Youll learn in Chapter 30 that a current can be induced in

What is magnetism?

What fields are especially important?

How do charges respond to magnetic fields?

How do currents respond to magnetic fields?

Why is magnetism important?

Does the compass needle rotate clockwise (cw), counterclockwise (ccw), or not at all?

A proton moves with velocity v u = 1.0 * 107 ni m/s. As it passes the origin, what is the magnetic field at the 1x, y, z2 positions (1 mm, 0 mm, 0 mm), (0 mm, 1 mm, 0 mm), and (1 mm, 1 mm, 0 mm)?

The magnetic field at position P points I P a. Up. b. Down. c. Into the page. d. Out of the page.

The electron in FIGURE 29.12 is moving to the right. What is the direction of the electrons magnetic field at the dot?

The positive charge is moving straight out of the page. What is the direction of the magnetic field at the dot? v out of page u a. Up b. Down c. Left d. Right

A long, straight wire carries current I in the positive x-direction. Find the magnetic field at distance r from the wire.

A 1.0-m-long, 1.0-mm-diameter nichrome heater wire is connected to a 12 V battery. What is the magnetic field strength 1.0 cm away from the wire?

What is the current direction in this loop? And which side of the loop is the north pole? a. Current cw; north pole on top b. Current cw; north pole on bottom c. Current ccw; north pole on top d. Current ccw; north pole on bottom

FIGURE 29.16a shows a current loop, a circular loop of wire with radius R that carries current I. Find the magnetic field of the current loop at distance z on the axis of the loop.

An electron moves perpendicular to a magnetic field. What is the direction of B u ? a. Left b. Up c. Into the page d. Right e. Down f. Out of the page

What current is needed in a 5-turn, 10-cm-diameter coil to cancel the earths magnetic field at the center of the coil?

What is the current direction in the loop? a. Out of the page at the top of the loop, into the page at the bottom b. Out of the page at the bottom of the loop, into the page at the top

Youll learn in Chapter 30 that a current can be induced in a closed loop of wire. If the loop happens to be made of a superconducting material, with zero resistance, the induced current willin principlepersist forever. The current cannot be measured with an ammeter because any real ammeter has resistance that will quickly stop the current. Instead, physicists measure the persistent current in a superconducting loop by measuring its magnetic field. What is the current in a 3.0-mmdiameter superconducting loop if the axial magnetic field is 9.0 mT at a distance of 2.5 cm?

Which magnet or magnets induced this magnetic dipole? N S (a) S N (b) N S (c) S

A wire of radius R carries current I. Find the magnetic field inside the wire at distance r 6 R from the axis

A 1.0-m-long MRI solenoid generates a 1.2 T magnetic field. To produce such a large field, the solenoid is wrapped with superconducting wire that can carry a 100 A current. How many turns of wire does the solenoid need?

A long wire carries a 10 A current from left to right. An electron 1.0 cm above the wire is traveling to the right at a speed of 1.0 * 107 m/s. What are the magnitude and the direction of the magnetic force on the electron?

In FIGURE 29.38, an electron is accelerated from rest through a potential difference of 500 V, then injected into a uniform magnetic field. Once in the magnetic field, it completes half a revolution in 2.0 ns. What is the radius of its orbit?

A Hall probe consists of a strip of the metal bismuth that is 0.15 mm thick and 5.0 mm wide. Bismuth is a poor conductor with charge-carrier density 1.35 * 1025 m-3 . The Hall voltage on the probe is 2.5 mV when the current through it is 1.5 A. What is the strength of the magnetic field, and what is the electric field strength inside the bismuth?

The 0.10 T uniform magnetic field of FIGURE 29.44 is horizontal, parallel to the floor. A straight segment of 1.0-mm-diameter copper wire, also parallel to the floor, is perpendicular to the magnetic field. What current through the wire, and in which direction, will allow the wire to float in the magnetic field?

Two stiff, 50-cm-long, parallel wires are connected at the ends by metal springs. Each spring has an unstretched length of 5.0 cm and a spring constant of 0.025 N/m. The wires push each other apart when a current travels around the loop. How much current is required to stretch the springs to lengths of 6.0 cm?

