In deriving the force on one of the long, current-carrying

A \(+6.00-\mu \mathrm{C}\) point charge is moving at a constant \(8.00\times 10^6\mathrm{\ m}/\mathrm{s}\) in the \(+y \text {-direction }\), relative to a reference frame. At the instant when the point charge is at the origin of this reference frame, what is the magnetic-field vector \(\mathbf{\vec{B}}\) it produces at the following points: \(\text{ (a) }x=0.500\mathrm{\ m},\ y=0,\ z=0\text{; }\text{ (b) }x=0\text{, }\text{ (c) }x=0,\quad y=0,\ z=+\ 0.500\mathrm{\ m};\ \left(d\right)\ x=0,\ y=-0.500\mathrm{\ m},\ z=+0.500\mathrm{\ m}?\)

Fields Within the Atom. In the Bohr model of the hydrogen atom, the electron moves in a circular orbit of radius with a speed of If we are viewing the atom in such a way that the electrons orbit is in the plane of the paper with the electron moving clockwise, find the magnitude and direction of the electric and magnetic fields that the electron produces at the location of the nucleus (treated as a point).

An electron moves at 0.100c as shown in Fig. E28.3. Find the magnitude and direction of the magnetic field this electron produces at the following points, each \(2.00\ \mu \mathrm{m}\) from the electron: (a) points A and B; (b) point C; (c) point D.

An alpha particle (charge ) and an electron move in opposite directions from the same point, each with the speed of (Fig. E28.4). Find the magnitude and direction of the total magnetic field these charges produce at point P, which is 1.75 nm from each of them.

A charge is moving at a constant speed of in the relative to a reference frame. At the instant when the point charge is at the origin, what is the magnetic-field vector it produces at the following points:

Positive point charges and are moving relative to an observer at point P, as shown in Fig. E28.6. The distance d is 0.120 m, and (a) When the two charges are at the locations shown in the figure, what are the magnitude and direction of the net magnetic field they produce at point P? (b) What are the magnitude and direction of the electric and magnetic forces that each charge exerts on the other, and what is the ratio of the magnitude of the electric force to the magnitude of the magnetic force? (c) If the direction of is reversed, so both charges are moving in the same direction, what are the magnitude and direction of the magnetic forces that the two charges exert on each other?

Figure E28.6 shows two point charges, \(q \text { and } q^{\prime} \text {, }\) moving relative to an observer at point P. Suppose that the lower charge is actually negative, with \(q^{\prime}=-q\). (a) Find the magnetic field (magnitude and direction) produced by the two charges at point P if (i) \(v^{\prime}=v / 2\); (ii) \(v^{\prime}=v\); (iii) \(v^{\prime}=2 v\). (b) Find the direction of the magnetic force that \(q \text { exerts on } q^{\prime} \text {, }\) and find the direction of the magnetic force that \(q^{\prime} \text { exerts on } q\). (c) If \(v=v^{\prime}=3.00 \times 10^5 \mathrm{\ m}/ \mathrm{s}\), what is the ratio of the magnitude of the magnetic force acting on each charge to that of the Coulomb force acting on each charge?

An electron and a proton are each moving at in perpendicular paths as shown in Fig. E28.8. At the instant when they are at the positions shown in the figure, find the magnitude and direction of (a) the total magnetic field they produce at the origin; (b) the magnetic field the electron produces at the location of the proton; (c) the total electric force and the total magnetic force that the electron exerts on the proton.

A negative charge \(q = -3.60 \times 10^{-6} \mathrm{\ C}\) is located at the origin and has velocity \(\overrightarrow{\boldsymbol{v}}=\left(7.50 \times 10^{4} \mathrm{~m} / \mathrm{s}\right) \hat{\boldsymbol{\imath}}+(-4.90 \times \left.10^{4} \mathrm{~m} / \mathrm{s}\right) \hat{\jmath}\). At this instant what is the magnitude and direction of the magnetic field produced by this charge at the point \(x=0.200\mathrm{\ m},y=-0.300\mathrm{\ m},z=0\)? Text Transcription: q = -3.60 times 10^-6 C overrightarrow v = (7.50 times 10^4 m/s) hat i+(-4.90 times 10^4 m/s) hat j x = 0.200 m, y = -0.300 m, z=0

A short current element carries a current of 8.20 A in the same direction as . Point P is located at . Use unit vectors to express the magnetic field at P produced by this current element.

A straight wire carries a 10.0-A current (Fig. E28.11). ABCD is a rectangle with point D in the middle of a 1.10-mm segment of the wire and point C in the wire. Find the magnitude and direction of the magnetic field due to this segment at (a) point A; (b) point B; (c) point C.

A long, straight wire, carrying a current of 200 A, runs through a cubical wooden box, entering and leaving through holes in the centers of opposite faces (Fig. E28.12). The length of each side of the box is 20.0 cm. Consider an element dl of the wire 0.100 cm long at the center of the box. Compute the magnitude dB of the magnetic field produced by this element at the points a, b, c, d, and e in Fig. E28.12. Points a, c, and d are at the centers of the faces of the cube; point b is at the midpoint of one edge; and point e is at a corner. Copy the figure and show the directions and relative magnitudes of the field vectors. (Note: Assume that the length dl is small in comparison to the distances from the current element to the points where the magnetic field is to be calculated.)

A long, straight wire lies along the z-axis and carries a 4.00-A current in the Find the magnetic field (magnitude and direction) produced at the following points by a 0.500-mm segment of the wire centered at the origin: (a) x = 2.00 m, y = 0, z = 0; (b) x = 0, y = 2.00 m, z = 0; (c) x = 2.00 m, y = 2.00 m, z = 0; (d) x = 0, y = 0, z = 2.00 m.

Two parallel wires are 5.00 cm apart and carry currents in opposite directions, as shown in Fig. E28.14. Find the magnitude and direction of the magnetic field at point P due to two 1.50-mm segments of wire that are opposite each other and each 8.00 cm from P.

A wire carrying a 28.0-A current bends through a right angle. Consider two 2.00-mm segments of wire, each 3.00 cm from the bend (Fig. E28.15). Find the magnitude and direction of the magnetic field these two segments produce at point P, which is midway between them

A square wire loop 10.0 cm on each side carries a clockwise current of 15.0 A. Find the magnitude and direction of the magnetic field at its center due to the four 1.20-mm wire segments at the midpoint of each side

The Magnetic Field from a Lightning Bolt. Lightning bolts can carry currents up to approximately 20 kA. We can model such a current as the equivalent of a very long, straight wire. (a) If you were unfortunate enough to be 5.0 m away from such a lightning bolt, how large a magnetic field would you experience? (b) How does this field compare to one you would experience by being 5.0 cm from a long, straight household current of 10 A?

A very long, straight horizontal wire carries a current such that \(3.50 \times 10^{18}\) electrons per second pass any given point going from west to east. What are the magnitude and direction of the magnetic field this wire produces at a point 4.00 cm directly above it

Currents in the Heart. The body contains many small currents caused by the motion of ions in the organs and cells. Measurements of the magnetic field around the chest due to currents in the heart give values of about Although the actual currents are rather complicated, we can gain a rough understanding of their magnitude if we model them as a long, straight wire. If the surface of the chest is 5.0 cm from this current, how large is the current in the heart?

Bacteria Navigation. Certain bacteria (such as Aquaspirillum magnetotacticum) tend to swim toward the earths geographic north pole because they contain tiny particles, called magnetosomes, that are sensitive to a magnetic field. If a transmission line carrying 100 A is laid underwater, at what range of distances would the magnetic field from this line be great enough to interfere with the migration of these bacteria? (Assume that a field less than 5 percent of the earths field would have little effect on the bacteria. Take the earths field to be and ignore the effects of the seawater.)

(a) How large a current would a very long, straight wire have to carry so that the magnetic field 2.00 cm from the wire is equal to 1.00 G (comparable to the earths northward-pointing magnetic field)? (b) If the wire is horizontal with the current running from east to west, at what locations would the magnetic field of the wire point in the same direction as the horizontal component of the earths magnetic field? (c) Repeat part (b) except the wire is vertical with the current going upward.

Two long, straight wires, one above the other, are separated by a distance 2a and are parallel to the x-axis. Let the +y-axis be in the plane of the wires in the direction from the lower wire to the upper wire. Each wire carries current I in the +x-direction. What are the magnitude and direction of the net magnetic field of the two wires at a point in the plane of the wires (a) midway between them; (b) at a distance a above the upper wire; (c) at a distance a below the lower wire?

A long, straight wire lies along the y-axis and carries a current \(\mathit{\mathbf{I=8.00\mathrm{\ A}}}\) in the \(-y \text {-direction }\) (Fig. E28.23). In addition to the magnetic field due to the current in the wire, a uniform magnetic field \(\overrightarrow{\boldsymbol{B}}_{0}\) with magnitude \(1.50 \times 10^{-6}\mathrm{\ T}\) is in the \(+\ x\text{-direction }\) What is the total field (magnitude and direction) at the following points in the xz-plane: \(\text{ (a) }x=0,\ z=1.00\mathrm{\ m}\text{; }\text{ (b) }x=1.00\mathrm{\ m},\ z=0\text{; }\text{ (c) }x=0,\ z=-0.25\mathrm{m}\text{?}\)

Currents in dc transmission lines can be 100 A or more. Some people have expressed concern that the electromagnetic fields (EMFs) from such lines near their homes could cause health dangers. For a line with current 150 A and at a height of 8.0 m above the ground, what magnetic field does the line produce at ground level? Express your answer in teslas and as a percent of the earths magnetic field, which is 0.50 gauss. Does this seem to be cause for worry?

