Solved: A 20-cm-long conducting rod with mass 18 g is suspended by wires of negligible

A charged particle moves through a region containing only a magnetic field. Under what condition will it experience no force?

An electron moving with velocity v ! through a magnetic field B S experiences a magnetic force F S . Which of the vectors F S , v ! , and B S must be at right angles?

A magnetic field points out of this page. Will a positively charged particle moving in the plane of the page circle clockwise or counterclockwise as viewed from above?

Do particles in a cyclotron gain energy from the electric field, the magnetic field, or both? Explain

An electron and a proton moving at the same speed enter a region containing a uniform magnetic field. Which is deflected more from its original path?

Two identical particles carrying equal charge are moving in opposite directions, perpendicular to a uniform magnetic field, when they collide elastically head-on. Describe their subsequent motion.

In what two senses does a current loop behave like a magnetic dipole?

The BiotSavart law shows that the magnetic field of a current element decreases as 1/r2 . Could you put together a complete circuit whose field exhibits this decrease? Why or why not?

Do currents in the same direction attract or repel? Explain.

If a current is passed through an unstretched spring, will the spring contract or expand? Explain.

Figure 26.38 shows some magnetic field lines associated with two parallel wires carrying equal currents perpendicular to the page. Are the currents in the same or opposite directions? How can you tell? Note: The only currents in Fig. 26.38 are those in the two wires.

Why is a piece of iron attracted into a solenoid?

Would there be a magnetic force on a piece of iron deep inside a long solenoid? Explain.

An unmagnetized piece of iron has no net magnetic dipole moment, yet its attracted to either pole of a bar magnet. Why?

Find (a) the minimum magnetic field needed to exert a 5.4-fN force on an electron moving at 21 Mm/s and (b) the field strength required if the field were at 45 to the electrons velocity.

An electron moving at right angles to a 0.10-T magnetic field experiences an acceleration of 6.0*1015 m/s2 . (a) Whats its speed? (b) By how much does its speed change in 1 ns?

Find the magnitude of the magnetic force on a proton moving at 2.5*105 m/s (a) perpendicular; (b) at 30; (c) parallel to a 0.50-T magnetic field.

The magnitude of Earths magnetic field is about 0.5 gauss near Earths surface. Whats the maximum possible magnetic force on an electron with kinetic energy of 1 keV? Compare with the gravitational force on the electron

A velocity selector uses a 60-mT magnetic field perpendicular to a 24-kN/C electric field. At what speed will charged particles pass through the selector undeflected?

Find the radius of the path described by a proton moving at 15 km/s in a plane perpendicular to a 400-G magnetic field.

How long does it take an electron to complete a circular orbit perpendicular to a 1.0-G magnetic field?

Radio astronomers detect electromagnetic radiation at a frequency of 42 MHz from an interstellar gas cloud. If the radiation results from electrons spiraling in a magnetic field, whats the field strength?

In a microwave oven, electrons describe circular motion in a magnetic field within a special tube called a magnetron; as youll learn in Chapter 29, the electrons motion results in the production of micowaves. (a) If the electrons circle at a frequency of 2.45 GHz, whats the magnetic field strength? (b) If the magnetron can accommodate electron orbits with maximum diameter 2.72 mm, whats the electrons energy in eV?

Two protons, moving in a plane perpendicular to a uniform 500-G magnetic field, undergo an elastic head-on collision. How much time elapses before they collide again?

Find the magnitude of the force on a 65.5-cm-long wire carrying 12.0 A at right angles to a 475-G magnetic field.

A wire carrying 15 A makes a 25 angle with a uniform magnetic field. The magnetic force per unit length of wire is 0.31 N/m. Find (a) the magnetic field strength and (b) the maximum force per unit length that could be achieved by reorienting the wire.

Youre on a team performing a high-magnetic-field experiment. A conducting bar carrying 4.1 kA will pass through a 1.3-m-long region containing a 12-T magnetic field, making a 60 angle with the field. A colleague proposes resting the bar on wooden blocks. You argue that it will have to be clamped in place, and to back up your argument you claim that the magnetic force will exceed 10,000 pounds. Are you right?

A wire with mass per unit length 75 g/m runs horizontally at right angles to a horizontal magnetic field. A 6.2-A current in the wire results in its being suspended against gravity. Whats the magnetic field strength?

