A toroid is a solenoid in the shape of a donut (Fig. 2060). Use Ampres law along the

(I) (a) What is the force per meter of length on a straight wire carrying a 6.40-A current when perpendicular to a \(0.90-\mathrm{T}\) uniform magnetic field? (b) What if the angle between the wire and field is \(35.0^{\circ}\) ? Equation Transcription: Text Transcription: 0.90-T 35.0^circ

How much current is flowing in a wire 4.80 m long if the maximum force on it is 0.625 N when placed in a uniform 0.0800-T field?

(I) A 240-m length of wire stretches between two towers and carries a 120-A current. Determine the magnitude of the force on the wire due to the Earth's magnetic field of \(5.0 \times 10^{-5} \mathrm{~T}\) which makes an angle of \(68^{\circ}\) with the wire. Equation Transcription: Text Transcription: 5.0 times 10^-5 T 68^circ

(I) A 2.6-m length of horizontal wire carries a 4.5-A current toward the south. The dip angle of the Earth's magnetic field makes an angle of \(41^{\circ}\) to the wire. Estimate the magnitude of the magnetic force on the wire due to the Earth's magnetic field of \(5.5 \times 10^{-5} \mathrm{~T}\). Equation Transcription: Text Transcription: 41^circ 5.5 times 10^-5 T

The magnetic force per meter on a wire is measured to be only 45% of its maximum possible value. What is the angle between the wire and the magnetic field?

The force on a wire carrying 6.45 A is a maximum of 1.28 N when placed between the pole faces of a magnet. If the pole faces are 55.5 cm in diameter, what is the approximate strength of the magnetic field?

(II) The force on a wire is a maximum of \(8.50 \times 10^{-2} \mathrm{~N}\) when placed between the pole faces of a magnet. The current flows horizontally to the right and the magnetic field is vertical. The wire is observed to "jump" toward the observer when the current is turned on. (a) What type of magnetic pole is the top pole face? (b) If the pole faces have a diameter of 10.0 cm, estimate the current in the wire if the field is 0.220 T. (c) If the wire is tipped so that it makes an angle of \(10.0^{\circ}\) with the horizontal, what force will it now feel? [Hint: What length of wire will now be in the field?] Equation Transcription: Text Transcription: 8.50 times 10^-2 N 10.0^circ

(II) Suppose a straight 1.00-mm-diameter copper wire could just "float" horizontally in air because of the force due to the Earth's magnetic field \(\overrightarrow{\mathbf{B}}\), which is horizontal, perpendicular to the wire, and of magnitude \(5.0 \times 10^{-5} \mathrm{~T}\). What current would the wire carry? Does the answer seem feasible? Explain briefly. Equation Transcription: Text Transcription: overrightarrow^B 5.0 times 10^-5 T

(I) Determine the magnitude and direction of the force on an electron traveling \(7.75 \times 10^{5} \mathrm{~m} / \mathrm{s}\) horizontally to the east in a vertically upward magnetic field of strength 0.45 T. Equation Transcription: Text Transcription: 7.75 times 10^5 m/s

(I) An electron is projected vertically upward with a speed of \(1.70 \times 10^{6} \mathrm{~m} / \mathrm{s}\) into a uniform magnetic field of 0.640 T that is directed horizontally away from the observer. Describe the electron's path in this field. Equation Transcription: Text Transcription: 1.70 times 10^6 m/s

(I) Alpha particles (charge \(q=+2 e\), mass \(m=6.6 \times 10^{-27} \mathrm{~kg}\) ) move at \(1.6 \times 10^{6} \mathrm{~m} / \mathrm{s}\). What magnetic field strength would be required to bend them into a circular path of radius \(r=0.14 \mathrm{~m}\) ? Equation Transcription: Text Transcription: q=+2e m=6.6 times 10^-27 kg 1.6 times 10^6 m/s r=0.14 m

(I) Find the direction of the force on a negative charge for each diagram shown in Fig. 20–52, where \(\overrightarrow{v}\) (green) is the velocity of the charge and \(\overrightarrow{B}\) (blue) is the direction of the magnetic field. (\(\otimes\) means the vector points inward. \(\odot\) means it points outward, toward you.)