The lightweight glass sphere in FIGURE Q29.1 hangs by a thread. The north pole of a bar magnet is brought near the sphere. a. Suppose the sphere is electrically neutral. Is it attracted to, repelled by, or not affected by the magnet? Explain. b. Answer the same question if the sphere is positively charged.

The metal sphere in FIGURE Q29.2 hangs by a thread. When the north pole of a magnet is brought near, the sphere is strongly attracted to the magnet. Then the magnet is reversed and its south pole is brought near the sphere. How does the sphere respond? Explain.

You have two electrically neutral metal cylinders that exert strong attractive forces on each other. You have no other metal objects. Can you determine if both of the cylinders are magnets, or if one is a magnet and the other is just a piece of iron? If so, how? If not, why not?

What is the current direction in the wire of FIGURE Q29.4? Explain.

What is the current direction in the wire of FIGURE Q29.5? Explain

What is the initial direction of deflection for the charged particles entering the magnetic fields shown in FIGURE Q29.6?

What is the initial direction of deflection for the charged particles entering the magnetic fields shown in FIGURE Q29.7?

Determine the magnetic field direction that causes the charged particles shown in FIGURE Q29.8 to experience the indicated magnetic force.

Determine the magnetic field direction that causes the charged particles shown in FIGURE Q29.9 to experience the indicated magnetic force

You have a horizontal cathode-ray tube (CRT) for which the controls have been adjusted such that the electron beam should make a single spot of light exactly in the center of the screen. You observe, however, that the spot is deflected to the right. It is possible that the CRT is broken. But as a clever scientist, you realize that your laboratory might be in either an electric or a magnetic field. Assuming that you do not have a compass, any magnets, or any charged rods, how can you use the CRT itself to determine whether the CRT is broken, is in an electric field, or is in a magnetic field? You cannot remove the CRT from the room.

The south pole of a bar magnet is brought toward the current loop of FIGURE Q29.11. Does the bar magnet attract, repel, or have no effect on the loop? Explain

Give a step-by-step explanation, using both words and pictures, of how a permanent magnet can pick up a piece of nonmagnetized iron.

What is the magnetic field strength at points 2 to 4 in FIGURE EX29.1? Assume that the wires overlap closely at 2 and 3, that each point is the same distance from nearby wires, and that all other wires are too far away to contribute to the field.

Points 1 and 2 in FIGURE EX29.2 are the same distance from the wires as the point where B = 2.0 mT. What are the strength and direction of B u at points 1 and 2?

A proton moves along the x-axis with vx = 1.0 * 107 m/s. As it passes the origin, what are the strength and direction of the magnetic field at the 1x, y, z2 positions (a) (1 cm, 0 cm, 0 cm), (b) (0 cm, 1 cm, 0 cm), and (c) (0 cm, -2 cm, 0 cm)?

An electron moves along the z-axis with vz = 2.0 * 107 m/s. As it passes the origin, what are the strength and direction of the magnetic field at the 1x, y, z2 positions (a) (1 cm, 0 cm, 0 cm), (b) (0 cm, 0 cm, 1 cm), and (c) (0 cm, 1 cm, 1 cm)?

What is the magnetic field at the position of the dot in FIGURE EX29.5? Give your answer as a vector.

What is the magnetic field at the position of the dot in FIGURE EX29.6? Give your answer as a vector.

What currents are needed to generate the magnetic field strengths of Table 29.1 at a point 1.0 cm from a long, straight wire?

The element niobium, which is a metal, is a superconductor (i.e., no electrical resistance) at temperatures below 9 K. However, the superconductivity is destroyed if the magnetic field at the surface of the metal reaches or exceeds 0.10 T. What is the maximum current in a straight, 3.0-mm-diameter superconducting niobium wire?

Although the evidence is weak, there has been concern in recent years over possible health effects from the magnetic fields generated by electric transmission lines. A typical high-voltage transmission line is 20 m above the ground and carries a 200 A current at a potential of 110 kV. a. What is the magnetic field strength on the ground directly under such a transmission line? b. What percentage is this of the earths magnetic field of 50 mT?

