P6.28.f^Chapter 28 Ending^914^924^,,^38460^38990%
WHAT WE HAVE LEARNED
EXAM STUDY GUIDE
KEY TERMS
Biot-Savart Law, p. 894 magnetic permeability of free space, p. 894 Ampere's Law, p. 903 Amperian loop, p. 903 Helmholtz coil, p. 904 solenoid, p. 905 toroid, p. 907 magnetization, p. 910 magnetic field strength, p. 911 magnetic susceptibility, p. 911 diamagnetism, p. 912 paramagnetism, p. 912 relative magnetic permeability, p. 912 ferromagnetism, p. 912 domain, p. 912
NEW SYMBOLS AND EQUATIONS

μ0 = 4π · 10−7 T m/A, magnetic permeability of free space

, vector direction of integration in Ampere's Law

ienc, enclosed current inside an Amperian loop

, Ampere's Law

, orbital magnetic dipole moment for an electron in circular orbit

, orbital angular momentum for an electron moving in a circular orbit in an atom

, magnetization

, magnetic field strength

χm, magnetic susceptibility

κm, relative magnetic permeability

ANSWERS TO SELF-TEST OPPORTUNITIES
Page 916
PROBLEM-SOLVING PRACTICE
Problem-Solving Guidelines
  1. When using the Biot-Savart Law, you should always draw a diagram of the situation, with the current element highlighted. Check for simplifying symmetries before proceeding with calculations; you can save yourself a significant amount of work.

  2. When applying Ampere's Law, choose an Amperian loop that has some geometrical symmetry, in order to simplify the evaluation of the integral. Often, you can use right-hand rule 3 to choose the direction of integration along the loop: Point your thumb in the direction of the net current through the loop and your fingers curl in the direction of integration. This method will also remind you to sum the currents through the Amperian loop to determine the enclosed current.

  3. Remember the superposition principle for magnetic fields: The net magnetic field at any point in space is the vector sum of the individual magnetic fields generated by different objects. Make sure you do not simply add the magnitudes. Instead, you generally need to add the spatial components of the different sources of magnetic field separately.

  4. All of the principles governing motion of charged particles in magnetic fields and all of the problem-solving guidelines presented in Chapter 27 still apply. It does not matter if the magnetic field is due to a permanent magnet or an electromagnet.

  5. In order to calculate the magnetic field in a material, you can use the formulas derived from Ampere's Law and Biot-Savart's Law, but you have to replace μ0 with μκmμ0 ≡ (1+ χm)μ0.

SOLVED PROBLEM28.4 Magnetic Field from Four Wires
FIGURE 28.32
Four wires located at the corners of a square. Two of the wires are carrying current into the page, and the other two are carrying current out of the page.

Four wires are each carrying a current of magnitude i =1.00 A. The wires are located at the four corners of a square with side a = 3.70 cm. Two of the wires are carrying current into the page, and the other two are carrying current out of the page (Figure 28.32).

PROBLEM

What is the y-component of the magnetic field at the center of the square?

SOLUTION
THINK

The magnetic field at the center of the square is the vector sum of the magnetic fields from the four current-carrying wires. The magnitude of the magnetic field from all four wires is the same. The direction of the magnetic field from each wire is determined using right-hand rule 3.

SKETCH
FIGURE 28.33
The magnetic fields from the four current-carrying wires.

Figure 28.33 shows the magnetic fields from the four wires: is the magnetic field from wire 1, is the magnetic field from wire 2, is the magnetic field from wire 3, and is the magnetic field from wire 4. Note that and are equal and and are equal.

RESEARCH

The magnitude of the magnetic field from each of the four wires is given by

where is the distance from each wire to the center of the square.

Right-hand rule 3 gives us the directions of the magnetic fields, which are shown in Figure 28.33. The y-component of each of the magnetic fields is given by

SIMPLIFY

The sum of the y-components of the four magnetic fields is

where we have used .

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CALCULATE

Putting in the numerical values gives us

ROUND

We report our result to three significant figures:

DOUBLE-CHECK

To double-check our result, we calculate the magnitude of the magnetic field from one wire at the center of the square:

The sum of the y-components is then

which agrees with our result.

SOLVED PROBLEM28.5 Electron Motion in a Solenoid

An ideal solenoid has 200.0 turns/cm. An electron inside the coil of a solenoid moves in a circle with radius r = 3.00 cm perpendicular to the solenoid's axis. The electron moves with a speed of v = 0.0500c, where c is the speed of light.