(I) Determine the direction of \(\overrightarrow{\mathbf{B}}\) for each case in Fig. 20-53, where \(\overrightarrow{\mathbf{F}}\) represents the maximum magnetic force on a positively charged particle moving with velocity \(\overrightarrow{\mathbf{v}}\). Equation Transcription: Text Transcription: overrightarrow^B overrightarrow^F overrightarrow^v

(II) What is the velocity of a beam of electrons that goes undeflected when moving perpendicular to an electric and to a magnetic field. \(\overrightarrow{\mathbf{E}}\) and \(\overrightarrow{\mathbf{B}}\) are also perpendicular to each other and have magnitudes \(7.7 \times 10^{3} \mathrm{~V} / \mathrm{m}\) and \(7.5 \times 10^{-3} \mathrm{~T}\), respectively. What is the radius of the electron orbit if the electric field is turned off? Equation Transcription: Text Transcription: overrightarrow^E overrightarrow^B 7.7 times 10^3 V/m 7.5 times 10^-3T

(II) A helium ion \((Q=+2 e)\) whose mass is \(6.6 \times 10^{-27} \mathrm{~kg}\) is accelerated by a voltage of 3700 V. (a) What is its speed? (b) What will be its radius of curvature if it moves in a plane perpendicular to a uniform 0.340-T field? (c) What is its period of revolution? Equation Transcription: Text Transcription: (Q=+2e) 6.6 times 10^-27 kg

(II) For a particle of mass \(m\) and charge \(q\) moving in a circular path in a magnetic field \(B,(a)\) show that its kinetic energy is proportional to \(r^{2}\), the square of the radius of curvature of its path. (b) Show that its angular momentum is \(L=q B r^{2}\), around the center of the circle. Equation Transcription: Text Transcription: m q B,(a) r^2 L=qBr^2

A 1.5-MeV (kinetic energy) proton enters a 0.30-T field, in a plane perpendicular to the field. What is the radius of its path? See Section 174.

(II) An electron experiences the greatest force as it travels \(2.8 \times 10^{6} \mathrm{~m} / \mathrm{s}\) in a magnetic field when it is moving northward. The force is vertically upward and of magnitude \(6.2 \times 10^{-13} \mathrm{~N}\). What is the magnitude and direction of the magnetic field? Equation Transcription: Text Transcription: 2.8 times 10^6 m/s 6.2 times 10^-13 N

A proton and an electron have the same kinetic energy upon entering a region of constant magnetic field. What is the ratio of the radii of their circular paths?

(III) A proton (mass \(m_{\mathrm{p}}\) ), a deuteron \(\left(m=2 m_{\mathrm{p}}, Q=e\right)\), and an alpha particle \(\left(m=4 m_{\mathrm{p}}, Q=2 e\right)\) are accelerated by the same potential difference \(V\) and then enter a uniform magnetic field \(\overrightarrow{\mathbf{B}}\), where they move in circular paths perpendicular to \(\overrightarrow{\mathbf{B}}\). Determine the radius of the paths for the deuteron and alpha particle in terms of that for the proton. Equation Transcription: Text Transcription: m_p (m=2m_p,Q=e) (m=4m_p,Q=2e) V overrightarrow^B overrightarrow^B

(III) A 3.40-g bullet moves with a speed of 155 m/s perpendicular to the Earth's magnetic field of \(5.00 \times 10^{-5} \mathrm{~T}\). If the bullet possesses a net charge of \(18.5 \times 10^{-9} \mathrm{C}\), by what distance will it be deflected from its path due to the Earth's magnetic field after it has traveled 1.50 km ? Equation Transcription: Text Transcription: 5.00 times 10^-5 T 18.5 times 10^-9 T

(III) A Hall probe, consisting of a thin rectangular slab of current-carrying material, is calibrated by placing it in a known magnetic field of magnitude 0.10 T. When the field is oriented normal to the slab's rectangular face, a Hall emf of 12 mV is measured across the slab's width. The probe is then placed in a magnetic field of unknown magnitude \(B\), and a Hall emf of 63 mV is measured. Determine \(B\) assuming that the angle \(\theta\) between the unknown field and the plane of the slab's rectangular face is \((a) \theta=90^{\circ}\), and \((b) \theta=60^{\circ} .\) Equation Transcription: Text Transcription: B B theta (a)theta =90^circ (b) theta=96^circ

The Hall effect can be used to measure blood flow rate because the blood contains ions that constitute an electric current. (a) Does the sign of the ions influence the emf? Explain. (b) Determine the flow velocity in an artery 3.3 mm in diameter if the measured emf across the width of the artery is 0.13 mV and B is 0.070 T. (In actual practice, an alternating magnetic field is used.)