A wire carries 6.71 A. You form it into a single-turn circular loop and measure a magnetic field of 42.8 T at the loop center. (a) Whats the loops radius? (b) Whats the field strength on the loop axis at 10.0 cm from the loop center?

A single-turn wire loop is 2.0 cm in diameter and carries a 650-mA current. Find the magnetic field strength (a) at the loop center and (b) on the loop axis, 20 cm from the center.

A 2.2-m-long wire carrying 3.5 A is wound into a tight coil 5.0 cm in diameter. Find the magnetic field at its center.

Whats the current in a long wire if the magnetic field strength 1.2 cm from the wires axis is 67 T?

In standard household wiring, parallel wires about 1 cm apart carry currents of about 15 A. Whats the force per unit length between these wires?

Earths magnetic dipole moment is 8.0*1022 A# m2 . Find the magnetic field strength at Earths magnetic poles.

A single-turn square wire loop 18.0 cm on a side carries a 1.25-A current. (a) Whats the loops magnetic dipole moment? (b) Whats the magnitude of the torque the loop experiences when its in a 2.12-T magnetic field with the loops dipole moment vector at 65.0 to the field?

An electric motor contains a 250-turn circular coil 6.2 cm in diameter. If it develops a maximum torque of 1.2 N# m at a current of 3.3 A, whats the magnetic field strength?

The line integral of the magnetic field on a closed path surrounding a wire has the value 8.8 T# m. Find the current in the wire.

The magnetic field shown in Fig. 26.39 has uniform magnitude 75 T, but its direction reverses abruptly. Find the current encircled by the rectangular loop shown.

Number 12 gauge wire, commonly used in household wiring, is 2.053 mm in diameter and can safely carry currents of up to 20.0 A. For a wire carrying this maximum current, find the magnetic field strength (a) 0.150 mm from the wires axis, (b) at the wires surface, and (c) 0.375 mm beyond the wires surface

Show that Equations 26.18 and 26.19 give the same results when evaluated at the wires surface.

A superconducting solenoid has 3300 turns per meter and carries 4.1 kA. Find the magnetic field strength in the solenoid.

A particle carrying a 50@C charge moves with velocity v ! = 5.0ni + 3.2k n m/s through a magnetic field given by B S = 9.4 ni + 6.7 nj T. (a) Find the magnetic force on the particle. (b) Form the dot products F S # v ! and F S # B S to show explicitly that the force is perpendicular to both v ! and B

Jupiter has the strongest magnetic field in our solar system, about 14 G at its poles. Approximating the field as that of a dipole, find Jupiters magnetic dipole moment. (Hint: Consult Appendix E.)

A proton moving with velocity v ! 1 = 3.6*104 nj m/s experiences a magnetic force of 7.4*10-16 ni N. A second proton moving on the x-axis experiences a magnetic force of 2.8*10-16 nj N. Find the magnitude and direction of the magnetic field (assumed uniform), and the velocity of the second proton.

A simplified model of Earths magnetic field has it originating in a single current loop at the outer edge of the planets liquid core (radius 3000 km). What current would give the 62@T field measured at the north magnetic pole?

A beam of electrons moving in the x-direction at 8.7 Mm/s enters a region where a uniform 180-G magnetic field points in the ydirection. The boundary of the field region is perpendicular to the beam. How far into the field region does the beam penetrate?

Show that the orbital radius of a charged particle moving at right angles to a magnetic field B can be written r = 12Km/qB, where K is the kinetic energy in joules, m the particles mass, and q its charge

A 90-cm-diameter cyclotron with a 2.0-T magnetic field is used to accelerate deuterium nuclei (one proton plus one neutron). (a) At what frequency should the dee voltage be alternated? (b) Whats the maximum kinetic energy of the deuterons? (c) If the magnitude of the potential difference between the dees is 1500 V, how many orbits do the deuterons complete before reaching maximum energy?

An electron is moving in a uniform 0.25-T magnetic field; its velocity components parallel and perpendicular to the field are both 3.1 Mm/s. (a) Whats the radius of the electrons spiral path? (b) How far does it move along the field direction in the time it takes to complete a full orbit about the field?