PROBLEM

What is the current in the solenoid?

SOLUTION
THINK

The solenoid produces a uniform magnetic field, which is proportional to the current flowing in the solenoid. The radius of circular motion of the electron is related to the speed of the electron and the magnetic field inside the solenoid.

SKETCH
FIGURE 28.34
Electron traveling in a circular path inside a solenoid.

Figure 28.34 shows the circular path of the electron in the uniform magnetic field of the solenoid.

RESEARCH

The magnitude of the magnetic field inside the solenoid is given by

(i)

where i is the current in the solenoid and n is the number of turns per unit length. The magnetic force provides the centripetal force needed for the electron to move in a circle and so the radius of the electron's path can be related to B:

(ii)

where m is the electron's mass, v is its speed, and e is the magnitude of the electron's charge.

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SIMPLIFY

Combining equations (i) and (ii), we have

Solving this equation for the current in the solenoid, we obtain

(iii)
CALCULATE

The speed of the electron was specified in terms of the speed of light:

Putting this and the other numerical values into equation (iii), we get

ROUND

We report our result to three significant figures:

DOUBLE-CHECK

To double-check our result, we use it to calculate the magnitude of the magnetic field inside the solenoid:

This magnitude of magnetic field seems reasonable. Thus, our calculated value for the current in the solenoid seems reasonable.

MULTIPLE-CHOICE QUESTIONS
QUESTIONS
Page 920
PROBLEMS

A blue problem number indicates a worked-out solution is available in the Student Solutions Manual. One and two •• indicate increasing level of problem difficulty.

Sections 28.1 and 28.2
  • 28.22 Two long parallel wires are separated by 3.0 mm. The current flowing in one of the wires is twice that in the other wire. If the magnitude of the force on a 1.0-m length of one of the wires is 7.0 μN, what are the values of the two currents?

  • 28.23 An electron is shot from an electron gun with a speed of 4.0 · 105 m/s and moves parallel to and a distance of 5.0 cm above a long, straight wire carrying a current of 15 A. Determine the magnitude and the direction of the acceleration of the electron the instant it leaves the electron gun.

  • 28.24 An electron moves in a straight line at a speed of 5 · 106 m/s. What is the magnitude and the direction of the magnetic field created by the moving electron at a distance d = 5 m ahead of it on its line of motion? How does the answer change if the moving particle is a proton?

  • 28.25 Suppose that the magnetic field of the Earth were due to a single current moving in a circle of radius 2.00 · 103 km through the Earth's molten core. The strength of the Earth's magnetic field on the surface near a magnetic pole is about 6.00 · 10−5 T. About how large a current would be required to produce such a field?

  • 28.26 A square ammeter has sides of length 3.00 cm. The sides of the ammeter are capable of measuring the magnetic field they are subject to. When the ammeter is clamped around a wire carrying a DC current, as shown in the figure, the average value of the magnetic field measured in the sides is 3.00 G. What is the current in the wire?

  • •27.27 A long, straight wire carrying a 2.00-A current lies along the x-axis. A particle with charge q = –3.00 μC passes parallel to the z-axis through the point (x,y,z) = (0,2,0). Where in the xy-plane should another long, straight wire be placed so that there is no magnetic force on the particle at the point where it crosses the plane?

    Answer

  • 28.28 Find the magnetic field in the center of a wire semicircle like that shown in the figure, with radius R = 10.0 cm, if the current is i = 12.0 A.

  • 28.29 Two very long wires run parallel to the z-axis, as shown in the figure. They each carry a current, i1 = i2 = 25.0 A, in the direction of the positive z-axis. The magnetic field of the Earth is given by T (in the xy-plane and pointing due north). A magnetic compass needle is placed at the origin. Determine the angle θ between the compass needle and the x-axis. (Hint: The compass needle will align its axis along the direction of the net magnetic field.)

    Answer

  • 28.30 Two identical coaxial coils of wire of radius 20.0 cm are directly on top of each other, separated by a 2.00-mm gap. The lower coil is on a flat table and has a current i in the clockwise direction; the upper coil carries an identical current and has a mass of 0.0500 kg. Determine the magnitude and the direction that the current in the upper coil has to have to keep the coil levitated at its current height.

  • •28.31 A long, straight wire lying along the x-axis carries a current, i, flowing in the positive x-direction. A second long, straight wire lies along the y-axis and has a current i in the positive y-direction. What is the magnitude and the direction of the magnetic field at point z = b on the z-axis?