Two long, straight, parallel wires, 10.0 cm apart, carry equal 4.00-A currents in the same direction, as shown in Fig. E28.25. Find the magnitude and direction of the magnetic field at (a) point midway between the wires; (b) point 25.0 cm to the right of (c) point 20.0 cm directly above

A rectangular loop with dimensions 4.20 cm by 9.50 cm carries current I. The current in the loop produces a magnetic field at the center of the loop that has magnitude and direction away from you as you view the plane of the loop. What are the magnitude and direction (clockwise or counterclockwise) of the current in the loop?

Four, long, parallel power lines each carry 100-A currents. A cross-sectional diagram of these lines is a square, 20.0 cm on each side. For each of the three cases shown in Fig. E28.27, calculate the magnetic field at the center of the square.

Four very long, current-carrying wires in the same plane intersect to form a square 40.0 cm on each side, as shown in Fig. E28.28. Find the magnitude and direction of the current I so that the magnetic field at the center of the square is zero

Two insulated wires perpendicular to each other in the same plane carry currents as shown in Fig. E28.29. Find the magnitude of the net magnetic field these wires produce at points P and Q if the 10.0 A-current is (a) to the right or (b) to the left.

Three parallel wires each carry current I in the directions shown in Fig. E28.30. If the separation between adjacent wires is d, calculate the magnitude and direction of the net magnetic force per unit length on each wire.

Two long, parallel wires are separated by a distance of 0.400 m (Fig. E28.31) The currents \(I_1\) and \(I_2\) have the directions shown. (a) Calculate the magnitude of the force exerted by each wire on a 1.20-m length of the other. Is the force attractive or repulsive? (b) Each current is doubled, so that \(I_1\) becomes 10.0 A and \(I_2\) becomes 4.00 A. Now what is the magnitude of the force that each wire exerts on a 1.20-m length of the other?

Two long, parallel wires are separated by a distance of 2.50 cm. The force per unit length that each wire exerts on the other is and the wires repel each other. The current in one wire is 0.600 A. (a) What is the current in the second wire? (b) Are the two currents in the same direction or in opposite directions?

Lamp Cord Wires. The wires in a household lamp cord are typically 3.0 mm apart center to center and carry equal currents in opposite directions. If the cord carries current to a 100-W light bulb connected across a 120-V potential difference, what force per meter does each wire of the cord exert on the other? Is the force attractive or repulsive? Is this force large enough so it should be considered in the design of the lamp cord? (Model the lamp cord as a very long straight wire.)

A long, horizontal wire AB rests on the surface of a table and carries a current I. Horizontal wire CD is vertically above wire AB and is free to slide up and down on the two vertical metal guides C and D (Fig. E28.34). Wire CD is connected through the sliding contacts to another wire that also carries a current I, opposite in direction to the current in wire AB. The mass per unit length of the wire CD is To what equilibrium height h will the wire CD rise, assuming that the magnetic force on it is due entirely to the current in the wire AB?

Currents in the Brain. The magnetic field around the head has been measured to be approximately Although the currents that cause this field are quite complicated, we can get a rough estimate of their size by modeling them as a single circular current loop 16 cm (the width of a typical head) in diameter. What is the current needed to produce such a field at the center of the loop?

Calculate the magnitude and direction of the magnetic field at point P due to the current in the semicircular section of wire shown in Fig. E28.36. (Hint: Does the current in the long, straight section of the wire produce any field at P?)

Calculate the magnitude of the magnetic field at point P of Fig. E28.37 in terms of R, \(I_1\), and \(I_2\). What does your expression give when \(I_1\) = \(I_2\)?

A closely wound, circular coil with radius 2.40 cm has 800 turns. (a) What must the current in the coil be if the magnetic field at the center of the coil is 0.0580 T? (b) At what distance x from the center of the coil, on the axis of the coil, is the magnetic field half its value at the center?

A closely wound, circular coil with a diameter of 4.00 cm has 600 turns and carries a current of 0.500 A. What is the magnitude of the magnetic field (a) at the center of the coil and (b) at a point on the axis of the coil 8.00 cm from its center?

A closely wound coil has a radius of 6.00 cm and carries a current of 2.50 A. How many turns must it have if, at a point on the coil axis 6.00 cm from the center of the coil, the magnetic field

Two concentric circular loops of wire lie on a tabletop, one inside the other. The inner wire has a diameter of 20.0 cm and carries a clockwise current of 12.0 A, as viewed from above, and the outer wire has a diameter of 30.0 cm. What must be the magnitude and direction (as viewed from above) of the current in the outer wire so that the net magnetic field due to this combination of wires is zero at the common center of the wires?

Figure E28.42 shows, in cross section, several conductors that carry currents through the plane of the figure. The currents have the magnitudes \(I_{1}=4.0 \mathrm{~A}\), and \(I_{2}=6.0, I_{2}=6.0 \mathrm{~A}\), and \(I_{3}=2.0 \mathrm{~A}\), and the directions shown. Four paths, labeled a through d, are shown. What is the line integral \(\oint \overrightarrow{\boldsymbol{B}} \cdot d \overrightarrow{\boldsymbol{l}}\) for each path? Each integral involves going around the path in the counterclockwise direction. Explain your answers.

A closed curve encircles several conductors. The line integral around this curve is (a) What is the net current in the conductors? (b) If you were to integrate around the curve in the opposite direction, what would be the value of the line integral? Explain.

As a new electrical technician, you are designing a large solenoid to produce a uniform 0.150-T magnetic field near the center of the solenoid. You have enough wire for 4000 circular turns. This solenoid must be 1.40 m long and 2.80 cm in diameter. What current will you need to produce the necessary field?

Coaxial Cable. A solid conductor with radius a is supported by insulating disks on the axis of a conducting tube with inner radius b and outer radius c (Fig. E28.45). The central conductor and tube carry equal currents I in opposite directions. The currents are distributed uniformly over the cross sections of each conductor. Derive an expression for the magnitude of the magnetic field (a) at points outside the central, solid conductor but inside the tube and (b) at points outside the tube 1r 7 c2

Repeat Exercise 28.45 for the case in which the current in the central, solid conductor is \(I_1\), the current in the tube is \(I_2\) and these currents are in the same direction rather than in opposite directions.

A long, straight, cylindrical wire of radius R carries a current uniformly distributed over its cross section. At what locations is the magnetic field produced by this current equal to half of its largest value? Consider points inside and outside the wire

A 15.0-cm-long solenoid with radius 0.750 cm is closely wound with 600 turns of wire. The current in the windings is 8.00 A. Compute the magnetic field at a point near the center of the solenoid.

A solenoid is designed to produce a magnetic field of 0.0270 T at its center. It has radius 1.40 cm and length 40.0 cm, and the wire can carry a maximum current of 12.0 A. (a) What minimum number of turns per unit length must the solenoid have? (b) What total length of wire is required?

A toroidal solenoid has an inner radius of 12.0 cm and an outer radius of 15.0 cm. It carries a current of 1.50 A. How many equally spaced turns must it have so that it will produce a magnetic field of 3.75 mT at points within the coils 14.0 cm from its center?

A magnetic field of 37.2 T has been achieved at the MIT Francis Bitter National Magnetic Laboratory. Find the current needed to achieve such a field (a) 2.00 cm from a long, straight wire; (b) at the center of a circular coil of radius 42.0 cm that has 100 turns; (c) near the center of a solenoid with radius 2.40 cm, length 32.0 cm, and 40,000 turns

A toroidal solenoid (see Example 28.10) has inner radius and outer radius The solenoid has 250 turns and carries a current of 8.50 A. What is the magnitude of the magnetic field at the following distances from the center of the torus: (a) 12.0 cm; (b) 16.0 cm; (c) 20.0 cm?

A wooden ring whose mean diameter is 14.0 cm is wound with a closely spaced toroidal winding of 600 turns. Compute the magnitude of the magnetic field at the center of the cross section of the windings when the current in the windings is 0.650 A.

A toroidal solenoid with 400 turns of wire and a mean radius of 6.0 cm carries a current of 0.25 A. The relative permeability of the core is 80. (a) What is the magnetic field in the core? (b) What part of the magnetic field is due to atomic currents?

A toroidal solenoid with 500 turns is wound on a ring with a mean radius of 2.90 cm. Find the current in the winding that is required to set up a magnetic field of 0.350 T in the ring (a) if the ring is made of annealed iron and (b) if the ring is made of silicon steel

The current in the windings of a toroidal solenoid is 2.400 A. There are 500 turns, and the mean radius is 25.00 cm. The toroidal solenoid is filled with a magnetic material. The magnetic field inside the windings is found to be 1.940 T. Calculate (a) the relative permeability and (b) the magnetic susceptibility of the material that fills the toroid.