A biophysics experiment uses a very sensitive magnetic field probe to determine the current associated with a nerve impulse traveling along an axon. If the peak field strength 1.0 mm from an axon is 8.0 pT, what is the peak current carried by the axon?

The magnetic field at the center of a 1.0-cm-diameter loop is 2.5 mT. a. What is the current in the loop? b. A long straight wire carries the same current you found in part a. At what distance from the wire is the magnetic field 2.5 mT?

What are the magnetic fields at points a to c in FIGURE EX29.12? Give your answers as vectors.

A wire carries current I into the junction shown in FIGURE EX29.13. What is the magnetic field at the dot?

What are the magnetic field strength and direction at points a to c in FIGURE EX29.14?

Point A in FIGURE EX29.15 is 2.0 mm from the wire. What is the magnetic field strength at point A? You can assume that the wire is very long and that all other wires are too far away to contribute to the magnetic field.

The on-axis magnetic field strength 10 cm from a small bar magnet is 5.0 mT. a. What is the bar magnets magnetic dipole moment? b. What is the on-axis field strength 15 cm from the magnet?

A 100 A current circulates around a 2.0-mm-diameter superconducting ring. a. What is the rings magnetic dipole moment? b. What is the on-axis magnetic field strength 5.0 cm from the ring?

A small, square loop carries a 25 A current. The on-axis magnetic field strength 50 cm from the loop is 7.5 nT. What is the edge length of the square?

The earths magnetic dipole moment is 8.0 * 1022 Am2 . a. What is the magnetic field strength on the surface of the earth at the earths north magnetic pole? How does this compare to the value in Table 29.1? You can assume that the current loop is deep inside the earth. b. Astronauts discover an earth-size planet without a magnetic field. To create a magnetic field at the north pole with the same strength as earths, they propose running a current through a wire around the equator. What size current would be needed?

What is the line integral of B u between points i and f in FIGURE EX29.20?

What is the line integral of B u between points i and f in FIGURE EX29.21?

The value of the line integral of B u around the closed path in FIGURE EX29.22 is 3.77 * 10-6 Tm. What is I3?

The value of the line integral of B u around the closed path in FIGURE EX29.23 is 1.38 * 10-5 Tm. What are the direction (in or out of the page) and magnitude of I3?

What is the line integral of B u between points i and f in FIGURE EX29.24?

Magnetic resonance imaging needs a magnetic field strength of 1.5 T. The solenoid is 1.8 m long and 75 cm in diameter. It is tightly wound with a single layer of 2.0-mm-diameter superconducting wire. What size current is needed?

A proton moves in the magnetic field B u = 0.50 ni T with a speed of 1.0 * 107 m/s in the directions shown in FIGURE EX29.26. For each, what is magnetic force F u on the proton? Give your answers in component form.

An electron moves in the magnetic field B u = 0.50 ni T with a speed of 1.0 * 107 m/s in the directions shown in FIGURE EX29.27. For each, what is magnetic force F u on the electron? Give your answers in component form.

Radio astronomers detect electromagnetic radiation at 45 MHz from an interstellar gas cloud. They suspect this radiation is emitted by electrons spiraling in a magnetic field. What is the magnetic field strength inside the gas cloud?

To five significant figures, what are the cyclotron frequencies in a 3.0000 T magnetic field of the ions (a) O2 +, (b) N2 +, and (c) CO+? The atomic masses are shown in the table; the mass of the missing electron is less than 0.001 u and is not relevant at this level of precision. Although N2 + and CO+ both have a nominal molecular mass of 28, they are easily distinguished by virtue of their slightly different cyclotron frequencies. Use the following constants: 1 u = 1.6605 * 10-27 kg, e = 1.6022 * 10-19 C. Atomic masses 12 C 12.000 14 N 14.003 16 O 15.995

For your senior project, you would like to build a cyclotron that will accelerate protons to 10% of the speed of light. The largest vacuum chamber you can find is 50 cm in diameter. What magnetic field strength will you need?