A wire of negligible resistance is bent into a rectangle as in Fig. 26.40, and a battery and resistor are connected as shown. The right-hand side of the circuit extends into a region containing a uniform 38-mT magnetic field pointing into the page. Find the magnitude and direction of the net force on the circuit.

Youre designing a prosthetic ankle that includes a miniature electric motor containing a 150-turn circular coil 15 mm in diameter. The motor needs to develop a maximum torque of 3.1 mN# m. The strongest magnets available that will fit in the prosthesis produce a 220-mT field. What current do you need in your motors coil?

A 20-cm-long conducting rod with mass 18 g is suspended by wires of negligible mass (Fig. 26.41). A uniform magnetic field of 0.15 T points horizontally into the page, as shown. An external circuit supplies current between the supports A and B. (a) Whats the minimum current necessary to move the bar to the upper position, so its supported against gravity? (b) What direction should the current flow?

A rectangular copper strip measures 1.0 mm in the direction of a uniform 2.4-T magnetic field. When the strip carries a 6.8-A current perpendicular to the field, a 1.2@V Hall potential develops across the strip. Find the number density of free electrons in the copper.

A single-turn wire loop 10 cm in diameter carries a 12-A current. It experiences a 0.015 N# m torque when the normal to the loop plane makes a 25 angle with a uniform magnetic field. Find the magnetic field strength.

A simple electric motor consists of a 220-turn coil, 4.2 cm in diameter, mounted between the poles of a magnet that produces a 95-mT field. When a 15-A current flows in the coil, what are (a) the coils magnetic dipole moment and (b) the motors maximum torque?

Nuclear magnetic resonance (NMR) is a technique for analyzing chemical structures and also the basis of magnetic resonance imaging used for medical diagnosis. NMR relies on sensitive measurements of the energy needed to flip atomic nuclei by 180 in a given magnetic field. In an apparatus with a 9.4-T magnetic field, what energy is needed to flip a proton (m = 1.41*10-26 A# m2 ) from parallel to antiparallel to the field?

A wire carrying 1.5 A passes through a 48-mT magnetic field. The wire is perpendicular to the field and makes a quarter-circle turn of radius 21 cm in the field region, as shown in Fig. 26.42. Find the magnitude and direction of the magnetic force on the curved section of wire.

Your company is developing a device incorporating a 20-cmdiameter coil carrying 0.50 A that, when properly oriented, will just cancel Earths 50@T magnetic field at the coils center. How much wire must you requisition for each coil?

A single piece of wire carrying current I is bent so it includes a circular loop of radius a, as shown in Fig. 26.43. Find an expression for the magnetic field at the loop center.

You and a friend get lost while hiking, so your friend pulls out a magnetic compass to get re-oriented. However, youre standing right under a power line carrying 1.5 kA toward magnetic north; its 10 m above the compass. The horizontal component of Earths magnetic field at your latitude points northward and has magnitude 0.24 G. Will the compass help you find your way?

Part of a long wire carrying current I is bent into a semicircle of radius a, as in Fig. 26.44. Use the BiotSavart law to find the magnetic field at P, the center of the semicircle.

Three parallel wires of length l each carry current I in the same direction. Theyre positioned at the vertices of an equilateral triangle of side a, and oriented perpendicular to the triangle. Find an expression for the magnitude of the force on each wire.

A long, straight wire carries a 25-A current. A 10-cm by 15-cm rectangular wire loop carrying 850 mA is 3.0 cm from the wire, as shown in Fig. 26.45. Find the magnitude and direction of the net magnetic force on the loop.

A long conducting rod of radius R carries a nonuniform current density J = J0r/R, where J0 is a constant and r is the radial distance from the rods axis. Find expressions for the magnetic field strength (a) inside and (b) outside the rod.

A long, hollow conducting pipe of radius R carries a uniform current I along the pipe, as shown in Fig. 26.46. Use Ampres law to find the magnetic field strength (a) inside and (b) outside the pipe.

The coaxial cable shown in Fig. 26.47 consists of a solid inner conductor of radius a and a hollow outer conductor of inner radius b and thickness c. The two carry equal but opposite currents I, uniformly distributed. Find expressions for the magnetic field as a function of radial position r (a) within the inner conductor, (b) between the inner and outer conductors, and (c) beyond the outer conductor.