(III) A long copper strip 1.8 cm wide and 1.0 mm thick is placed in a 1.2-T magnetic field as in Fig. 20-21a. When a steady current of 15 A passes through it, the Hall emf is measured to be \(1.02 \mu \mathrm{V}\). Determine (a) the drift velocity of the electrons and (b) the density of free (conducting) electrons (number per unit volume) in the copper. [Hint: See also Section 18-8.] Equation Transcription: Text Transcription: 1.02 mu V

(I) Jumper cables used to start a stalled vehicle often carry a 65 -A current. How strong is the magnetic field 4.5 cm from one cable? Compare to the Earth's magnetic field \(\left(5.0 \times 10^{-5} \mathrm{~T}\right)\). Equation Transcription: Text Transcription: (5.0 times 10^-5 T)

(I) If an electric wire is allowed to produce a magnetic field no larger than that of the Earth \(\left(0.50 \times 10^{-4} \mathrm{~T}\right)\) at a distance of 12 cm from the wire, what is the maximum current the wire can carry? Equation Transcription: Text Transcription: (0.50 times10^-4 T)

Determine the magnitude and direction of the force between two parallel wires 25 m long and 4.0 cm apart, each carrying 25 A in the same direction.

(I) A vertical straight wire carrying an upward 28-A current exerts an attractive force per unit length of \(7.8 \times 10^{-4} \mathrm{~N} / \mathrm{m}\) on a second parallel wire 9.0 cm away. What current (magnitude and direction) flows in the second wire? Equation Transcription: Text Transcription: 7.8 times 10^-4 N/m

(II) In Fig. 20-54, a long straight wire carries current \(I\) out of the page toward you. Indicate, with appropriate arrows, the direction and (relative) magnitude of \(\overrightarrow{\mathbf{B}}\) at each of the points C, D, and E in the plane of the page. Equation Transcription: Text Transcription: I overrightarrow^B

(II) An experiment on the Earth's magnetic field is being carried out 1.00 m from an electric cable. What is the maximum allowable current in the cable if the experiment is to be accurate to \(\pm 3.0 \%\) ? Equation Transcription: Text Transcription: pm 3.0%

(II) A rectangular loop of wire is placed next to a straight wire, as shown in Fig. 20–55. There is a current of 3.5 A in both wires. Determine the magnitude and direction of the net force on the loop.

(II) A horizontal compass is placed 18 cm due south from a straight vertical wire carrying a 48 -A current downward. In what direction does the compass needle point at this location? Assume the horizontal component of the Earth's field at this point is \(0.45 \times 10^{-4} \mathrm{~T}\) and the magnetic declination is \(0^{\circ}\). Equation Transcription: Text Transcription: 0.45 times 10^-4 T 0^circ

(II) A long horizontal wire carries 24.0 A of current due north. What is the net magnetic field 20.0 cm due west of the wire if the Earth's field there points downward, \(44^{\circ}\) below the horizontal, and has magnitude \(5.0 \times 10^{-5} \mathrm{~T} ?\) Equation Transcription: Text Transcription: 44^circ 5.0 times 10^-5 T

(II) A straight stream of protons passes a given point in space at a rate of \(2.5 \times 10^{9}\) protons/s. What magnetic field do they produce 1.5 m from the beam? Equation Transcription: Text Transcription: 2.5 times 10^9

Determine the magnetic field midway between two long straight wires 2.0 cm apart in terms of the current I in one when the other carries 25 A. Assume these currents are (a) in the same direction, and (b) in opposite directions

Two straight parallel wires are separated by 7.0 cm. There is a 2.0-A current flowing in the first wire. If the magnetic field strength is found to be zero between the two wires at a distance of 2.2 cm from the first wire, what is the magnitude and direction of the current in the second wire?