A long solenoid with 60 turns of wire per centimeter carries a current of 0.15 A. The wire that makes up the solenoid is wrapped around a solid core of silicon steel \(\left(K_{\mathrm{m}}=5200\right)\). (The wire of the solenoid is jacketed with an insulator so that none of the current flows into the core.) (a) For a point inside the core, find the magnitudes of (i) the magnetic field \(\overrightarrow{\boldsymbol{B}}_{0}\) due to the solenoid current; (ii) the magnetization \(\vec{M}\) (iii) the total magnetic field \(\overrightarrow{\boldsymbol{B}}\). (b) In a sketch of the solenoid and core, show the directions of the vectors \(\overrightarrow{\boldsymbol{B}}, \overrightarrow{\boldsymbol{B}}_{0}\), and \(\vec{M}\) inside the core.

When a certain paramagnetic material is placed in an external magnetic field of 1.5000 T, the field inside the material is measured to be 1.5023 T. Find (a) the relative permeability and (b) the magnetic permeability of this material.

A pair of point charges, and are moving as shown in Fig. P28.59 with speeds and When the charges are at the locations shown in the figure, what are the magnitude and direction of (a) the magnetic field produced at the origin and (b) the magnetic force that exerts on q?

At a particular instant, charge is at the point and has velocity . Charge is at the point and has velocity . At this instant, what are the magnitude and direction of the magnetic force that exerts on

Two long, parallel transmission lines, 40.0 cm apart, carry 25.0-A and 75.0-A currents. Find all locations where the net magnetic field of the two wires is zero if these currents are in (a) the same direction and (b) the opposite direction.

A long, straight wire carries a current of 5.20 A. An electron is traveling in the vicinity of the wire. At the instant when the electron is 4.50 cm from the wire and traveling with a speed of directly toward the wire, what are the magnitude and direction (relative to the direction of the current) of the force that the magnetic field of the current exerts on the electron?

A long, straight wire carries a 13.0-A current. An electron is fired parallel to this wire with a velocity of in the same direction as the current, 2.00 cm from the wire. (a) Find the magnitude and direction of the electrons initial acceleration. (b) What should be the magnitude and direction of a uniform electric field that will allow the electron to continue to travel parallel to the wire? (c) Is it necessary to include the effects of gravity? Justify your answer.

Two very long, straight wires carry currents as shown in Fig. P28.64. For each case, find all locations where the net magnetic field is zero.

CP Two identical circular, wire loops 40.0 cm in diameter each carry a current of 3.80 A in the same direction. These loops are parallel to each other and are 25.0 cm apart. Line is normal to the plane of the loops and passes through their centers. A proton is fired at perpendicular to line from a point midway between the centers of the loops. Find the magnitude of the magnetic force these loops exert on the proton just after it is fired.

A negative point charge \(q=-7.20\ \mathrm{mC}\) is moving in a reference frame. When the point charge is at the origin, the magnetic field it produces at the point x = 25.0 cm, y = 0, z = 0 is \(\vec{\boldsymbol{B}}=(6.00 \ \mu \mathrm{T}) \hat{\boldsymbol{j}}\), and its speed is 800 m/s. (a) What are the x-, y-, and z-components of the velocity \(\vec{\boldsymbol{v}}_{0}\) of the charge? (b) At this same instant, what is the magnitude of the magnetic field that the charge produces at the point x = 0, y = 25.0 cm, z = 0?

Two long, straight, parallel wires are 1.00 m apart (Fig. P28.67). The wire on the left carries a current of 6.00 A into the plane of the paper. (a) What must the magnitude and direction of the current be for the net field at point P to be zero? (b) Then what are the magnitude and direction of the net field at Q? (c) Then what is the magnitude of the net field at S?

Figure P28.68 shows an end view of two long, parallel wires perpendicular to the xyplane, each carrying a current I but in opposite directions. (a) Copy the diagram, and draw vectors to show the field of each wire and the net field at point P. (b) Derive the expression for the magnitude of at any point on the x-axis in terms of the x-coordinate of the point. What is the direction of ? (c) Graph the magnitude of at points on the x-axis. (d) At what value of x is the magnitude of a maximum? (e) What is the magnitude of when

Refer to the situation in Problem 28.68. Suppose that a third long, straight wire, parallel to the other two, passes through point P (see Fig. P28.68) and that each wire carries a current Let and Find the magnitude and direction of the force per unit length on the third wire, (a) if the current in it is directed into the plane of the figure, and (b) if the current in it is directed out of the plane of the figure.

A pair of long, rigid metal rods, each of length L, lie parallel to each other on a perfectly smooth table. Their ends are connected by identical, very light conducting springs of force constant k (Fig. P28.70) and negligible unstretched length. If a current I runs through this circuit, the springs will stretch. At what separation will the rods remain at rest? Assume that k is large enough so that the separation of the rods will be much less than L.

Two long, parallel wires hang by 4.00-cm-long cords from a common axis (Fig. P28.71). The wires have a mass per unit length of 0.0125 kg/m and carry the same current in opposite directions. What is the current in each wire if the cords hang at an angle of \(6.00^{\circ}\) with the vertical?

The long, straight wire AB shown in Fig. P28.72 carries a current of 14.0 A. The rectangular loop whose long edges are parallel to the wire carries a current of 5.00 A. Find the magnitude and direction of the net force exerted on the loop by the magnetic field of the wire.

A flat, round iron ring 5.00 cm in diameter has a current running through it that produces a magnetic field of \(75.4\ \mu\mathrm{T}\) at its center. This ring is placed in a uniform external magnetic field of 0.375 T. What is the maximum torque the external field can exert on the ring? Show how the ring should be oriented relative to the field for the torque to have its maximum value. Text Transcription: 75.4 mu T

The wire semicircles shown in Fig. P28.74 have radii a and b. Calculate the net magnetic field (magnitude and direction) that the current in the wires produces at point P.

Helmholtz Coils. Figure 28.75 is a sectional view of two circular coils with radius a, each wound with N turns of wire carrying a current I, circulating in the same direction in both coils. The coils are separated by a distance a equal to their radii. In this configuration the coils are called Helmholtz coils; they produce a very uniform magnetic field in the region between them. (a) Derive the expression for the magnitude B of the magnetic field at a point on the axis a distance x to the right of point P, which is midway between the coils. (b) Graph B versus x for to Compare this graph to one for the magnetic field due to the right-hand coil alone. (c) From part (a), obtain an expression for the magnitude of the magnetic field at point P. (d) Calculate the magnitude of the magnetic field at P if turns, and (e) Calculate and at Discuss how your results show that the field is very uniform in the vicinity of P

A circular wire of diameter D lies on a horizontal table and carries a current I. In Fig. P28.76 point A marks the center of the circle and point C is on its rim. (a) Find the magnitude and direction of the magnetic field at point A. (b) The wire is now unwrapped so it is straight, centered on point C, and perpendicular to the line AC, but the same current is maintained in it. Now find the magnetic field at point A. (c) Which field is greater: the one in part (a) or in part (b)? By what factor? Why is this result physically reasonable?

A long, straight wire with a circular cross section of radius R carries a current I. Assume that the current density is not constant across the cross section of the wire, but rather varies as where is a constant. (a) By the requirement that J integrated over the cross section of the wire gives the total current I, calculate the constant in terms of I and R. (b) Use Amperes law to calculate the magnetic field for (i) and (ii) Express your answers in terms of I

The wire shown in Fig. P28.78 is infinitely long and carries a current I. Calculate the magnitude and direction of the magnetic field that this current produces at point P.

A conductor is made in the form of a hollow cylinder with inner and outer radii a and b, respectively. It carries a current I uniformly distributed over its cross section. Derive expressions for the magnitude of the magnetic field in the regions (a)

A circular loop has radius R and carries current \(I_{2}\) in a clockwise direction (Fig. P28.80). The center of the loop is a distance D above a long, straight wire. What are the magnitude and direction of the current \(I_{1}\) in the wire if the magnetic field at the center of the loop is zero?

CALC A long, straight, solid cylinder, oriented with its axis in the z-direction, carries a current whose current density is The current density, although symmetric about the cylinder axis, is not constant but varies according to the relationship where a is the radius of the cylinder, r is the radial distance from the cylinder axis, and is a constant having units of amperes. (a) Show that is the total current passing through the entire cross section of the wire. (b) Using Amperes law, derive an expression for the magnitude of the magnetic field in the region (c) Obtain an expression for the current I contained in a circular cross section of radius and centered at the cylinder axis. (d) Using Amperes law, derive an expression for the magnitude of the magnetic field in the region How do your results in parts (b) and (d) compare for

A long, straight, solid cylinder, oriented with its axis in the z-direction, carries a current whose current density is \(\vec{J}\). The current density, although symmetric about the cylinder axis, is not constant and varies according to the relationship \(\vec{J}=\left(\frac{b}{r}\right) e^{(r-a) / \delta \hat{k}}\) for \(r \leq a\) = 0 for \(r \geq a\) where the radius of the cylinder is a = 5.00 cm, r is the radial distance from the cylinder axis, b is a constant equal to 600 A/m, and \(\delta\) is a constant equal to 2.50 cm. (a) Let \(I_0\) be the total current passing through the entire cross section of the wire. Obtain an expression for \(I_0\) in terms of b, \(\delta\) and a. Evaluate your expression to obtain a numerical value for \(I_0\). (b) Using Ampere's law, derive an expression for the magnetic field \(\vec{B}\) in the region \(r \geq a\). Express your answer in terms of \(I_0\) rather than b. (c) Obtain an expression for the current I contained in a circular cross section of radius \(r \leq a\) and centered at the cylinder axis. Express your answer in terms of \(I_0\) rather than b. (d) Using Ampere's law, derive an expression for the magnetic field \(\vec{B}\) in the region \(r \leq a\) (e) Evaluate the magnitude of the magnetic field at \(r=\delta\), r = a, and r = 2a.