A solenoid used in a plasma physics experiment is 10 cm in diameter, is 1.0 m long, and carries a 35-A current to produce a 100-mT magnetic field. (a) How many turns are in the solenoid? (b) If the solenoid resistance is 2.7 , how much power does it dissipate?

You have 10 m of 0.50-mm-diameter copper wire and a battery capable of passing 15 A through the wire. What magnetic field strengths could you obtain (a) inside a 2.0-cm-diameter solenoid wound with the wire as closely spaced as possible and (b) at the center of a single circular loop made from the wire?

Derive Equation 26.21 for the solenoid field by considering the solenoid to be made of infinitesimal current loops. Use Equation 26.9 for the loop fields, and integrate over all loops.

The largest lightning strikes have peak currents of around 250 kA, flowing in essentially cylindrical channels of ionized air. How far from such a flash would the resulting magnetic field be equal to Earths magnetic field strength, about 50 T?

A coaxial cable (see Fig. 26.47) consists of a 1.0-mm-diameter inner conductor and a 0.20-mm-thick outer conductor with interior diameter 1.0 cm. A 100-mA current flows down the inner conductor and back along the outer conductor. Find the magnetic field strength (a) 0.10 mm, (b) 5.0 mm, and (c) 2.0 cm from the cable axis.

Indium antimonide (InSb) is a semiconductor commonly used in Hall-effect devices because of its relatively large Hall coefficient. A magnetic-field sensor is made from a 50@m-thick strip of InSb, with Hall coefficient 228 cm3 /C. The table below shows the Hall potential as a function of current when the sensor is oriented with its current perpendicular to the unknown magnetic field. Plot the Hall potential against a quantity that should give a straight line, determine a best-fit line, and from it find the magnetic field strength.

Suppose the current sheet in Example 26.9 is actually a slab with non-negligible thickness d and that the current is distributed uniformly throughout its volume. Find an expression for the magnetic field inside the slab as a function of the perpendicular distance x from the center plane of the slab. Show that your result agrees with that of Example 26.9 at the surface of the slab.

A circular wire loop of radius 15 cm and negligible thickness carries a 2.0-A current. Use suitable approximations to find the magnetic field of this loop (a) in the loop plane, 1.0 mm outside the loop, and (b) on the loop axis, 3.0 m from the loop center.

A long, flat conducting bar of width w carries a total current I distributed uniformly, as shown in Fig. 26.48. Use approximations to write expressions for the magnetic field strength (a) near the conductor surface 1r V w2 but not near its edges and (b) far from the conductor 1r W w2.

A long, hollow conducting pipe of radius R and length l carries a uniform current I flowing around the pipe (Fig. 26.49). Find expressions for the magnetic field (a) inside and (b) outside the pipe. (Hint: What configuration does this resemble?)

A solid conducting wire of radius R runs parallel to the z-axis and carries a current density given by J S = J011 - r/R2k n, where J0 is a constant and r is the distance from the wire axis. Find expressions for (a) the total current in the wire and (b) the magnetic field for r 7 R and (c) r 6 R.

A disk of radius a carries uniform surface charge density s and rotates with angular speed v about the disk axis. Show that the magnetic field at the disks center is 1 2m0sva.

Youre developing a system to orient an orbiting telescope. The system uses three perpendicular coils, with torques developed in Earths magnetic field when current passes through them. Weight limitations restrict you to a length l of wire for each coil. A colleague argues youll get the greatest dipole moment and therefore the most torque with a multi-turn coil. You say a 1-turn coil is best. Whos right?

The structure shown in Fig. 26.50 is made from conducting rods. The upper horizontal rod (mass 22 g, length 95 cm) is free to slide vertically on the uprights while maintaining electrical contact. A battery connected across the insulating gap at the bottom of the left-hand upright drives 66 A through the structure. At what height h will the upper wire be in equilibrium?

A long, flat conducting ribbon of width w is parallel to a long, straight wire; its near edge is a distance a from the wire (Fig. 26.51). Wire and ribbon carry the same current I; its distributed uniformly over the ribbon. Use integration to show that the force per unit length between the two has magnitude m0I 2 2pw ln a a + w a b

Find an expression for the magnetic field at the center of a square loop of side a carrying current I

Repeat the calculation in Problem 69 for a solenoid of finite length l and cross-sectional radius a to find the magnetic field strength at the center of the solenoids axis

A magnetic dipole m ! = mni is on the axis of a circular current loop of radius a oriented as shown in Fig. 26.17a, a distance x from the center. Differentiate Equation 26.16 to find the force on the dipole, and evaluate its magnitude for x = a. Is the force attractive or repulsive?