The microwaves in a microwave oven are produced in a special tube called a magnetron. The electrons orbit the magnetic field at 2.4 GHz, and as they do so they emit 2.4 GHz electromagnetic waves. a. What is the magnetic field strength? b. If the maximum diameter of the electron orbit before the electron hits the wall of the tube is 2.5 cm, what is the maximum electron kinetic energy?

The Hall voltage across a conductor in a 55 mT magnetic field is 1.9 mV. When used with the same current in a different magnetic field, the voltage across the conductor is 2.8 mV. What is the strength of the second field?

What magnetic field strength and direction will levitate the 2.0 g wire in FIGURE EX29.33?

The two 10-cm-long parallel wires in FIGURE EX29.34 are separated by 5.0 mm. For what value of the resistor R will the force between the two wires be 5.4 * 10-5 N?

The right edge of the circuit in FIGURE EX29.35 extends into a 50 mT uniform magnetic field. What are the magnitude and direction of the net force on the circuit?

What is the net force (magnitude and direction) on each wire in FIGURE EX29.36?

FIGURE EX29.37 is a cross section through three long wires with linear mass density 50 g/m. They each carry equal currents in the directions shown. The lower two wires are 4.0 cm apart and are attached to a table. What current I will allow the upper wire to float so as to form an equilateral triangle with the lower wires?

A square current loop 5.0 cm on each side carries a 500 mA current. The loop is in a 1.2 T uniform magnetic field. The axis of the loop, perpendicular to the plane of the loop, is 30 away from the field direction. What is the magnitude of the torque on the current loop?

A small bar magnet experiences a 0.020 Nm torque when the axis of the magnet is at 45 to a 0.10 T magnetic field. What is the magnitude of its magnetic dipole moment?

a. What is the magnitude of the torque on the current loop in FIGURE EX29.40? b. What is the loops equilibrium orientation?

A long wire carrying a 5.0 A current perpendicular to the xy-plane intersects the x-axis at x = -2.0 cm. A second, parallel wire carrying a 3.0 A current intersects the x-axis at x = +2.0 cm. At what point or points on the x-axis is the magnetic field zero if (a) the two currents are in the same direction and (b) the two currents are in opposite directions?

The two insulated wires in FIGURE P29.42 cross at a 30 angle but do not make electrical contact. Each wire carries a 5.0 A current. Points 1 and 2 are each 4.0 cm from the intersection and equally distant from both wires. What are the magnitude and direction of the magnetic fields at points 1 and 2?

What are the strength and direction of the magnetic field at the center of the loop in FIGURE P29.43?

At what distance on the axis of a current loop is the magnetic field half the strength of the field at the center of the loop? Give your answer as a multiple of R.

Find an expression for the magnetic field strength at the center (point P) of the circular arc in FIGURE P29.45.

What are the strength and direction of the magnetic field at point P in FIGURE P29.46?

A scientist measuring the resistivity of a new metal alloy left her ammeter in another lab, but she does have a magnetic field probe. So she creates a 6.5-m-long, 2.0-mm-diameter wire of the material, connects it to a 1.5 V battery, and measures a 3.0 mT magnetic field 1.0 mm from the surface of the wire. What is the materials resistivity?

A 2.5-m-long, 2.0-mm-diameter aluminum wire has a potential difference of 1.5 V between its ends. Consider an electron halfway between the center of the wire and the surface that is moving parallel to the wire at the drift speed. What is the ratio of the electric force on the electron to the magnetic force on the electron?

Your employer asks you to build a 20-cm-long solenoid with an interior field of 5.0 mT. The specifications call for a single layer of wire, wound with the coils as close together as possible. You have two spools of wire available. Wire with a #18 gauge has a diameter of 1.02 mm and has a maximum current rating of 6 A. Wire with a #26 gauge is 0.41 mm in diameter and can carry up to 1 A. Which wire should you use, and what current will you need?

