Answer: The current in the windings of a toroidal solenoid

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?

Streams of charged particles emitted from the sun during periods of solar activity create a disturbance in the earths magnetic field. How does this happen?

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?

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

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?

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.

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 didnt we use the total magnetic field due to both conductors?

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.

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

What are the relative advantages and disadvantages of Amperes law and the law of Biot and Savart for practical calculations of magnetic fields?

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.

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.

In the circuit shown in Fig. Q28.13, 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?

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?

Show that the units \(\mathrm{A} \cdot \ \mathrm{m}^2\) and J/T for the Bohr magneton are equivalent.

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

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 doesnt attract molecules of oxygen gas to its poles.

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 doesnt attract molecules of oxygen gas to its poles.

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

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 Einsteinde Haas effect.) Explain why the cylinder begins to rotate.

A +6.00@mC point charge is moving at a constant 8.00 * 106 m>s in the +y-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 B S 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, y = 0, z = +0.500 m; (d) x = 0, y = -0.500 m, z = +0.500 m?

A +6.00@mC point charge is moving at a constant 8.00 * 106 m>s in the +y-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 B S 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, y = 0, z = +0.500 m; (d) x = 0, y = -0.500 m, z = +0.500 m?

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 mm from the electron: (a) points A and B; (b) point C; (c) point D

An alpha particle (charge +2e) and an electron move in opposite directions from the same point, each with the speed of 2.50 * 105 m>s (Fig. E28.4). Find the magnitude and direction of the total magnetic field these charges produce at point P, which is 8.65 nm from each charge.

A -4.80@mC 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?

Positive point charges q = +8.00 mC and q = +3.00 mC 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 * 106 m>s, and v = 9.00 * 106 m>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 v S 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?

A negative charge q = -3.60 * 10-6 C is located at the origin and has velocity v S =17.50 * 104 m>s2nd +1-4.90 * 104 m>s2ne. 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?

An electron and a proton are each moving at 735 km>s in perpendicular paths as shown in Fig. E28.8. At the instant when they are at the positions shown, 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 straight wire carries a 10.0-A current (Fig. E28.9). 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 short current element dl S = 10.500 mm2ne carries a current of 5.40 A in the same direction as dl S . Point P is located at r S = 1-0.730 m2nd + 10.390 m2k n. Use unit vectors to express the magnetic field at P produced by this current element

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

Two parallel wires are 5.00 cm apart and carry currents in opposite directions, as shown in Fig. E28.12. 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. E28.13). 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 8.00 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 \(8.20 \times 10^{18}\) 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 10 mG. 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% of the earths field would have little effect on the bacteria. Take the earths field to be 5.0 * 10-5 T, 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 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.

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. E28.21). In addition to the magnetic field due to the current in the wire, a uniform magnetic field B S 0 with magnitude 1.50 * 10-6 T is in the +x-direction. What is the total field (magnitude and direction) at the following points in the xz-plane: (a) x = 0, z = 1.00 m; (b) x = 1.00 m, z = 0; (c) x = 0, z = -0.25 m?

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. E28.23. Find the magnitude and direction of the magnetic field at (a) point P1, midway between the wires; (b) point P2, 25.0 cm to the right of P1; (c) point P3, 20.0 cm directly above P1.

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. E28.25, calculate the magnetic field at the center of the square.

Four very long, currentcarrying wires in the same plane intersect to form a square 40.0 cm on each side, as shown in Fig. E28.26. Find the magnitude and direction of the current I so that the magnetic field at the center of the square is zero.

Two very long insulated wires perpendicular to each other in the same plane carry currents as shown in Fig. E28.27. 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 very long parallel wires each carry current I in the directions shown in Fig. E28.28. 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. E28.29). The currents I1 and I2 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 I1 becomes 10.0 A and I2 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 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?

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. E28.32). 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?

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?

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. E28.34. (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. E28.35 in terms of R, I1, and I2. What does your expression give when I1 = I2?

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 0.0770 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 single circular current loop 10.0 cm in diameter carries a 2.00-A current. (a) What is the magnetic field at the center of this loop? (b) Suppose that we now connect 1000 of these loops in series within a 500-cm length to make a solenoid 500 cm long. What is the magnetic field at the center of this solenoid? Is it 1000 times the field at the center of the loop in part (a)? Why or why not?

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 \(6.39 \times 10^{-4} \ \mathrm {T}\)?

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?

An alpha particle \((\text { charge }+2 e)\) and an electron move in opposite directions from the same point, each with the speed of \(2.50\times 10^5\mathrm{\ m}/\mathrm{s}\) (Fig. E28.4). Find the magnitude and direction of the total magnetic field these charges produce at point P, which is 8.65 nm from each charge.

A closed curve encircles several conductors. The line integral AB S ~ dl S around this curve is 3.83 * 10-4 T # 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.

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 55.0 cm 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. E28.43). 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 1a 6 r 6 b2 and (b) at points outside the tube 1r 7 c2.