An Infinite Current Sheet. Long, straight conductors with square cross sections and each carrying current I are laid side by side to form an infi- nite current sheet (Fig. P28.83). The conductors lie in the xy-plane, are parallel to the y-axis, and carry current in the +y-direction. There are n conductors per unit length measured along the x-axis. (a) What are the magnitude and direction of the magnetic field a distance a below the current sheet? (b) What are the magnitude and direction of the magnetic field a distance a above the current sheet?

Long, straight conductors with square cross section, each carrying current I, are laid side by side to form an infinite current sheet with current directed out of the plane of the page (Fig. P28.84). A second infinite current sheet is a distance d below the first and is parallel to it. The second sheet carries current into the plane of the page. Each sheet has n conductors per unit length. (Refer to Problem 28.83.) Calculate the magnitude and direction of the net magnetic field at (a) point P (above the upper sheet); (b) point R (midway between the two sheets); (c) point S (below the lower sheet).

CP A piece of iron has magnetization Find the average magnetic dipole moment per atom in this piece of iron. Express your answer both in and in Bohr magnetons. The density of iron is given in Table 14.1, and the atomic mass of iron (in grams per mole) is given in Appendix D. The chemical symbol for iron is Fe

A wide, long, insulating belt has a uniform positive charge per unit area on its upper surface. Rollers at each end move the belt to the right at a constant speed Calculate the magnitude and direction of the magnetic field produced by the moving belt at a point just above its surface. (Hint: At points near the surface and far from its edges or ends, the moving belt can be considered to be an infinite current sheet like that in Problem 28.83.)

Two long, straight conducting wires with linear mass density \(\lambda\) are suspended from cords so that they are each horizontal, parallel to each other, and a distance d apart. The back ends of the wires are connected to each other by a slack, low-resistance connecting wire. A charged capacitor (capacitance C) is now added to the system; the positive plate of the capacitor (initial charge \(+Q_{0}\)) is connected to the front end of one of the wires, and the negative plate of the capacitor (initial charge \(-Q_{0}\)) is connected to the front end of the other wire (Fig. P28.87). Both of these connections are also made by slack, low-resistance wires. When the connection is made, the wires are pushed aside by the repulsive force between the wires, and each wire has an initial horizontal velocity of magnitude \(v_0\). Assume that the time constant for the capacitor to discharge is negligible compared to the time it takes for any appreciable displacement in the position of the wires to occur. (a) Show that the initial speed \(v_0\) of either wire is given by \(v_{0}=\frac{\mu_{0} Q_{0}^{2}}{4 \pi \lambda R C d}\) where R is the total resistance of the circuit. (b) To what height h will each wire rise as a result of the circuit connection?

CALC A wire in the shape of a semicircle with radius a is oriented in the yz-plane with its center of curvature at the origin (Fig. P28.88). If the current in the wire is I, calculate the magnetic-field components produced at point P, a distance x out along the x-axis. (Note: Do not forget the contribution from the straight wire at the bottom of the semicircle that runs from to You may use the fact that the fields of the two antiparallel currents at cancel, but you must explain z 7 a why they cancel.)

Problem 2E Fields Within the Atom. In the Bohr model of the hydrogen atom, the electron moves in a circular orbit of radius 5.3 × 10-11 m with a speed of 2.2 × 106 m/s. If we are viewing the atom in such a way that the electron’s orbit is in the plane of the paper with the electron moving clockwise, find the magnitude and direction of the electric and magnetic fields that the electron produces at the location of the nucleus (treated as a point).

Problem 3DQ The text discussed the magnetic field of an infinitely long, straight conductor carrying a current. Of course, there is no such thing as an infinitely long anything. How do you decide whether a particular wire is long enough to be considered infinite?

An electron moves at 0.100c as shown in Fig. E28.3. Find the magnitude and direction of the magnetic field this electron produces at the following points, each 2.00 \mu \mathrm{m} from the electron: (a) points A and B; (b) point C; (c) point D. Equation Transcription: Text Transcription: mum Vector v 90deg 60deg 60deg

Problem 5E A -4.80-µC charge is moving at a constant speed of 6.80 × 105 m/s in the +x-direction relative to a reference frame. At the instant when the point charge is at the origin, what is the magnetic-field vector it produces at the following points: (a) x = 0.500 m, y = 0, z = 0; (b) x = 0, y = 0.500 m, z = 0; (c) x = 0.500 m, y = 0.500 m, z = 0; (d) x = 0, y = 0, z = 0.500 m?

Problem 6DQ Suppose you have three long, parallel wires arranged so that in cross section they are at the corners of an equilateral triangle. Is there any way to arrange the currents so that all three wires attract each other? So that all three wires repel each other? Explain.

Positive point charges \(q=+8.00 \mu \mathrm{C}\) and \(q^{\prime}=+3.00 \mu \mathrm{C}\) are moving relative to an observer at point P, as shown in Fig. E28.6. The distance d is 0.120 m, \(v=4.50 \times 10^{\circ} \mathrm{m} / \mathrm{s}\), and \(v^{\prime}=9.00 \times 10^{6} \mathrm{~m} / \mathrm{s}\). (a) When the two charges are at the locations shown in the figure, what are the magnitude and direction of the net magnetic field they produce at point P? (b) What are the magnitude and direction of the electric and magnetic forces that each charge exerts on the other, and what is the ratio of the magnitude of the electric force to the magnitude of the magnetic force? (c) If the direction of \(\vec{v}^{\prime}\) is reversed, so both charges are moving in the same direction, what are the magnitude and direction of the magnetic forces that the two charges exert on each other? Equation Transcription: Text Transcription: q=+8.00 muC q’=+3.00 muC v=4.50x10^6 m/s v'=9.00x10^6 m/s Vector v’ v’ q’

In deriving the force on one of the long, current-carrying conductors in Section 28.4, why did we use the magnetic field due to only one of the conductors? That is, why didn’t we use the total magnetic field due to both conductors?

Figure E28.6 shows two point charges, q and \(q^{\prime}\), moving relative to an observer at point P. Suppose that the lower charge is actually negative, with \(q^{\prime}=-q\). (a) Find the magnetic field (magnitude and direction) produced by the two charges at point P if (i) \(v^{\prime}=v / 2 \text {; (ii) } v^{\prime}=v \text {; (iii) } v^{\prime}=2v\). b) Find the direction of the magnetic force that q exerts on \(q^{\prime}\), and find the direction of the magnetic force that \(q^{\prime}\) exerts on q. (c) If \(v=v^{\prime}=3.00 \times 10^5 \mathrm{\ m}/ \mathrm{s}\), what is the ratio of the magnitude of the magnetic force acting on each charge to that of the Coulomb force acting on each charge?

Problem 8DQ Two concentric, coplanar, circular loops of wire of different diameter carry currents in the same direction. Describe the nature of the force exerted on the inner loop by the outer loop and on the outer loop by the inner loop.

An electron and a proton are each moving at \(845 \mathrm{~km} / \mathrm{s}\) in perpendicular paths as shown in Fig. E28.8. At the instant when they are at the positions shown in the figure, find the magnitude and direction of (a) the total magnetic field they produce at the origin; (b) the magnetic field the electron produces at the location of the proton; (c) the total electric force and the total magnetic force that the electron exerts on the proton. Equation Transcription: Text Transcription: 845 km/s 5.00 nm 4.00 nm

Problem 9DQ A current was sent through a helical coil spring. The spring contracted, as though it had been compressed. Why?

A negative charge \(q=-3.60 \times 10^{-6}\) C is located at the origin and has velocity \(\vec{v}=\left(7.50 \times 10^{4} \mathrm{~m} / \mathrm{s}\right) \widehat{i}+\left(-4.90 \times 10^{4} \mathrm{~m} / \mathrm{s}\right) \widehat{j}\). At this instant what are the magnitude and direction of the magnetic field produced by this charge at the point \(x=0.200\) m, \(y=-0.300\) m, \(z=0\)? Equation Transcription: Text Transcription: q=-3.60x10^-6 Vector v=(7.50x10^4 m/s)i hat+(-4.90x10^4 m/s)j hat x=0.200 y=-300 z=0

A straight wire carries a 10.0-A current (Fig. E28.11). ABCD is a rectangle with point D in the middle of a 1.10-mm segment of the wire and point C in the wire. Find the magnitude and direction of the magnetic field due to this segment at (a) point A; (b) point B; (c) point C.

Problem 12DQ If the magnitude of the magnetic field a distance R from a Very long, straight, current-carrying wire is B, at what distance from the wire will the field have magnitude 3B?