The magnetic field strength at the north pole of a 2.0-cmdiameter, 8-cm-long Alnico magnet is 0.10 T. To produce the same field with a solenoid of the same size, carrying a current of 2.0 A, how many turns of wire would you need?

The earths magnetic field, with a magnetic dipole moment of 8.0 * 1022 Am2 , is generated by currents within the molten iron of the earths outer core. Suppose we model the core current as a 3000-km-diameter current loop made from a 1000-km-diameter wire. The loop diameter is measured from the centers of this very fat wire. a. What is the current in the current loop? b. What is the current density J in the current loop? c. To decide whether this is a large or a small current density, compare it to the current density of a 1.0 A current in a 1.0-mm-diameter wire

Weak magnetic fields can be measured at the surface of the brain. Although the currents causing these fields are quite complicated, we can estimate their size by modeling them as a current loop around the equator of a 16-cm-diameter (the width of a typical head) sphere. What current is needed to produce a 3.0 pT fieldthe strength measured for one subjectat the pole of this sphere?

The heart produces a weak magnetic field that can be used to diagnose certain heart problems. It is a dipole field produced by a current loop in the outer layers of the heart. a. It is estimated that the field at the center of the heart is 90 pT. What current must circulate around an 8.0-cm-diameter loop, about the size of a human heart, to produce this field? b. What is the magnitude of the hearts magnetic dipole moment?

What is the magnetic field strength at the center of the semicircle in FIGURE P29.54?

The toroid of FIGURE P29.55 is a coil of wire wrapped around a doughnut-shaped ring (a torus). Toroidal magnetic fields are used to confine fusion plasmas. a. From symmetry, what must be the shape of the magnetic field in this toroid? Explain. b. Consider a toroid with N closely spaced turns carrying current I. Use Ampres law to find an expression for the magnetic field strength at a point inside the torus at distance r from the axis. c. Is a toroidal magnetic field a uniform field? Explain.

The coaxial cable shown in FIGURE P29.56 consists of a solid inner conductor of radius R1 surrounded by a hollow, very thin outer conductor of radius R2. The two carry equal currents I, but in opposite directions. The current density is uniformly distributed over each conductor. a. Find expressions for three magnetic fields: within the inner conductor, in the space between the conductors, and outside the outer conductor. b. Draw a graph of B versus r from r = 0 to r = 2R2 if R1 = 1 3 R2.

A long, hollow wire has inner radius R1 and outer radius R2. The wire carries current I uniformly distributed across the area of the wire. Use Ampres law to find an expression for the magnetic field strength in the three regions 0 6 r 6 R1, R1 6 r 6 R2, and R2 6 r.

A proton moving in a uniform magnetic field with v u 1 = 1.00 * 106 ni m/s experiences force F u 1 = 1.20 * 10-16 k n N. A second proton with v u 2 = 2.00 * 106 j n m/s experiences F u 2 = -4.16 * 10-16 k n N in the same field. What is B u ? Give your answer as a magnitude and an angle measured ccw from the +x@ axis

An electron travels with speed 1.0 * 107 m/s between the two parallel charged plates shown in FIGURE P29.59. The plates are separated by 1.0 cm and are charged by a 200 V battery. What magnetic field strength and direction will allow the electron to pass between the plates without being deflected?

An electron in a cathode-ray tube is accelerated through a potential difference of 10 kV, then passes through the 2.0-cmwide region of uniform magnetic field in FIGURE P29.60. What field strength will deflect the electron by 10?

An antiproton (same properties as a proton except that q = -e) is moving in the combined electric and magnetic fields of FIGURE P29.61. What are the magnitude and direction of the antiprotons acceleration at this instant?

a. A 65-cm-diameter cyclotron uses a 500 V oscillating potential difference between the dees. What is the maximum kinetic energy of a proton if the magnetic field strength is 0.75 T? b. How many revolutions does the proton make before leaving the cyclotron?