Repeat Exercise 28.43 for the case in which the current in the central, solid conductor is I1, the current in the tube is I2, and these currents are in the same direction rather than in opposite directions.

A solenoid that is 35 cm long and contains 450 circular coils 2.0 cm in diameter carries a 1.75-A current. (a) What is the magnetic field at the center of the solenoid, 1.0 cm from the coils? (b) Suppose we now stretch out the coils to make a very long wire carrying the same current as before. What is the magnetic field 1.0 cm from the wires center? Is it the same as that in part (a)? Why or why not?

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.

An ideal toroidal solenoid (see Example 28.10) has inner radius r1 = 15.0 cm and outer radius r2 = 18.0 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?

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 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 \(\overrightarrow{\boldsymbol{M}}\); (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 \(\overrightarrow{\boldsymbol{M}}\) inside the core.

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 pair of point charges, q = +8.00 mC and q = -5.00 mC, are moving as shown in Fig. P28.55 with speeds v = 9.00 * 104 m>s and v = 6.50 * 104 m>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 q exerts on q?

At a particular instant, charge q1 = +4.80 * 10-6 C is at the point 10, 0.250 m, 02 and has velocity v S 1 =19.20 * 105 m>s2nd. Charge q2 = -2.90 * 10-6 C is at the point 10.150 m, 0, 02 and has velocity v S 2 = 1-5.30 * 105 m>s2ne. At this instant, what are the magnitude and direction of the magnetic force that q1 exerts on q2?

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 8.60 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 at a speed of 6.00 * 104 m>s 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 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.

An electron is moving in the vicinity of a long, straight wire that lies along the x-axis. The wire has a constant current of 9.00 A in the -x-direction. At an instant when the electron is at point (0, 0.200 m, 0) and the electrons velocity is v S = 15.00 * 104 m>s2nd 13.00 * 104 m>s2ne, what is the force that the wire exerts on the electron? Express the force in terms of unit vectors, and calculate its magnitude.

An electric bus operates by drawing direct current from two parallel overhead cables, at a potential difference of 600 V, and spaced 55 cm apart. When the power input to the buss motor is at its maximum power of 65 hp, (a) what current does it draw and (b) what is the attractive force per unit length between the cables?

Figure P28.62 shows an end view of two long, parallel wires perpendicular to the xy-plane, each carrying a current I but in opposite directions. (a) Copy the diagram, and draw vectors to show the B S field of each wire and the net B S field at point P. (b) Derive the expression for the magnitude of B S at any point on the x-axis in terms of the x-coordinate of the point. What is the direction of B S ? (c) Graph the magnitude of B S at points on the x-axis. (d) At what value of x is the magnitude of B S a maximum? (e) What is the magnitude of B S when x W a?

Two long, straight, parallel wires are 1.00 m apart (Fig. P28.63). The wire on the left carries a current I1 of 6.00 A into the plane of the paper. (a) What must the magnitude and direction of the current I2 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?

The long, straight wire AB shown in Fig. P28.64 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.

Two long, parallel wires hang by 4.00-cm-long cords from a common axis (Fig. P28.65). 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 with the vertical?

The wire semicircles shown in Fig. P28.66 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 P28.67 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 x = 0 to x = a>2. Compare this graph to one for the magnetic field due to the righthand 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 N = 300 turns, I = 6.00 A, and a = 8.00 cm. (e) Calculate dB>dx and d2 B>dx2 at P1x = 02. Discuss how your results show that the field is very uniform in the vicinity of P

Calculate the magnetic field (magnitude and direction) at a point P due to a current I = 12.0 A in the wire shown in Fig. P28.68. Segment BC is an arc of a circle with radius 30.0 cm, and point P is at the center of curvature of the arc. Segment DA is an arc of a circle with radius 20.0 cm, and point P is at its center of curvature. Segments CD and AB are straight lines of length 10.0 cm each.

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 \(J=\alpha r\), where \(\alpha\) 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 \(\alpha\) in terms of I and R. (b) Use Ampere’s law to calculate the magnetic field B(r) for \(\text { (i) } r \leq R \text { and (ii) } r \geq R\). Express your answers in terms of I.

The wire shown in Fig. P28.70 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 long, straight, solid cylinder, oriented with its axis in the z-direction, carries a current whose current density is J S . The current density, although symmetric about the cylinder axis, is not constant but varies according to the relationship J S = 2I0 pa2 c 1 - a r a b 2 d k n for r a = 0 for r a where a is the radius of the cylinder, r is the radial distance from the cylinder axis, and I0 is a constant having units of amperes. (a) Show that I0 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 B S in the region r a. (c) Obtain an expression for the current I contained in a circular cross section of radius r a and centered at the cylinder axis. (d) Using Amperes law, derive an expression for the magnitude of the magnetic field B S in the region r a. How do your results in parts (b) and (d) compare for r = a?