    Answer

  • 28.32 A square loop of wire with a side length of 10.0 cm carries a current of 0.300 A. What is the magnetic field in the center of the square loop?

  • 28.33 The figure shows the cross section through three long wires with a linear mass distribution of 100. g/m. They carry currents i1, i2, and i3 in the directions shown. Wires 2 and 3 are 10.0 cm apart and are attached to a vertical surface, and each carries a current of 600. A. What current, i1, will allow wire 1 to “float” at a perpendicular distance d from the vertical surface of 10.0 cm? (Neglect the thickness of the wires.)

    Answer

  • 28.34 A hairpin configuration is formed of two semi-infinite straight wires that are 2.00 cm apart and joined by a semicircular piece of wire (whose radius must be 1.00 cm and whose center is at the origin of xyz-coordinates). The top straight wire is along the line y = 1.00 cm, and the bottom straight wire is along the line y = –1.00 cm; these two wires are in the left side (x < 0) of the xy-plane. The current in the hairpin is 3.00 A, and it is directed toward the right in the top wire, clockwise around the semicircle, and to the left in the bottom wire. Find the magnetic field at the origin of the coordinate system.

  • •28.35 A long, straight wire is located along the x-axis (y = 0 and z = 0). The wire carries a current of 7.00 A in the positive x-direction. What is the magnitude and the direction of the force on a particle with a charge of 9.00 C located at (+1.00 m,+2.00 m,0), when it has a velocity of 3000. m/s in each of the following directions?

    1. the positive x-direction

    2. the positive y-direction

    3. the negative z-direction

    Answer

  • 28.36 A long, straight wire has a 10.0-A current flowing in the positive x-direction, as shown in the figure. Close to the wire is a square loop of copper wire that carries a 2.00-A current in the direction shown. The near side of the loop is d = 0.50 m away from the wire. The length of each side of the square is a = 1.00 m.

    1. Find the net force between the two current-carrying objects.

    2. Find the net torque on the loop.

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  • 28.37 A square box with sides of length 1.00 m has one corner at the origin of a coordinate system, as shown in the figure. Two coils are attached to the outside of the box. One coil is on the box face that is in the xz-plane at y = 0, and the second is on the box face in the yz-plane at x = 1.00 m. Each of the coils has a diameter of 1.00 m and contains 30.0 turns of wire carrying a current of 5.00 A in each turn. The current in each coil is clockwise when the coil is viewed from outside the box. What is the magnitude and the direction of the magnetic field at the center of the box?

    Answer

Section 28.3
  • 28.38 A square loop, with sides of length L, carries current i. Find the magnitude of the magnetic field from the loop at the center of the loop, as a function of i and L.

  • 28.39 The figure shows a cross section across the diameter of a long, solid, cylindrical conductor. The radius of the cylinder is R = 10.0 cm. A current of 1.35 A is uniformly distributed through the conductor and is flowing out of the page. Calculate the direction and the magnitude of the magnetic field at positions ra = 0.0 cm, rb = 4.00 cm, rc = 10.00 cm, and rd = 16.0 cm.

    Answer

  • 28.40 Parallel wires, a distance D apart, carry a current, i, in opposite directions as shown in the figure. A circular loop, of radius R = D/2, has the same current flowing in a counterclockwise direction. Determine the magnitude and the direction of the magnetic field from the loop and the parallel wires at the center of the loop as a function of i and R.

  • 28.41 A current of constant density, J0, flows through a very long cylindrical conducting shell with inner radius a and outer radius b. What is the magnetic field in the regions r < a, a < r < b, and r > b? Does Ba<r<b = Br>b for r = b?

    Answer

  • 28.42 The current density in a cylindrical conductor of radius R, varies as J(r) = J0r/R (in the region from zero to R). Express the magnitude of the magnetic field in the regions r < R and r > R. Produce a sketch of the radial dependence, B(r).

  • •28.43 A very large sheet of a conductor located in the xy-plane, as shown in the figure, has a uniform current flowing in the y-direction. The current density is 1.5 A/cm. Use Ampere's Law to calculate the direction and the magnitude of the magnetic field just above the center of the sheet (not close to any edges).