An antiproton is identical to a proton except it has the opposite charge, -e. To study antiprotons, they must be confined in an ultrahigh vacuum because they will annihilateproducing gamma raysif they come into contact with the protons of ordinary matter. One way of confining antiprotons is to keep them in a magnetic field. Suppose that antiprotons are created with a speed of 1.5 * 104 m/s and then trapped in a 2.0 mT magnetic field. What minimum diameter must the vacuum chamber have to allow these antiprotons to circulate without touching the walls?

FIGURE P29.64 shows a mass spectrometer, an analytical instrument used to identify the various molecules in a sample by measuring their charge-to-mass ratio q/m. The sample is ionized, the positive ions are accelerated (starting from rest) through a potential difference V, and they then enter a region of uniform magnetic field. The field bends the ions into circular trajectories, but after just half a circle they either strike the wall or pass through a small opening to a detector. As the accelerating voltage is slowly increased, different ions reach the detector and are measured. Consider a mass spectrometer with a 200.00 mT magnetic field and an 8.0000 cm spacing between the entrance and exit holes. To five significant figures, what accelerating potential differences V are required to detect the ions (a) O2 +, (b) N2 +, and (c) CO+? See Exercise 29 for atomic masses; the mass of the missing electron is less than 0.001 u and is not relevant at this level of precision. Although N2 + and CO+ both have a nominal molecular mass of 28, they are easily distinguished by virtue of their slightly different accelerating voltages. Use the following constants: 1 u = 1.6605 * 10-27 kg, e = 1.6022 * 10-19 C.

The uniform 30 mT magnetic field in FIGURE P29.65 points in the positive z-direction. An electron enters the region of magnetic field with a speed of 5.0 * 106 m/s and at an angle of 30 above the xy-plane. Find the radius r and the pitch p of the electrons spiral trajectory

Particle accelerators, such as the Large Hadron Collider, use magnetic fields to steer charged particles around a ring. Consider a proton ring with 36 identical bending magnets connected by straight segments. The protons move along a 1.0-m-long circular arc as they pass through each magnet. What magnetic field strength is needed in each magnet to steer protons around the ring with a speed of 2.5 * 107 m/s? Assume that the field is uniform inside the magnet, zero outside.

A particle of charge q and mass m moves in the uniform fields E u = E0 k n and B u = B0 k n. At t = 0, the particle has velocity v u 0 = v0ni. What is the particles speed at a later time t?

A Hall-effect probe to measure magnetic field strengths needs to be calibrated in a known magnetic field. Although it is not easy to do, magnetic fields can be precisely measured by measuring the cyclotron frequency of protons. A testing laboratory adjusts a magnetic field until the protons cyclotron frequency is 10.0 MHz. At this field strength, the Hall voltage on the probe is 0.543 mV when the current through the probe is 0.150 mA. Later, when an unknown magnetic field is measured, the Hall voltage at the same current is 1.735 mV. What is the strength of this magnetic field?

It is shown in more advanced courses that charged particles in circular orbits radiate electromagnetic waves, called cyclotron radiation. As a result, a particle undergoing cyclotron motion with speed v is actually losing kinetic energy at the rate dK dt = - 1 m0q4 6pcm2 2B2 v 2 How long does it take (a) an electron and (b) a proton to radiate away half its energy while spiraling in a 2.0 T magnetic field?

A proton in a cyclotron gains K = 2eV of kinetic energy per revolution, where V is the potential between the dees. Although the energy gain comes in small pulses, the proton makes so many revolutions that it is reasonable to model the energy as increasing at the constant rate P = dK/dt = K/T, where T is the period of the cyclotron motion. This is power input because it is a rate of increase of energy. a. Find an expression for r1t2, the radius of a protons orbit in a cyclotron, in terms of m, e, B, P, and t. Assume that r = 0 at t = 0. Hint: Start by finding an expression for the protons kinetic energy in terms of r. b. A relatively small cyclotron is 2.0 m in diameter, uses a 0.55 T magnetic field, and has a 400 V potential difference between the dees. What is the power input to a proton, in W? c. How long does it take a proton to spiral from the center out to the edge?