A circular loop has radius R and carries current I2 in a clockwise direction (Fig. P28.72). The center of the loop is a distance D above a long, straight wire. What are the magnitude and direction of the current I1 in the wire if the magnetic field at the center of the loop is zero?

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. P28.73). 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. P28.74). 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 28.73.) 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).

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 \(\begin{aligned}\vec{J} &=\left(\frac{b}{r}\right) e^{(r-a) / \delta} \hat{\boldsymbol{k}} & & \text { for } r \leq a \\&=\mathbf{0} & & \text { for } r \geq a\end{aligned}\) 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 \(600 \mathrm{~A} / \mathrm{m}\), 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 \(\overrightarrow{\boldsymbol{B}}\) 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 \(\overrightarrow{\boldsymbol{B}}\) in the region \(r \leq a\). (e) Evaluate the magnitude of the magnetic field at \(r=\delta, r=a\), and r = 2a.

As a summer intern at a research lab, you are given a long solenoid that has two separate windings that are wound close together, in the same direction, on the same hollow cylindrical form. You must determine the number of turns in each winding. The solenoid has length L = 40.0 cm and diameter 2.80 cm. You let a 2.00-mA current flow in winding 1 and vary the current I in winding 2; both currents flow in the same direction. Then you measure the magnetic-field magnitude B at the center of the solenoid as a function of I. You plot your results as BL>m0 versus I. The graph in Fig. P28.76 shows the best-fit straight line to your data. (a) Explain why the data plotted in this way should fall close to a straight line. (b) Use Fig. P28.76 to calculate N1 and N2, the number of turns in windings 1 and 2. (c) If the current in winding 1 remains 2.00 mA in its original direction and winding 2 has I = 5.00 mA in the opposite direction, what is B at the center of the solenoid?

You use a teslameter (a Hall-effect device) to measure the magnitude of the magnetic field at various distances from a long, straight, thick cylindrical copper cable that is carrying a large constant current. To exclude the earths magnetic field from the measurement, you first set the meter to zero. You then measure the magnetic field B at distances x from the surface of the cable and obtain these data: x 1cm2 2.0 4.0 6.0 8.0 10.0 B 1mT2 0.406 0.250 0.181 0.141 0.116 (a) You think you remember from your physics course that the magnetic field of a wire is inversely proportional to the distance from the wire. Therefore, you expect that the quantity Bx from your data will be constant. Calculate Bx for each data point in the table. Is Bx constant for this set of measurements? Explain. (b) Graph the data as x versus 1>B. Explain why such a plot lies close to a straight line. (c) Use the graph in part (b) to calculate the current I in the cable and the radius R of the cable.

A pair of long, rigid metal rods, each of length 0.50 m, lie parallel to each other on a frictionless table. Their ends are connected by identical, very lightweight conducting springs with unstretched length l0 and force constant k (Fig. P28.78). When a current I runs through the circuit consisting of the rods and springs, the springs stretch. You measure the distance x each spring stretches for certain values of I. When I = 8.05 A, you measure that x = 0.40 cm. When I = 13.1 A, you find x = 0.80 cm. In both cases the rods are much longer than the stretched springs, so it is accurate to use Eq. (28.11) for two infinitely long, parallel conductors. (a) From these two measurements, calculate l0 and k. (b) If I = 12.0 A, what distance x will each spring stretch? (c) What current is required for each spring to stretch 1.00 cm?

Two long, straight conducting wires with linear mass density l 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, lowresistance connecting wire. A charged capacitor (capacitance C) is now added to the system; the positive plate of the capacitor (initial charge +Q0) is connected to the front end of one of the wires, and the negative plate of the capacitor (initial charge -Q0) is connected to the front end of the other wire (Fig. P28.79). 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 v0. 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 v0 of either wire is given by v0 = m0Q 2 0 4plRCd 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?

A wide, long, insulating belt has a uniform positive charge per unit area s 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 28.73.)

What current is needed in the wire so that the magnetic field experienced by the bacteria has a magnitude of \(150 \ \mu \mathrm{T}\)? (a) 0.095 A; (b) 0.12 A; (c) 0.30 A; (d) 14 A.

To use a larger sample, the experimenters construct a solenoid that has the same length, type of wire, and loop spacing but twice the diameter of the original. How does the maximum possible magnetic torque on a bacterium in this new solenoid compare with the torque the bacterium would have experienced in the original solenoid? Assume that the currents in the solenoids are the same. The maximum torque in the new solenoid is (a) twice that in the original one; (b) half that in the original one; (c) the same as that in the original one; (d) one-quarter that in the original one.

The solenoid is removed from the enclosure and then used in a location where the earths magnetic field is 50 mT and points horizontally. A sample of bacteria is placed in the center of the solenoid, and the same current is applied that produced a magnetic field of 150 mT in the lab. Describe the field experienced by the bacteria: The field (a) is still 150 mT; (b) is now 200 mT; (c) is between 100 and 200 mT, depending on how the solenoid is oriented; (d) is between 50 and 150 mT, depending on how the solenoid is oriented.