(II) Two long straight wires each carry a current I out of the page toward the viewer, Fig. 20-56. Indicate, with appropriate arrows, the direction of \(\overrightarrow{B}\) at each of the points 1 to 6 in the plane of the page. State if the field is zero at any of the points.

II) A power line carries a current of 95 A west along the tops of 8.5-m-high poles. $(a)$ What is the magnitude and direction of the magnetic field produced by this wire at the ground directly below? How does this compare with the Earth's magnetic field of about \(\frac{1}{2} \mathrm{G}\) ? (b) Where would the wire's magnetic field cancel the Earth's field? Equation Transcription: Text Transcription: 1/2G

(II) A compass needle points \(17^{\circ}\) E of N outdoors. However, when it is placed 12.0 cm to the east of a vertical wire inside a building, it points \(32^{\circ}\) E of N. What is the magnitude and direction of the current in the wire? The Earth's field there is \(0.50 \times 10^{-4} \mathrm{~T}\) and is horizontal. Equation Transcription: Text Transcription: 17^circE 32^circE 0.5 times 10^-4 T

A long pair of insulated wires serves to conduct 24.5 A of dc current to and from an instrument. If the wires are of negligible diameter but are 2.8 mm apart, what is the magnetic field 10.0 cm from their midpoint, in their plane (Fig.2057)? Compare to the magnetic field of the Earth.

A third wire is placed in the plane of the two wires shown in Fig. 2057 parallel and just to the right. If it carries 25.0 A upward, what force per meter of length does it exert on each of the other two wires? Assume it is 2.8 mm from the nearest wire, center to center.

(III) Two long thin parallel wires 13.0 cm apart carry 28-A currents in the same direction. Determine the magnetic field vector at a point 10.0 cm from one wire and 6.0 cm from the other (Fig. 20–58).

(III) Two long wires are oriented so that they are perpendicular to each other. At their closest, they are 20.0 cm apart (Fig. 20–59). What is the magnitude of the magnetic field at a point midway between them if the top one carries a current of 20.0 A and the bottom one carries 12.0 A?

A thin 12-cm-long solenoid has a total of 460 turns of wire and carries a current of 2.0 A. Calculate the field inside the solenoid near the center.

A 30.0-cm-long solenoid 1.25 cm in diameter is to produce a field of 4.65 mT at its center. How much current should the solenoid carry if it has 935 turns of the wire?

A 30.0-cm-long solenoid 1.25 cm in diameter is to produce a field of 4.65 mT at its center. How much current should the solenoid carry if it has 935 turns of the wire?

(II) A 550-turn horizontal solenoid is 15 cm long. The current in its coils is 38 A. A straight wire cuts through the center of the solenoid, along a 3.0-cm diameter. This wire carries a 22-A current downward (and is connected by other wires that don’t concern us). What is the force on this wire assuming the solenoid’s magnetic field points due east?

You have 1.0 kg of copper and want to make a practical solenoid that produces the greatest possible magnetic field for a given voltage. Should you make your copper wire long and thin, short and fat, or something else? Consider other variables, such as solenoid diameter, length, and so on. Explain your reasoning

(III) A toroid is a solenoid in the shape of a donut (Fig. 20-60). Use Ampère's law along the circular paths, shown dashed in Fig. 20-60a, to determine that the magnetic field (a) inside the toroid is \(B=\mu_{0} N I / 2 \pi R\), where \(N\) is the total number of turns, and (b) outside the toroid is \(B=0\). (c) Is the field inside a toroid uniform like a solenoid's? If not, how does it vary? FIGURE 20–60 Problem 49. (a) A toroid or torus. (b) A section of the toroid showing direction of the current for three loops: \(\odot\) means current toward you, and \(\otimes\) means current away from you. Equation Transcription: Text Transcription: B=mu_0 NI/2pi R N B=0 odot otimes

(III) (a) Use Ampère's law to show that the magnetic field between the conductors of a coaxial cable (Fig. 20-61) is \(B=\mu_{0} I / 2 \pi r\) if \(r\) (distance from center) is greater than the radius of the inner wire and less than the radius of the outer cylindrical braid ( = ground). (b) Show that \(B=0\) outside the coaxial cable. Equation Transcription: Text Transcription: B=mu_0 I/2pi r r B=0

(I) A single square loop of wire 22.0 cm on a side is placed with its face parallel to the magnetic field as in Fig. 20–34b. When 5.70 A flows in the coil, the torque on it is \(0.325 \mathrm{~m} \cdot \mathrm{N}\). What is the magnetic field strength? Equation Transcription: Text Transcription: 0.325 m cdot N

If the current to a motor drops by 12%, by what factor does the output torque change?