    Answer

  • ••28.44 A coaxial wire consists of a copper core of radius 1.00 mm surrounded by a copper sheath of inside radius 1.50 mm and outside radius 2.00 mm. A current i, flows in one direction in the core and in the opposite direction in the sheath. Graph the magnitude of the magnetic field as a function of the distance from the center of the wire.

  • ••28.45 The current density of a cylindrical conductor of radius R varies as J(r) = J0er/R (in the region from zero to R). Express the magnitude of the magnetic field in the regions r < R and r > R. Produce a sketch of the radial dependence, B(r).

    Answer

Section 28.4
  • 28.46 A current of 2.00 A is flowing through a 1000-turn solenoid of length L = 40.0 cm. What is the magnitude of the magnetic field inside the solenoid?

  • 28.47 Solenoid A has twice the diameter, three times the length, and four times the number of turns of solenoid B. The two solenoids have currents of equal magnitudes flowing through them. Find the ratio of the magnitude of the magnetic field in the interior of solenoid A to that of solenoid B.

    Answer

  • 28.48 A long solenoid (diameter of 6.00 cm) is wound with 1000 turns per meter of thin wire through which a current of 0.250 A is maintained. A wire carrying a current of 10.0 A is inserted along the axis of the solenoid. What is the magnitude of the magnetic field at a point 1.00 cm from the axis?

  • 28.49 A long, straight wire carries a current of 2.5 A.

    1. What is the strength of the magnetic field at a distance of 3.9 cm from the wire?

    2. If the wire still carries 2.5 A, but is used to form a long solenoid with 32 turns per centimeter and a radius of 3.9 cm, what is the strength of the magnetic field at the center of the solenoid?

    Answer

  • 28.50 Figure 28.18a shows a Helmholtz coil used to generate uniform magnetic fields. Suppose the Helmholtz coil consists of two sets of coaxial wire loops with 15 turns of radius R = 75.0 cm, which are separated by R, and each coil carries a current of 0.123 A flowing in the same direction. Calculate the magnitude and the direction of magnetic field in the center between the coils.

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  • 28.51 A particle detector utilizes a solenoid that has 550 turns of wire per centimeter. The wire carries a current of 22 A. A cylindrical detector that lies within the solenoid has an inner radius of 0.80 m. Electron and positron beams are directed into the solenoid parallel to its axis. What is the minimum momentum perpendicular to the solenoid axis that a particle can have if it is to be able to enter the detector?

    Answer

Sections 28.5 through 28.7
  • 28.52 An electron has a spin magnetic moment of magnitude μ = 9.285 · 10−24A m2. Consequently, it has energy associated with its orientation in a magnetic field. If the difference between the energy of an electron that is “spin up” in a magnetic field of magnitude B and the energy of one that is “spin down” in the same magnetic field (where “up” and “down” refer to the direction of the magnetic field) is 9.460 · 10−25 J, what is the field magnitude, B?

  • 28.53 When a magnetic dipole is placed in a magnetic field, it has a natural tendency to minimize its potential energy by aligning itself with the field. If there is sufficient thermal energy present, however, the dipole may rotate so that it is no longer aligned with the field. Using kBT as a measure of the thermal energy, where kB is Boltzmann's constant and T is the temperature in kelvins, determine the temperature at which there is sufficient thermal energy to rotate the magnetic dipole associated with a hydrogen atom from an orientation parallel to an applied magnetic field to one that is antiparallel to the applied field. Assume that the strength of the field is 0.15 T.

    Answer

  • 28.54 Aluminum becomes superconducting at a temperature around 1.0 K if exposed to a magnetic field of magnitude less than 0.0105 T. Determine the maximum current that can flow in an aluminum superconducting wire with radius R = 1.0 mm.

  • 28.55 If you want to construct an electromagnet by running a current of 3.0 A through a solenoid with 500 windings and length 3.5 cm and you want the magnetic field inside the solenoid to have magnitude B = 2.96 T, you can insert a ferrite core into the solenoid. What value of the relative magnetic permeability should this ferrite core have in order to make this work?

    Answer

  • 28.56 What is the magnitude of the magnetic field inside a long, straight tungsten wire of circular cross section with diameter 2.4 mm and carrying a current of 3.5 A, at a distance of 0.60 mm from its central axis?

  • 28.57 You charge up a small rubber ball of mass 200. g by rubbing it over your hair. The ball acquires a charge of 2.00 μC. You then tie a 1.00-m-long string to it and swing it in a horizontal circle, providing a centripetal force of 25.0 N. What is the magnetic moment of the system?