A long, straight wire, carrying a current of 200 A, runs through a cubical wooden box, entering and leaving through holes in the centers of opposite faces (Fig. E28.12). The length of each side of the box is 20.0 cm. Consider an element dl of the wire 0.100 cm long at the center of the box. Compute the magnitude dB of the magnetic field produced by this element at the points a, b, c, d, and in Fig. E28.12. Points a, c, and d are at the centers of the faces of the cube; point b is at the midpoint of one edge; and point is at a corner. Copy the figure and show the directions and relative magnitudes of the field vectors. (Note: Assume that the length dl is small in comparison to the distances from the current element to the points where the magnetic field is to be calculated.)

Problem 13DQ Two very long, parallel wires carry equal currents in opposite directions. (a) Is there any place that their magnetic fields completely cancel? If so, where? If not, why not? (b) How would the answer to part (a) change if the currents were in the same direction?

Problem 13E A long, straight wire lies along the z-axis and carries a 4.00-A current in the +z-direction. Find the magnetic field (magnitude and direction) produced at the following points by a 0.500-mm segment of the wire centered at the origin: (a) x = 2.00 m, y = 0, z = 0; (b) x = 0, y = 2.00 m, z = 0; (c) x = 2.00 m, y = 2.00 m, z = 0; (d) x = 0, y = 0, z = 2.00 m,

In the circuit shown in Fig. Q28.14, when switch S is suddenly closed, the wire L is pulled toward the lower wire carrying current I. Which (a or b) is the positive terminal of the battery? How do you know?

Problem 17E The Magnetic Field from a Lightning Bolt. Lightning bolts can carry currents up to approximately 20 kA. We can model such a current as the equivalent of a very long, straight wire. (a) If you were unfortunate enough to be 5.0 m away from such a lightning bolt, how large a magnetic field would you experience? (b) How does this field compare to one you would experience by being 5.0 cm from a long, straight household current of 10 A?

Problem 18DQ What features of atomic structure determine whether an element is diamagnetic or paramagnetic? Explain.

Problem 18E A very long, straight horizontal wire carries a current such that 3.50 × 1018 electrons per second pass any given point going west to east. What are the magnitude and direction of the magnitude this wire produces at a point 4.00 cm directly above it?

Problem 20E BIO Bacteria Navigation. Certain bacteria (such as Aquaspirillum magnetotacticum) tend to swim toward the earth’s geographic north pole because they contain tiny particles, called magnetosomes, that are sensitive to a magnetic field. If a transmission line carrying 100 A is laid underwater, at what range of distances would the magnetic field from this line be great enough to interfere with the migration of these bacteria? (Assume that a field less than 5% of the earth’s field would have little effect on the bacteria. Take the earth’s field to be 5.0 × 10-5 T, and ignore the effects of the seawater.)

Problem 21DQ The discussion of magnetic forces on current loops in Section 27.7 ed that no net force is exerted on a complete loop in a uniform magnetic field. only a torque. Yet magnetized materials that contain atomic current loops certainly do experience net forces in magnetic fields. How is this discrepancy resolved?

Problem 21E (a) How large a current would a very long, straight wire have to carry so that the magnetic field 2.00 cm from the wire is equal to 1.00 G (comparable to the earth’s northward-pointing magnetic field)? (b) If the wire is horizontal with the current running from east to west, at what locations would the magnetic field of the wire point in the same direction as the horizontal component of the earth’s magnetic field? (c) Repeat part (b) except the wire is vertical with the current going upward.

Problem 22DQ Show that the units A ? m2 and J/T for the Bohr magneton are equivalent.

Problem 22E Two long, straight wires, one above the other, are separated by a distance 2a and are parallel to the x-axis. Let the +y-axis be in the plane of the wires in the direction from the lower wire to the upper wire. Each wire carries current I in the +x-direction. What are the magnitude and direction of the net magnetic field of the two wires at a point in the plane of the wires (a) midway between them; (b) at a distance a above the upper wire; (c) at a distance a below the lower wire?

A long, straight wire lies along the \(y\)-axis and carries a current \(I=8.00\) A in the \(?y\)-direction (Fig. E28.23). In addition to the magnetic field due to the current in the wire, a uniform magnetic field \(\vec{B}_{0}\) with magnitude \(1.50 \times 10^{-6}\) ???? is in the \(+x\)-direction. What is the total field (magnitude and direction) at the following points in the \(x z\)-plane: (a) \(x=0\), \(z=1.00\) m; (b) \(x=1.00\) m, \(z=0\); (c) \(x=0, z=-0.25\)? Equation Transcription: Text Transcription: y I=8.00 -y Vector B_0 1.50x10^-6 +x xz x=0 z=1.00 x=1.00 z=0 x=0,z=-0.25 Vector B_0

Problem 24E BIO Transmission Lines and Health. Currents in dc transmission lines can be 100 A or higher. Some people are concerned that the electromagnetic fields from such lines near their homes could pose health dangers. For a line that has current 150 A and a height of 8.0 m above the ground, what magnetic field does the line produce at ground level? Express your answer in teslas and as a percentage of the earth’s magnetic field, which is 0.50 G. Is this value cause for worry?

Two long, straight, parallel wires, 10.0 cm apart, carry equal 4.00-A currents in the same direction, as shown in Fig. E28.25. Find the magnitude and direction of the magnetic field at (a) point \(P_{1}\), midway between the wires; (b) point \(P_{2}\), 25.0 cm to the right of \(P_{1}\); (c) point \(P_{3}\), 20.0 cm directly above \(P_{1}\). Equation Transcription: Text Transcription: P_1 P_2 P_3 P_1

Problem 26E A rectangular loop with dimensions 4.20 cm by 9.50 cm carries current I. The current in the loop produces a magnetic field at the center of the loop that has magnitude 5.50 × 10-5 T and direction away from you as you view the plane of the loop. What are the magnitude and direction (clockwise or counterclockwise) of the current in the loop?

Four, long, parallel power lines each carry 100-A currents. A cross-sectional diagram of these lines is a square, 20.0 cm on each side. For each of the three cases shown in Fig. E28.27, calculate the magnetic field at the center of the square.

Four very long, current-carrying wires in the same plane intersect to form a square 40.0 cm on each side, as shown in Fig. E28.28. Find the magnitude and direction of the current I so that the magnetic field at the center of the square is zero.

Two insulated wires perpendicular to each other in the same plane carry currents as shown in Fig. E28.29. Find the magnitude of the net magnetic field these wires produce at points P and ???? if the 10.0 A-current is (a) to the right or (b) to the left.

Three parallel wires each carry current I in the directions shown in Fig. E28.30. If the separation between adjacent wires is d, calculate the magnitude and direction of the net magnetic force per unit length on each wire.

Two long, parallel wires are separated by a distance of 0.400 m (Fig. E28.31). The currents \(I_{1}\) and \(I_{2}\) have the directions shown. (a) Calculate the magnitude of the force exerted by each wire on a 1.20-m length of the other. Is the force attractive or repulsive? (b) Each current is doubled, so that \(I_{1}\) becomes 10.0 A and \(I_{2}\) becomes 4.00 A. Now what is the magnitude of the force that each wire exerts on a 1.20-m length of the other? Equation Transcription: Text Transcription: I_1 I_2 I_1 I_2 I_1=5.00 A I_2=2.00 A

Problem 32E Two long, parallel wires are separated by a distance of 2.50 cm. The force per unit length that each wire exerts on the other is 4.00 × 10-5 N/m, and the wires repel each other. The current in one wire is 0.600 A. (a) What is the current in the second wire? (b) Are the two currents in the same direction or in opposite directions?

Problem 33E Lamp Cord Wires. The wires in a household lamp cord are typically 3.0 mm apart center to center and carry equal currents in opposite directions. If the cord carries direct current to a 100-W light bulb connected across a 120-V potential difference, what force per meter does each wire of the cord exert on the other? Is the force attractive or repulsive? Is this force large enough so it should be considered in the design of the lamp cord? (Model the lamp cord as a very long straight wire.)

A long, horizontal wire AB rests on the surface of a table and carries a current I. Horizontal wire CD is vertically above wire AB and is free to slide up and down on the two vertical metal guides C and D (Fig. E28.34). Wire CD is connected through the sliding contacts to another wire that also carries a current I, opposite in direction to the current in wire AB. The mass per unit length of the wire CD is \(\lambda\). To what equilibrium height h will the wire CD rise, assuming that the magnetic force on it is due entirely to the current in the wire AB? Equation Transcription: Text Transcription: Lambda

Problem 35E BIO Currents in the Brain. The magnetic field around the head has been measured to be approximately 3.0 × 10-8 G. Although the currents that cause this field are quite complicated, we can get a rough estimate of their size by modeling them as a single circular current loop 16 cm (the width of a typical head) in diameter. What is the current needed to produce such a field at the center of the loop?