The 10-turn loop of wire shown in FIGURE P29.71 lies in a horizontal plane, parallel to a uniform horizontal magnetic field, and carries a 2.0 A current. The loop is free to rotate about a nonmagnetic axle through the center. A 50 g mass hangs from one edge of the loop. What magnetic field strength will prevent the loop from rotating about the axle?

The two springs in FIGURE P29.72 each have a spring constant of 10 N/m. They are compressed by 1.0 cm when a current passes through the wire. How big is the current?

each side, with its lower edge resting on a frictionless, horizontal surface. A 25 A current is flowing around the loop in the direction shown. What is the strength of a uniform, horizontal magnetic field for which the loop is in static equilibrium at the angle shown?

Magnetic fields are sometimes measured by balancing magnetic forces against known mechanical forces. Your task is to measure the strength of a horizontal magnetic field using a 12-cm-long rigid metal rod that hangs from two nonmagnetic springs, one at each end, with spring constants 1.3 N/m. You first position the rod to be level and perpendicular to the field, whose direction you determined with a compass. You then connect the ends of the rod to wires that run parallel to the field and thus experience no forces. Finally, you measure the downward deflection of the rod, stretching the springs, as you pass current through it. Your data are as follows: Current (A) Deflection (mm) 1.0 4 2.0 9 3.0 12 4.0 15 5.0 21 Use an appropriate graph of the data to determine the magnetic field strength.

A conducting bar of length l and mass m rests at the left end of the two frictionless rails of length d in FIGURE P29.75. A uniform magnetic field of strength B points upward. a. In which direction, into or out of the page, will a current through the conducting bar cause the bar to experience a force to the right? b. Find an expression for the bars speed as it leaves the rails at the right end.

a. In FIGURE P29.76, a long, straight, current-carrying wire of linear mass density m is suspended by threads. A magnetic field perpendicular to the wire exerts a horizontal force that deflects the wire to an equilibrium angle u. Find an expression for the strength and direction of the magnetic field B u . b. What B u deflects a 55 g/m wire to a 12 angle when the current is 10 A?

A wire along the x-axis carries current I in the negative xdirection through the magnetic field B u = B0 x l k n 0 x l 0 elsewhere a. Draw a graph of B versus x over the interval -3 2 l 6 x 6 3 2 l. b. Find an expression for the net force F u net on the wire. c. Find an expression for the net torque on the wire about the point x = 0.

A nonuniform magnetic field exerts a net force on a current loop of radius R. FIGURE P29.78 shows a magnetic field that is diverging from the end of a bar magnet. The magnetic field at the position of the current loop makes an angle u with respect to the vertical. a. Find an expression for the net magnetic force on the current. b. Calculate the force if R = 2.0 cm, I = 0.50 A, B = 200 mT, and u = 20.

You have a 1.0-m-long copper wire. You want to make an N-turn current loop that generates a 1.0 mT magnetic field at the center when the current is 1.0 A. You must use the entire wire. What will be the diameter of your coil?

a. Derive an expression for the magnetic field strength at distance d from the center of a straight wire of finite length l that carries current I. b. Determine the field strength at the center of a currentcarrying square loop having sides of length 2R. c. Compare your answer to part b to the field at the center of a circular loop of diameter 2R. Do so by computing the ratio Bsquare/Bcircl

A flat, circular disk of radius R is uniformly charged with total charge Q. The disk spins at angular velocity v about an axis through its center. What is the magnetic field strength at the center of the disk?

A long, straight conducting wire of radius R has a nonuniform current density J = J0r/R, where J0 is a constant. The wire carries total current I. a. Find an expression for J0 in terms of I and R. b. Find an expression for the magnetic field strength inside the wire at radius r. c. At the boundary, r = R, does your solution match the known field outside a long, straight current-carrying wire?

An infinitely wide flat sheet of charge flows out of the page in FIGURE CP29.83. The current per unit width along the sheet (amps per meter) is given by the linear current density Js. a. What is the shape of the magnetic field? To answer this question, you may find it helpful to approximate the current sheet as many parallel, closely spaced current-carrying wires. Give your answer as a picture showing magnetic field vectors. b. Find the magnetic field strength at distance d above or below the current sheet.