisector of the line connecting the wires. In unit-vector notation, what is the net magnetic field at P if the current in wire 2 is (a) out of the page and (b) into the page?

In Fig. 29-82, two infinitely long Fig. 29-81 wires carry equal currents i. Each follows Problem 76. a 90 arc on the circumference of the same circle of radius R. Show that the magnetic field lJ at the center of the circle is the same as the field lJ a distance R be- " , Iowan infinite straight wire carrying a current i to the left.

A long wire carrying 100 A is perpendicular to the magnetic field lines of a uniform magnetic field of magnitude 5.0 mT.At what distance from the wire is the net magnetic field equal to zero?

A long, hollow, cylindrical conductor (with inner radius 2.0 mm and 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. What is the magnitude of the magnetic field (a) 1.0 mm, (b) 3.0 mm, and (c) 5.0 mm from the central axis of the wire and cylinder?

A long wire is known to have a radius greater than 4.0 mm and to carry a current that is uniformly distributed over its cross section. The magnitude of the magnetic field due to that current is 0.28 mT at a point 4.0 mm from the axis of the wire, and 0.20 mT at a point 10 mm from the axis of the wire. What is the radius of the wire?

Figure 29-83 shows a cross section of an infinite conducting sheet carrying a current per unit x-length of A; the current emerges perpendicularly out of the page. (a) Use the Biot-Savart law and symmetry to show that for all Fig. 29-83 Problem 81. points P above the sheet and all points P' below it, the magnetic field lJ is parallel to the sheet and directed as shown. (b) Use Ampere's law to prove that B = ~/LoA at all points P and P'.

Figure 29-84 shows, in cross section, two long parallel wires that are separated by distance d = 18.6 cm. Each carries 4.23 A, out of the page in wire 1 and into the page in wire 2. In unit-vector notation, what is the net magnetic field at point P at distance R = 34.2 cm, due to the two currents?

In unit-vector notation, what is the magnetic field at point P in Fig. 29-85 if i = 10 A and a = 8.0 cm? (Note that the wires are not long.)

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. What is the magnitude of the force on a 3.0 m section of either of the a/ y - - ;1 1 ~~ Lx I I-a ,I Fig. 29-85 Problem 83. outer wires if the current in the center wire is 3.2 A (a) in the positive x direction and (b) in the negative x direction?

Figure 29-86 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 (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 current 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.

Show that the magnitude of the magnetic field produced at the center of a rectangular loop of wire of length L and width W, carrying a current i, is B = 2/Loi (U + W2)1I2 7T LW

Figure 29-87 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. Derive expressions for B(r) with radial distance r in the ranges (a) r < c, (b) c < r < b, (c) b < r < a, and (d) 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.

Figure 29-88 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 distance between 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 (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 square loop of wire of edge length a carries current i. Show that, at the center of the loop, the magnitude of the magnetic field produced by the current is B = 2V2/Loi. 7Ta

In Fig. 29-71, an arrangement known as Helmholtz coils consists of two circular coaxial coils, each of N turns and radius R, separated by distance s. The two coils carry equal currents i in the same direction. (a) Show that the first derivative of the magnitude of the net magnetic field of the coils (dB/dx) vanishes at the midpoint P regardless of the value of s. Why would you expect this to be true from symmetry? (b) Show that the second derivative (d2 B/dx2) also vanishes at P, provided s = R. This accounts for the uniformity of B near P for this particular coil separation.

A square loop of wire of edge length a carries current i. Show that the magnitude of the magnetic field produced at a point on the central perpendicular axis of the loop and a distance x from its center is Prove that this result is consistent with the result shown in Problem 89.

Show that if the thickness of a toroid is much smaller than its radius of curvature (a very skinny toroid), then Eq. 29-24 for the field inside a toroid reduces to Eq. 29-23 for the field inside a solenoid. Explain why this result is to be expected.

Show that a uniform magnetic field lJ cannot drop abruptly to zero (as is suggested by the lack of field lines to the right of point a in Fig. 29-89) as one moves perpendicular to lJ, say along the horizontal 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 lJ approaches zero in a gradual manner. Modify the field lines in the figure to indicate a more realistic situation.