Figure E28.42 shows, in cross sectio, several conductors that carry currents through the plane of the figure. The currents have the magnitudes \(I_{1}=4.0\) A, \(I_{2}=6.0\) A, and \(I_{3}=2.0\) A, and the directions shown. Four paths, labeled through d, are shown. What is the line integral \(\oint \vec{B} \cdot \overrightarrow{d l}\) each path? Each integral involves going around the path in the counterclockwise direction. Explain your answers. Equation Transcription: Text Transcription: I_1=4.0 I_2=6.0 I_3=2.0 ointegral vector B cdot dvector l I_1 I_2 I_3

A closed curve encircles several conductors. The line integral \(\oint \vec{B} \cdot \overrightarrow{d l}\) around this curve is \(3.83 \times 10^{-4} \mathrm{~T} \cdot \mathrm{m}\) (a) What is the net current in the conductors? (b) If you were to integrate around the curve in the opposite direction, what would be the value of the line integral? Explain. Equation Transcription: Text Transcription: ointegral vector B cdot vector l 3.83x10^-4 T cdot m

Problem 44E As a new electrical technician, you are designing a large solenoid to produce a uniform 0.150-T magnetic field near the center of the solenoid. You have enough wire for 4000 circular turns. This solenoid must be 1.40 m long and 2.80 cm in diameter. What current will you need to produce the necessary field?

Problem 48E A 15.0-cm-long solenoid with radius 0.750 cm is closely wound with 600 turns of wire. The current in the windings is 8.00 A. Compute the magnetic field at a point near the center of the solenoid.

Problem 49E A solenoid is designed to produce a magnetic field of 0.0270 T at its center. It has radius 1.40 cm and length 40.0 cm, and the wire can carry a maximum current of 12.0 A. (a) What minimum number of turns per unit length must the solenoid have? (b) What total length of wire is required?

Problem 50E A toroidal solenoid has an inner radius of 12.0 cm and an outer radius of 15.0 cm. It carries a current of 1.50 A. How many equally spaced turns must it have so that it will produce a magnetic field of 3.75 mT at points within the coils 14.0 cm from its center?

Problem 51E A magnetic field of 37.2 T has been achieved at the MIT Francis Bitter National Magnetic Laboratory. Find the current needed to achieve such a field (a) 2.00 cm from a long, straight wire; (b) at the center of a circular coil of radius 42.0 cm that has 100 turns; (c) near the center of a solenoid with radius 2.40 cm, length 32.0 cm, and 40,000 turns.

A toroidal solenoid (see Example 28.10) has inner radius \(r_{1}=15.0 \mathrm{~cm}\) and outer radius \(r_{2}=18.0 \mathrm{~cm}\). The solenoid has 250 turns and carries a current of 8.50 A. What is the magnitude of the magnetic field at the following distances from the center of the torus: (a) 12.0 cm; (b) 16.0 cm; (c) 20.0 cm? Equation Transcription: Text Transcription: r_1=15.0 cm r_2=18.0 cm

Problem 53E A wooden ring whose mean diameter is 14.0 cm is wound with a closely spaced toroidal winding of 600 turns. Compute the magnitude of the magnetic field at the center of the cross section of the windings when the current in the windings is 0.650 A.

Problem 54E A toroidal solenoid with 400 turns of wire and a mean radius of 6.0 cm carries a current of 0.25 A. The relative permeability of the core is 80. (a) What is the magnetic field in the core? (b) What part of the magnetic field is due to atomic currents?

Problem 55E A toroidal solenoid with 500 turns is wound on a ring with a mean radius of 2.90 cm. Find the current in the winding that is required to set up a magnetic field of 0.350 T in the ring (a) if the ring is made of annealed iron (Km = 1400) and (b) if the ring is made of silicon steel (Km = 5200).

Problem 56E The current in the windings of a toroidal solenoid is 2.400 A. There are 500 turns, and the mean radius is 25.00 cm. The toroidal solenoid is filled with a magnetic material. The magnetic field inside the windings is found to be 1.940 T. Calculate (a) the relative permeability and (b) the magnetic susceptibility of the material that fills the toroid.

At a particular instant, charge \(q_{1}=+4.80 \times 10^{-6}\) C is at the point0, (0.250 m, 0) and has velocity \(\vec{v}_{1}=\left(9.20 \times 10^{5} \mathrm{~m} / \mathrm{s}\right) \hat{i}\). Charge \(q_{2}=-2.90 \times 10^{-6}\) C is at the point (0.150 m, 0, 0) and has velocity \(\vec{v}_{2}=\left(-5.30 \times 10^{5} \mathrm{~m} / \mathrm{s}\right) \hat{j}\) this instant, what are the magnitude and direction of the magnetic force that \(q_{1}\) exerts on \(q_{2}\)? Equation Transcription: Text Transcription: q_1=+4.80x10^-6 Vector v_1=(9.20x10^5 m/s)i hat q_2=-2.90x10^-6 Vector v_2=(-5.30x10^5 m/s)j hat q_1 q_2

Problem 61P Two long, parallel transmission lines, 40.0 cm apart, carry 25.0-A and 75.0-A currents. Find all locations where the net magnetic field of the two wires is zero if these currents are in (a) the same direction and (b) the opposite direction.

Problem 62P A long, straight wire carries a current of 5.20 A. An electron is traveling in the vicinity of the wire. At the instant when the electron is 4.50 cm from the wire and traveling with a speed of 6.00 × 104 m/s directly to word the wire, what are the magnitude and direction (relative to the direction of the current) of the force that the magnetic field of the current exerts on the electron?

Problem 63P CP A long, straight wire carries a 13.0-A current. An electron is fired parallel to this wire with a velocity of 250 km/s in the same direction as the current, 2.00 cm from the wire. (a) Find the magnitude and direction of the electron’s initial acceleration. (b) What should be the magnitude and direction of a uniform electric field that will allow the electron to continue to travel parallel to the wire? (c) Is it necessary to include the effects of gravity? Justify your answer.

Two very long, straight wires carry currents as shown in Fig. P28.64. For each case, find all locations where the net magnetic field is zero.

Two identical circular, wire loops 40.0 cm in diameter each carry a current of 3.80 A in the same direction. These loops are parallel to each other and are 25.0 cm apart. Line ab is normal to the plane of the loops and passes through their centers. A proton is fired at 2400 km/s perpendicular to line ab from a point midway between the centers of the loops. Find the magnitude of the magnetic force these loops exert on the proton just after it is fired.

The long, straight wire AB shown in Fig. P28.72 carries a current of 14.0 A. The rectangular loop whose long edges are parallel to the wire carries a current of 5.00 A. Find the magnitude and direction of the net force exerted on the loop by the magnetic field of the wire. Equation Transcription: Text Transcription: I=14.0 A I=5.00 A

Problem 73P A flat, round iron ring 5.00 cm in diameter has a current running through it that produces a magnetic field of 75.4 ?T, its center. This ring is placed in a uniform external magnetic field of 0.375 T. What is the maximum torque the external field can exert on the ring? Show how the ring should be oriented relative to the field for the torque to have its maximum value.

The wire semicircles shown in Fig. P28.74 have radii a and b. Calculate the net magnetic field (magnitude and direction) that the current in the wires produces at point P.

CALC The wire shown in Fig. P28.78 is infinitely long and carries a current I. Calculate the magnitude and direction of the magnetic field that this current produces at point P.

Problem 79P Problem A conductor is made in the form of a hollow cylinder with inner and outer radii a and b. respectively. It carries a current I Uniformly distributed over its cross section. Derive expressions for the magnitude of the magnetic field in the regions (a) r < a; (b) a < r < b; (c) r > b.

A circular loop has radius R and carries current \(I_{2}\) in a clockwise direction (Fig. P28.80). The center of the loop is a distance D above a long, straight wire. What are the magnitude and direction of the current \(I_{1}\) in the wire if the magnetic field at the center of the loop is zero? Equation Transcription: Text Transcription: I_2 I_1 I_2 I_1

CALC A long, straight, solid cylinder, oriented with its axis in the \(z\)-direction, carries a current whose current density is \(\vec{j}\)The current density, although symmetric about the cylinder axis, is not constant but varies according to the relationship \(\vec{\jmath}=\frac{2 I_{0}}{\pi a^{2}}\left[1-\left(\frac{r}{\alpha}\right)^{2}\right] \hat{k} \text { for } r \leq \alpha=0\) for \(r \geq \alpha\) Where is the radius of the cylinder, \(r\) is the radial distance from the cylinder axis, and \(I_{0}\) is a constant having units of amperes. (a) Show that \(I_{0}\) is the total current passing through the entire cross section of the wire. (b) Using Ampere’s law, derive an expression for the magnitude of the magnetic field \(\vec{B}\) in the region \(r \geq \alpha\). (c) Obtain an expression for the current \(I\) contained in a circular cross section of radius \(r \leq \alpha\) and centered at the cylinder axis. (d) Using Ampere’s law, derive an expression for the magnitude of the magnetic field in the region \(r \leq \alpha\).How do your results in parts (b) and (d) compare for \(r=\alpha\) ? Equation Transcription: r = 0 r r r Text Transcription: z vec j vec j= 2I_0/pi alpha^²[1-(r/alpha)^2]hat k r leq alpha = 0 r geq alpha I I_0 r leq alpha vec B r = alpha