    Answer

  • 28.58 Consider a model of the hydrogen atom in which an electron orbits a proton in the plane perpendicular to the proton's spin angular momentum (and magnetic dipole moment) at a distance equal to the Bohr radius, a0 = 5.292 · 10−11 m. (This is an oversimplified classical model.) The spin of the electron is allowed to be either parallel to the proton's spin or antiparallel to it; the orbit is the same in either case. But since the proton produces a magnetic field at the electron's location, and the electron has its own intrinsic magnetic dipole moment, the energy of the electron differs depending on its spin. The magnetic field produced by the proton's spin may be modeled as a dipole field, like the electric field due to an electric dipole discussed in Chapter 22. Calculate the energy difference between the two electron-spin configurations. Consider only the interaction between the magnetic dipole moment associated with the electron's spin and the field produced by the proton's spin.

  • ••28.59 Consider an electron to be a uniformly dense sphere of charge, with a total charge of –e = –1.60 · 10−19 C, spinning at an angular frequency, ω.

    1. Write an expression for its classical angular momentum of rotation, L.

    2. Write an expression for its magnetic dipole moment, μ.

    3. Find the ratio, γe = μ/L, known as the gyromagnetic ratio.

    Answer

Additional Problems
  • 28.60 Two 50-turn coils, each with a diameter of 4.00 m, are placed 1.00 m apart, as shown in the figure. A current of 7.00 A is flowing in the wires of both coils; the direction of the current is clockwise for both coils when viewed from the left. What is the magnitude of the magnetic field in the center between the two coils?

  • 28.61 The wires in the figure are separated by a vertical distance d. Point B is at the midpoint between the two wires; point A is a distance d/2 from the lower wire. The horizontal distance between A and B is much larger than d. Both wires carry the same current, i. The strength of magnetic field at point A is 2.00 mT. What is the strength of the field at point B?

    Answer

  • 28.62 You are standing at a spot where the magnetic field of the Earth is horizontal, points due northward, and has magnitude 40.0 μT. Directly above your head, at a height of 12.0 m, a long, horizontal cable carries a steady DC current of 500.0 A due northward. Calculate the angle θ by which your magnetic compass needle is deflected from true magnetic north by the effect of the cable. Don't forget the sign of θ—is the deflection eastward or westward?

  • 28.63 The magnetic dipole moment of the Earth is approximately 8.0 · 1022 A m2. The source of the Earth's magnetic field is not known; one possibility might be the circulation of ions in the Earth's molten outer core. Assume that the circulating ions move a circular loop of radius 2500 km. What “current” must they produce to yield the observed field?

    Answer

  • Page 923
  • 28.64 A circular wire loop has radius R = 0.12 m and carries current i = 0.10 A. The loop is placed in the xy-plane in a uniform magnetic field given by , as shown in the figure. Determine the direction and the magnitude of the loop's magnetic moment and calculate the potential energy of the loop in the position shown. If the wire loop can move freely, how will it orient itself to minimize its potential energy, and what is the value of the lowest potential energy?

  • 28.65 A 0.90 m-long solenoid has a radius of 5.0 mm. When the wire carries a 0.20-A current, the magnetic field in the solenoid is 5.0 mT. How many turns of wire are there in the solenoid?

    Answer

  • 28.66 In a coaxial cable, the solid core carries a current i. The sheath also carries a current i but in the opposite direction and has an inner radius a and an outer radius b. The current density is equally distributed over each conductor. Find an expression for the magnetic field at a distance a < r < b from the center of the core.

  • •28.67 A 50-turn rectangular coil of wire of dimensions 10.0 cm by 20.0 cm lies in a horizontal plane, as shown in the figure. The axis of rotation of the coil is aligned north and south. It carries a current i = 1.00 A, and is in a magnetic field pointing from west to east. A mass of 50.0 g hangs from one side of the loop. Determine the strength the magnetic field has to have to keep the loop in the horizontal orientation.

    Answer

  • 28.68 Two long, straight parallel wires are separated by a distance of 20.0 cm. Each wire carries a current of 10.0 A in the same direction. What is the magnitude of the resulting magnetic field at a point that is 12.0 cm from each wire?

  • 28.69 A particle with a mass of 1.00 mg and a charge of q is moving at a speed of 1000. m/s along a horizontal path 10.0 cm below and parallel to a straight current-carrying wire. Determine q if the magnitude of the current in the wire is 10.0 A.