Youre an engineer at a nuclear power plant, and one of your colleagues has drawn up plans to reroute the conductors carrying current from the plants electric generator. Your colleague wants to carry this current on two parallel conducting rods 30 cm apart; each rod carries 15 kA with the currents flowing in opposite directions. The proposal calls for clamping the conductors in place every meter, with clamps capable of withstanding a maximum force of 100 N. Is the clamp design adequate?

Derive Equation 26.20 by considering the current sheet to be made of infinitely many infinitesimal line currents.

Your roommate is sold on magnet therapy, a sham treatment using small bar magnets attached to the body. You skeptically ask your roommate how this is supposed to work. He mumbles something about the Hall effect speeding blood flow. In reply, you estimate the Hall potential associated with typical blood parameters in the 100-G field of a bar magnet: red blood cells carrying 2-pC charge in a 12-cm/s flow through a 3.0-mm-diameter blood vessel containing 5 billion red blood cells per mL. To show that the Hall potential is negligible, you compare your estimate with the tens of mV typical of bioelectric activity. How do the two values compare?

A toroid is a solenoid-like coil bent into a circle (Fig. 26.52a). Toroids are the configuration of choice in magnetic-confinement nuclear fusion experiments, which, if successful, could provide us with an almost unlimited energy source using deuterium fuel extracted from seawater. The ITER consortium, an international collaboration, is building a large toroidal fusion experiment in France; its expected to be the first fusion device to produce energy on a large scale. Figure 26.52b shows a cross section of a toroid, with current emerging from the page at the inner edge and descending at the outer edge. The black circle is an Amprian loop The magnetic field associated with the toroid is nonzero a. only within the hole in the donut-shaped coil. b. only within the region bounded by the coils. c. only outside the coils. d. everywhere.

A toroid is a solenoid-like coil bent into a circle (Fig. 26.52a). Toroids are the configuration of choice in magnetic-confinement nuclear fusion experiments, which, if successful, could provide us with an almost unlimited energy source using deuterium fuel extracted from seawater. The ITER consortium, an international collaboration, is building a large toroidal fusion experiment in France; its expected to be the first fusion device to produce energy on a large scale. Figure 26.52b shows a cross section of a toroid, with current emerging from the page at the inner edge and descending at the outer edge. The black circle is an Amprian loop In Fig. 26.52b, the magnetic field lines must be a. straight, and pointing into the page. b. straight, and pointing out of the page. c. straight, and pointing radially. d. circular.

A toroid is a solenoid-like coil bent into a circle (Fig. 26.52a). Toroids are the configuration of choice in magnetic-confinement nuclear fusion experiments, which, if successful, could provide us with an almost unlimited energy source using deuterium fuel extracted from seawater. The ITER consortium, an international collaboration, is building a large toroidal fusion experiment in France; its expected to be the first fusion device to produce energy on a large scale. Figure 26.52b shows a cross section of a toroid, with current emerging from the page at the inner edge and descending at the outer edge. The black circle is an Amprian loop Doubling the total number of turns N in the toroid, without changing its size or the current, will a. double the magnetic field. b. quadruple the magnetic field. c. halve the magnetic field. d. not change the magnetic field.

A toroid is a solenoid-like coil bent into a circle (Fig. 26.52a). Toroids are the configuration of choice in magnetic-confinement nuclear fusion experiments, which, if successful, could provide us with an almost unlimited energy source using deuterium fuel extracted from seawater. The ITER consortium, an international collaboration, is building a large toroidal fusion experiment in France; its expected to be the first fusion device to produce energy on a large scale. Figure 26.52b shows a cross section of a toroid, with current emerging from the page at the inner edge and descending at the outer edge. The black circle is an Amprian loop The toroid has inner radius Rin and outer radius Rout, while r is the radial coordinate measured from the center. The toroid is made from wire wound into a total of N turns, and carries current I. Which of the following is the correct formula for the magnetic field within the coils? a. B = m0NI b. B = m0NI/2pRin c. B = m0NI/2pRout d. B = m0NI/2pr