(I) A galvanometer needle deflects full scale for a \(53.0-\mu \mathrm{A}\) current. What current will give full-scale deflection if the magnetic field weakens to 0.760 of its original value? Equation Transcription: Text Transcription: 53.0-mu A

(II) A circular coil 12.0 cm in diameter and containing nine loops lies flat on the ground. The Earth's magnetic field at this location has magnitude \(5.50 \times 10^{-5} \mathrm{~T}\) and points into the Earth at an angle of \(56.0^{\circ}\) below a line pointing due north. If a 7.20-A clockwise current passes through the coil, (a) determine the torque on the coil, and (b) which edge of the coil rises up: north, east, south, or west? Equation Transcription: Text Transcription: 5.50 times 10^-5T 56.0^circ

Protons move in a circle of radius 6.10 cm in a 0.566-T magnetic field. What value of electric field could make their paths straight? In what direction must the electric field point?

In a mass spectrometer, germanium atoms have radii of curvature equal to 21.0, 21.6, 21.9, 22.2, and 22.8 cm. The largest radius corresponds to an atomic mass of 76 u. What are the atomic masses of the other isotopes?

(II) Suppose the electric field between the electric plates in the mass spectrometer of Fig. 20-41 is \(2.88 \times 10^{4} \mathrm{~V} / \mathrm{m}\) and the magnetic fields are \(B=B^{\prime}=0.68 \mathrm{~T}\). The source contains carbon isotopes of mass numbers 12,13 , and 14 from a long-dead piece of a tree. (To estimate masses of the atoms, multiply by \(\left.1.67 \times 10^{-27} \mathrm{~kg} .\right)\) How far apart are the lines formed by the singly charged ions of each type on the photographic film? What if the ions were doubly charged? Equation Transcription: Text Transcription: 2.88 times 10^4 V/m B=B'=0.68T 1.67 times 10^-27 kg

(II) One form of mass spectrometer accelerates ions by a voltage \(V\) before they enter a magnetic field \(B\). The ions are assumed to start from rest. Show that the mass of an ion is \(m=q B^{2} R^{2} / 2 V\), where \(R\) is the radius of the ions' path in the magnetic field and \(q\) is their charge. Equation Transcription: Text Transcription: V B m=qB^2R^2/2V R q

(II) An unknown particle moves in a straight line through crossed electric and magnetic fields with \(E=1.5 \mathrm{kV} / \mathrm{m}\) and \(B=0.034 \mathrm{~T}\). If the electric field is turned off, the particle moves in a circular path of radius \(r=2.7 \mathrm{~cm}\). What might the particle be? Equation Transcription: Text Transcription: E=1.5 kV/m B=0.034 T r=2.7 cm

(III) A mass spectrometer is monitoring air pollutants. It is difficult, however, to separate molecules of nearly equal mass such as \(\mathrm{CO}(28.0106 \mathrm{u})\) and \(\mathrm{N}_{2}(28.0134 \mathrm{u})\). How large a radius of curvature must a spectrometer have (Fig. 20-41) if these two molecules are to be separated on the film by \(0.50 \mathrm{~mm}\) ? Equation Transcription: Text Transcription: CO (28.0106 u) N_2 (28.0134 u) 0.50 mm

(I) A long thin iron-core solenoid has 380 loops of wire per meter, and a \(350-\mathrm{m} \mathrm{A}\) current flows through the wire. If the permeability of the iron is \(3000 \mu_{0}\), what is the total field \(B\) inside the solenoid? Equation Transcription: Text Transcription: 350-mA 3000 mu_0 B

(II) An iron-core solenoid is 38 cm long and 1.8 cm in diameter, and has 780 turns of wire. The magnetic field inside the solenoid is 2.2 T when 48 A flows in the wire. What is the permeability \(\mu\) at this high field strength? Equation Transcription: Text Transcription: mu