    Answer

  • 28.70 A conducting coil consisting of n turns of wire is placed in a uniform magnetic field given by , as shown in the figure. The radius of the coil is R = 5.00 cm, and the angle between the magnetic field vector and the unit normal vector to the coil is θ = 60.0°. The current through the coil is i = 5.00 A.

    1. Specify the direction of the current in the coil, given the direction of the magnetic dipole moment, , in the figure.

    2. Calculate the number of turns, n, the coil must have for the torque on the loop to be 3.40 N m.

    3. If the radius of the loop is decreased to R = 2.50 cm, what should the number of turns, N, be for the torque to remain unchanged? Assume that i, B, and θ stay the same.

  • •28.71 A loop of wire of radius R = 25.0 cm has a smaller loop of radius r = 0.900 cm at its center such that the planes of the two loops are perpendicular to each other. When a current of 14.0 A is passed through both loops, the smaller loop experiences a torque due to the magnetic field produced by the larger loop. Determine this torque assuming that the smaller loop is sufficiently small that the magnetic field due to the larger loop is same across the entire surface.

    Answer

  • 28.72 Two wires, each 25.0 cm long, are connected to two separate 9.00-V batteries as shown in the figure. The resistance of the first wire is 5.00 Ω, and that of the other wire is unknown (R). If the separation between the wires is 4.00 mm, what value of R will produce a force of magnitude 4.00 · 10−5 N between them? Is the force attractive or repulsive?

  • 28.73 A proton is moving under the combined influence of an electric field (E = 1000. V/m) and a magnetic field (B = 1.20 T), as shown in the figure.

    1. What is the acceleration of the proton at the instant it enters the crossed fields?

    2. What would the acceleration be if the direction of the proton's motion was reversed?

    Answer

  • 28.74 A toy airplane of mass 0.175 kg, with a charge of 36 mC, is flying at a speed of 2.8 m/s at a height of 17.2 cm above and parallel to a wire, which is carrying a 25-A current; the airplane experiences some acceleration. Determine this acceleration.

  • Page 924
  • •28.75 An electromagnetic doorbell has been constructed by wrapping 70 turns of wire around a long, thin rod, as shown in the figure. The rod has a mass of 30.0 g, a length of 8.00 cm, and a cross-sectional area of 0.200 cm2. The rod is free to pivot about an axis through its center, which is also the center of the coil. Initially, the rod makes an angle of θ = 25.0° with the horizontal. When θ = 0.00°, the rod strikes a bell. There is a uniform magnetic field of 900.0 G directed along θ = 0.00°.

    1. If a current of 2.00 A is flowing in the coil, what is the torque on the rod when θ = 25.0°?

    2. What is the angular velocity of the rod when it strikes the bell?

    Answer

  • 28.76 Two long, parallel wires separated by a distance, d, carry currents in opposite directions. If the left-hand wire carries a current i/2, and the right-hand wire carries a current i, determine where the magnetic field is zero.

  • 28.77 A horizontally oriented coil of wire of radius 5.00 cm and carrying a current, i, is being levitated by the south pole of a vertically oriented bar magnet suspended above the center of the coil. If the magnetic field on all parts of the coil makes an angle θ of 45.0° with the vertical, determine the magnitude and the direction of the current needed to keep the coil floating in midair. The magnitude of the magnetic field is B = 0.0100 T, the number of turns in the coil is N = 10.0, and the total coil mass is 10.0 g.

    Answer

  • 28.78 As shown in the figure, a long, hollow, conducting cylinder of inner radius a and outer radius b carries a current that is flowing out of the page. Suppose that a = 5.00 cm, b = 7.00 cm, and the current i = 100. mA, uniformly distributed over the cylinder wall (between a and b). Find the magnitude of the magnetic field at each of the following distances r from the center of the cylinder:

    1. r = 4.00 cm

    2. r = 6.50 cm

    3. r = 9.00 cm

  • •28.79 A wire of radius R carries current i. The current density is given by J = J0(1 – r/R), where r is measured from the center of the wire and J0 is a constant. Use Ampere's Law to find the magnetic field inside the wire at a distance r < R from the central axis.

    Answer

  • 28.80 A circular wire of radius 5.0 cm has a current of 3.0 A flowing in it. The wire is placed in a uniform magnetic field of 5.0 mT.

    1. Determine the maximum torque on the wire.

    2. Determine the range of the magnetic potential energy of the wire.