A long, straight, solid cylinder, oriented with its axis in the \(z\)-direction, carries a current whose current density is \(\vec{J}\).The current density, although symmetric about the cylinder axis, is not constant and varies according to the relationship \(\vec{J}=\left(\frac{b}{r}\right) e^{(r-a) / \delta_{\widehat{k}}} \text { for } r \leq a\) \(=0 \text { for } r \geq a\) where the radius of the cylinder is \(a=5.00 \mathrm{~cm}\), r is the radial distance from the cylinder axis, b is a constant equal to 600 A/m, and \(\delta\) is a constant equal to 2.50 cm. (a) Let be the total \(I_{0}\) current passing through the entire cross section of the wire. Obtain an expression for \(I_{0}\) in terms of \(b\), \(\delta\), and \(a\). Evaluate your expression to obtain a numerical value for \(I_{0}\). (b) Using Ampere’s law, derive an expression for the magnetic field \(\vec{B}\) in the region \(r \geq a\). Express your answer in terms of \(I_{0}\) rather than \(b\). (c) Obtain an expression for the current I contained in a circular cross section of radius \(r \leq a\) and centered at the cylinder axis. Express your answer in terms of \(I_{0}\) rather than \(b\). (d) Using Ampere’s law, derive an expression for the magnetic field \(\vec{B}\) in the region \(r \leq a\). (e) Evaluate the magnitude of the magnetic field at \(r=\delta a, r=a\), \(r=2 a\). Equation Transcription: Text Transcription: z Vector J Vector J=(b over r)e^(r-a)over delta_k hat for </= a =0 for r >/= a a=5.00 cm Delta I_0 I_0 Delta b a I_0 Vector B r >/= a I_0 b r </=a Vector B r </=a r=delta,r=a r=2a

An Infinite Current Sheet. Long, straight conductors with square cross sections and each carrying current I are laid side by side to form an infinite current sheet (Fig. P28.83). The conductors lie in the \(x y\)-plane, are parallel to the \(y\)-axis, and carry current in the \(+y\)-direction. There are n conductors per unit length measured along the \(x\)-axis. (a) What are the magnitude and direction of the magnetic field a distance \(a\) below the current sheet? (b) What are the magnitude and direction of the magnetic field a distance \(a\) above the current sheet? Equation Transcription: Text Transcription: xy y +y x a a

Long, straight conductors with square cross section, each carrying current I, are laid side by side to form an infinite current sheet with current directed out of the plane of the page (Fig. P28.84). A second infinite current sheet is a distance d below the first and is parallel to it. The second sheet carries current into the plane of the page. Each sheet has n conductors per unit length. (Refer to Problem 28.83.) Calculate the magnitude and direction of the net magnetic field at (a) point P (above the upper sheet); (b) point R (midway between the two sheets); (c) point S (below the lower sheet).

CP A piece of iron has magnetization \(M=6.50 \times 10^{4} \mathrm{~A} / \mathrm{m}\). Find the average magnetic dipole moment per atom in this piece of iron. Express your answer both in \(\mathrm{A} \cdot \mathrm{m}^{2}\) and in Bohr magnetons. The density of iron is given in Table 14.1, and the atomic mass of iron (in grams per mole) is given in Appendix D. The chemical symbol for iron is \(\mathrm{Fe}\). Equation Transcription: Text Transcription: M=6.50x10^4 A/m A cdot m^2 Fe

A wide, long, insulating belt has a uniform positive charge per unit area \(\sigma\) on its upper surface. Rollers at each end move the belt to the right at a constant speed \(v\). Calculate the magnitude and direction of the magnetic field produced by the moving belt at a point just above its surface. (Hint: At points near the surface and far from its edges or ends, the moving belt can be considered to be an infinite current sheet like that in Problem 28.83.) Equation Transcription: Text Transcription: Sigma v

CALC A wire in the shape of a semicircle with radius \(a\) is oriented in the \(y z\)-plane with its center of curvature at the origin (Fig. P28.88). If the current in the wire is I, calculate the magnetic-field components produced at point P, a distance \(x\) out along the \(x\)-axis. (Note: Do not forget the contribution from the straight wire at the bottom of the semicircle that runs from \(z=-a\) to \(z=+a\).You may use the fact that the fields of the two antiparallel currents at \(z>a\) cancel, but you must explain why they cancel.) Equation Transcription: Text Transcription: a yz x x z=-a z=+a z>a

Problem 1DQ A topic of current interest in physics research is the search (thus far unsuccessful) for an isolated magnetic pole, or magnetic monopole. If such an entity were found, how could it be recognized? What would its properties be?

\(A+6.00-\mu C\) point charge is moving at a constant \(8.00 \times 10^{6} \mathrm{~m} / \mathrm{s}\) in the -direction, relative to a reference frame. At the instant when the point charge is at the origin of this reference frame, what is the magnetic-field vector \(\bar{B}\) it produces at the following points: (a) (b) (c) (d) Equation transcription: Text transcription: A+6.00-\mu C 8.00 \times 10^{6}{~m} /{s} bar{B}

The streams of changed particles emitted from the sun will have their own magnetic field around them. So their magnetic field will interact with earth’s magnetic field and create disturbance.

Problem 4DQ Two parallel conductors carrying current in the same direction attract each other. If they are permitted to move toward each other, the forces of attraction do work. From where does the energy come? Does this contradict the assertion in Chapter 27 that magnetic forces on moving charges do no work? Explain.

Problem 10DQ What are the relative advantages and disadvantages of Ampere’s law and the law of Biot and Savart for practical calculations of magnetic fields?

Problem 5DQ Pairs of conductors carrying current into or out of the power-supply components of electronic equipment are sometimes twisted together to reduce magnetic-field effects. Why does this help?

An alpha particle \((\text { charge }+2 e)\) and an electron move in opposite directions from the same point, each with the speed of \(2.50\times10^5\mathrm{\ m}/\mathrm{s}\) (Fig. E28.4). Find the magnitude and direction of the total magnetic field these charges produce at point P, which is 1.75 nm from each of them.

A short current element \(\overline{d I}=(0.500 \mathrm{~mm}) \hat{J}\) carries a current of in the same direction as \(\overline{d I}\). Point is located at \(\bar{r}=(-0.730 \mathrm{~m}) \widehat{\mathrm{i}}+(0.390 \mathrm{~m}) \hat{\mathcal{K}}\). Use unit vectors to express the magnetic field at produced by this current element. Equation transcription: Text transcription: \overline{d I}=(0.500 \mathrm{~mm}) \hat{J} \overline{d I} \bar{r}=(-0.730{~m}) \widehat{{i}}+(0.390{~m}) \hat{{K}}

Problem 11DQ Magnetic field lines never have a beginning or an end. Use this to explain why it is reasonable for the field of an ideal toroidal solenoid to be confined entirely to its interior, while a straight solenoid must have some field outside.

Two parallel wires are 5.00 cm apart and carry currents in opposite directions, as shown in Fig. E28.14. Find the magnitude and direction of the magnetic field at point P due to two 1.50-mm segments of wire that are opposite each other and each 8.00" " cm from P.

Problem 15DQ A metal ring carries a current that causes a magnetic field B0 at the center of the ring and a field B at point P a distance x from the center along the axis of the ring. If the radius of the ring is doubled, find the magnetic field at the center. Will the field at point P change by the same factor? Why?

A wire carrying a 28.0-A current bends through a right angle. Consider two segments of wire, each from the bend (Fig. E28.15). Find the magnitude and direction of the magnetic field these two segments produce at point , which is midway between them.

Problem 16DQ Why should the permeability of a paramagnetic material be expected to decrease with increasing temperature?

A square wire loop 10.0 cm on each side carries a clockwise current of 15.0 A. Find the magnitude and direction of the magnetic field at its center due to the four 1.20-mm wire segments at the midpoint of each side.

Problem 17DQ If a magnet is suspended over a container of liquid air, it attracts droplets to its poles. The droplets contain only liquid oxygen; even though nitrogen is the primary constituent of air, it is not attracted to the magnet. Explain what this tells you about the magnetic susceptibilities of oxygen and nitrogen, and explain why a magnet in ordinary, room-temperature air doesn’t attract molecules of oxygen gas to its poles.

Problem 19DQ The magnetic susceptibility of paramagnetic materials is quite strongly temperature dependent, but that of diamagnetic materials is nearly independent of temperature. Why the difference?

Problem 19E BIO Currents in the Heart. The body contains many small currents caused by the motion of ions in the organs and cells. Measurements of the magnetic field around the chest due to currents in the heart give values of about 10 µG. Although the actual currents are rather complicated, we can gain a rough understanding of their magnitude if we model them as a long, straight wire. If the surface of the chest is 5.0 cm from this current, how large is the current in the heart?

Problem 20DQ A cylinder of iron is placed so that it is free to rotate around its axis. Initially the cylinder is at rest, and a magnetic field is applied to the cylinder so that it is magnetized in a direction parallel to its axis. If the direction of the external field is suddenly reversed, the direction of magnetization will also reverse and the cylinder will begin rotating around its axis. (This is called the Einstein–de Haas effect.) Explain why the cylinder begins to rotate.

Calculate the magnitude and direction of the magnetic field at point due to the current in the semicircular section of wire shown in Fig. E28.36. (Hint: Does the current in the long, straight section of the wire produce any field at )

Calculate the magnitude of the magnetic field at point P of Fig. E28.37 in terms of R, \(I_{1}, \text { and } I_{2}\). What does your expression give when \(I_{1}=I_{2}\)?

Problem 38E A closely wound, circular coil with radius 2.40 cm has 800 turns. (a) what must the current in the coil be if the magnetic field at the center of the coil is 0.0580 T? (b)At what distance from the centre of the coil on the axis of the coil, is the magnetic field half its value at the center?

Problem 39E A closely wound, circular coil with a diameter of 4.00 cm has 600 turns and carries a current of 0.500 A. What is the magnitude of the magnetic field. (a) At the center of the coil and (b) At a point on the axis of the coil is 8.00 cm from its center?

Problem 40E A closely wound coil has a radius of 6.00 cm and carries a current of 2.50 A. How many turns must it have if, at a point on the coil axis 6.00 cm from the center of the coil, the magnetic field is 6.39 × 10-4 T?

Problem 41E Two concentric circular loops of wire lie on a tabletop, one inside the other. The inner wire has a diameter of 20.0 cm and carries a clockwise current of 12.0 A, as viewed from above, and the outer wire has a diameter of 30.0 cm. What must be the magnitude and direction (as viewed from above) of the current in the outer wire so that the net magnetic field due to this combination of wires is zero at the common center of the wires?

Coaxial Cable. A solid conductor with radius is supported by insulating disks on the axis of a conducting tube with inner radius and outer radius (Fig. E28.45). The central conductor and tube carry equal currents in opposite directions. The currents are distributed uniformly over the cross sections of each conductor. Derive an expression for the magnitude of the magnetic field (a) at points outside the central, solid conductor but inside the tube and at points outside the tube .

Repeat Exercise for the case in which the current in the central, solid conductor is \(I_{1}\), the current in the tube is \(I_{2}\), and these currents are in the same direction rather than in opposite directions. Equation transcription: Text transcription: I{1} I{2}

Problem 47E A long straight, cylindrical wire of radius R carries a current uniformly distributed over its cross section. At what locations is the magnitic field produced by this current is equal to half of its largest value? Consider points inside and outside the wire.

A long solenoid with 60 turns of wire per centimeter carries a current of 0.15 A. The wire that makes up the solenoid is wrapped around a solid core of silicon steel (\(K_{\mathrm{m}}=5200\)). (The wire of the solenoid is jacketed with an insulator so that none of the current flows into the core.) (a) For a point inside the core, find the magnitudes of (i) the magnetic field \(\overrightarrow{\boldsymbol{B}}_{0}\) due to the solenoid current; (ii) the magnetization \(\vec{M}\); (iii) the total magnetic field \(\vec{B}\). (b) In a sketch of the solenoid and core, show the directions of the vectors \(\overrightarrow{\boldsymbol{B}}, \overrightarrow{\boldsymbol{B}}_{0} \text {, and } \vec{M}\) inside the core.

Problem 58E When a certain paramagnetic material is placed in an external magnetic field of 1.5000 T, the field inside the material is measured to be 1.5023 T. Find (a) the relative permeability and (b) the magnetic permeability of this material.

A pair of point charges, \(q=+8.00 \mu C\) and \(q^{i}=-5.00 \mu C\), are moving as shown in Fig. P28.59 with speeds \(v=9.00 \times 10^{4} \mathrm{~m} / \mathrm{s}\) and \(v^{i}=650 \times 10^{4} \mathrm{~m} / \mathrm{s}\). When the charges are at the locations shown in the figure, what are the magnitude and direction of (a) the magnetic field produced at the origin and (b) the magnetic force that exerts on ? Equation transcription: Text transcription: q=+8.00 \mu C q^{i}=-5.00 \mu C v=9.00 \times 10^{4}{~m} /{s} v^{i}=650 \times 10^{4}{~m} /{s}

A negative point charge \(q=-7.20\mathrm{\ mC}\) is moving in a reference frame. When the point charge is at the origin, the magnetic field it produces at the point \(x=25.0\mathrm{\ cm},\ y=0,z=0\) is \(\overrightarrow{\boldsymbol{B}}=(6.00 \mu \mathrm{T}) \hat{\boldsymbol{J}}\), and its speed is \(800\mathrm{\ m}/\mathrm{s}\). (a) What are the \(x-, y-\), and z-components of the velocity \(\overrightarrow{\boldsymbol{v}}_0\) of the charge? (b) At this same instant, what is the magnitude of the magnetic field that the charge produces at the point \(x=0,\ y=25.0\mathrm{\ cm},\ z=0\)?

Two long, straight, parallel wires are apart (Fig. ). The wire on the left carries a current \(I_{1}\) of into the plane of the paper. (a) What must the magnitude and direction of the current \(I_{2}\) be for the net field at point to be zero? (b) Then what are the magnitude and direction of the net field at (c) Then what is the magnitude of the net field at ? Equation transcription: Text transcription: I{1} I{2}

Figure P28.68 shows an end view of two long, parallel wires perpendicular to the plane, each carrying a current but in opposite directions. (a) Copy the diagram, and draw vectors to show the \(\vec{B}\) field of each wire and the net \(\vec{B}\ field at point . (b) Derive the expression for the magnitude of \(\vec{B}\ at any point on the -axis in terms of the -coordinate of the point. What is the direction of \(\vec{B}\? (c) Graph the magnitude of \(\vec{B}\ at points on the -axis. (d) At what value of is the magnitude of a maximum? (e) What is the magnitude of \(\vec{B}\ when ? Equation transcription: Text transcription: \vec{B}

Refer to the situation in Problem 28.68. Suppose that a third long, straight wire, parallel to the other two, passes through point P (see Fig. P28.68) and that each wire carries a current I = 6.00 A. Let a = 40.0 cm and x = 60.0 cm. Find the magnitude and direction of the force per unit length on the third wire, (a) if the current in it is directed into the plane of the figure, and (b) if the current in it is directed out of the plane of the figure.

CP A pair of long, rigid metal rods, each of length , lie parallel to each other on a perfectly smooth table. Their ends are connected by identical, very light conducting springs of force constant (Fig. P28.70) and negligible unstretched length. If a current runs through this circuit, the springs will stretch. At what separation will the rods remain at rest? Assume that is large enough so that the separation of the rods will be much less than .

CP Two long, parallel wires hang by -cm-long cords from a common axis (Fig. P28.71). The wires have a mass per unit length of and carry the same current in opposite directions. What is the current in each wire if the cords hang at an angle of with the vertical?

CALC Helmholtz Coils. Figure is a sectional view of two circular coils with radius , each wound with turns of wire carrying a current , circulating in the same direction in both coils. The coils are separated by a distance equal to their radii. In this configuration the coils are called Helmholtz coils; they produce a very uniform magnetic field in the region between them. (a) Derive the expression for the magnitude of the magnetic field at a point on the axis a distance to the right of point , which is midway between the coils. (b) Graph versus for to . Compare this graph to one for the magnetic field due to the right-hand coil alone. (c) From part (a), obtain an expression for the magnitude of the magnetic field at point (d) Calculate the magnitude of the magnetic field at if turns, , and (e) Calculate and \(d^{2} B / d x^{2}\) at Discuss how your results show that the field is very uniform in the vicinity of . Equation transcription: Text transcription: d^{2} B / d x^{2}

A circular wire of diameter D lies on a horizontal table and carries a current I. In Fig. 28.76 point A marks the center of the circle and point C is on its rim. (a) Find the magnitude and direction of the magnetic field at point A. (b) The wire is now unwrapped so it is straight, centered on point C, and perpendicular to the line AC, but the same current is maintained in it. Now find the magnetic field at point A. (c) Which field is greater: the one in part (a) or in part (b)? By what factor? Why is this result physically reasonable?

Problem 77P Problem CALC A long, straight wire with a circular cross section of radius R carries a current I. Assume that the current density is not constant across the cross section of the wire, but rather varies as J = ?r, where ? is a constant. (a) By the requirement that J integrated over the cross section of the wire gives the total current I, calculate the constant ? in terms of I and R. (b) Use Ampere’s law to calculate the magnetic field B(r) for (i) r ? R and (ii) r ? R. Express your answers in terms of I.

CP Two long, straight conducting wires with linear mass density \(\lambda\) are suspended from cords so that they are each horizontal,parallel to each other, and a distance \(d\) apart. The backends of the wires are connected to each other by a slack, low resistance connecting wire. A charged capacitor (capacitance \(C\)) is now added to the system; the positive plate of the capacitor (initial charge \(+Q_{0}\)) is connected to the front end of one of the wires, and the negative plate of the capacitor (initial charge \(-Q_{0}\)) is connected to the front end of the other wire (Fig. P28.87). Both of these connections are also made by slack, low-resistance wires. When the connection is made, the wires are pushed aside by the repulsive force between the wires, and each wire has an initial horizontal velocity of magnitude \(v_{0}\). Assume that the time constant for the capacitor to discharge is negligible compared to the time it takes for any appreciable displacement in the position of the wires to occur. (a) Show That the initial speed \(v_{0}\) of either wire is given by \(\nu_{0}=\frac{\mu o Q o^{2}}{4 \pi \lambda R C d}\) where R is the total resistance of the circuit. (b) To what height h will each wire rise as a result of the circuit connection? Equation Transcription: Text Transcription: \lambda d C\ +Q_{0 -Q_{0} v_{0} \nu_{0}=\frac{\mu o Q o^{2}}{4 \pi \lambda R C d}