(II) The following are some values of \(B\) and \(B_{0}\) for a piece of iron as it is being magnetized (note different units): Determine the magnetic permeability \(\mu\) for each value and plot a graph of \(\mu\) versus \(B_{0}\). Equation Transcription: Text Transcription: B B_0 mu mu B_0

Two long straight parallel wires are 15 cm apart. Wire A carries 2.0-A current. Wire Bs current is 4.0 A in the same direction. (a) Determine the magnetic field magnitude due to wire A at the position of wire B. (b) Determine the magnetic field due to wire B at the position of wire A. (c) Are these two magnetic fields equal and opposite? Why or why not? (d) Determine the force on wire A due to wire B, and the force on wire B due to wire A. Are these two forces equal and opposite? Why or why not?

Protons with momentum \(4.8 \times 10^{-21} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\) are magnetically steered clockwise in a circular path 2.2 m in diameter. Determine the magnitude and direction of the field in the magnets surrounding the beam pipe. Equation Transcription: Text Transcription: 4.8 times 10^-21 kg cdot m/s

A small but rigid \(\text { U-shaped }\) wire carrying a 5.0-A current (Fig. 20–62) is placed inside a solenoid. The solenoid is 15.0 cm long and has 700 loops of wire, and the current in each loop is 7.0 A. What is the net force on the \(\text { U-shaped }\) wire? Equation Transcription: Text Transcription: U-shaped U-shaped

The power cable for an electric trolley (Fig. 20-63) carries a horizontal current of 330 A toward the east. The Earth's magnetic field has a strength \(5.0 \times 10^{-5} \mathrm{~T}\) and makes an angle of dip of \(22^{\circ}\) at this location. Calculate the magnitude and direction of the magnetic force on an 18-m length of this cable. Equation Transcription: Text Transcription: 5.0 times 10^-5 T 22^circ

A particle of charge q moves in a circular path of radius r perpendicular to a uniform magnetic field B. Determine its linear momentum in terms of the quantities given.

An airplane has acquired a net charge of \(1280 \mu \mathrm{C}\). If the Earth's magnetic field of \(5.0 \times 10^{-5} \mathrm{~T}\) is perpendicular to the airplane's velocity of magnitude 120 m/s, determine the force on the airplane. Equation Transcription: Text Transcription: 1280 mu C 5.0 times 10^-5 T

A 32-cm-long solenoid, 1.8 cm in diameter, is to produce a 0.050-T magnetic field at its center. If the maximum current is 6.4 A, how many turns must the solenoid have

Near the equator, the Earth's magnetic field points almost horizontally to the north and has magnitude \(B=0.50 \times 10^{-4} \mathrm{~T}\). What should be the magnitude and direction for the velocity of an electron if its weight is to be exactly balanced by the magnetic force? Equation Transcription: Text Transcription: B=0.50 times 10^-4 T

A doubly charged helium atom, whose mass is \(6.6 \times 10^{-27}\ kg\), is accelerated by a voltage of 3200 V. (a) What will be its radius of curvature in a uniform 0.240-T field? (b) What is its period of revolution?

A doubly charged helium atom, whose mass is is accelerated by a voltage of 3200 V. (a) What will be its radius of curvature in a uniform 0.240-T field? (b) What is its period of revolution?

(a) What value of magnetic field would make a beam of electrons, traveling to the west at a speed of \(4.8 \times 10^{6} \mathrm{~m} / \mathrm{s}\), go undeflected through a region where there is a uniform electric field of 12,000 V/m pointing south? (b) What is the direction of the magnetic field if it is perpendicular to the electric field? (c) What is the frequency of the circular orbit of the electrons if the electric field is turned off? Equation Transcription: Text Transcription: 4.8 times 10^6 m/s

Magnetic fields are very useful in particle accelerators for “beam steering”; that is, the magnetic fields can be used to change the direction of the beam of charged particles without altering their speed (Fig. 20–65). Show how this could work with a beam of protons. What happens to protons that are not moving with the speed for which the magnetic field was designed? If the field extends over a region 5.0 cm wide and has a magnitude of 0.41 T, by approximately what angle \(\theta\) will a beam of protons traveling at \(2.5 \times 10^{6} \mathrm{~m} / \mathrm{s}\) be bent? Equation Transcription: Text Transcription: theta 2.5 times 10^6 m/s

The magnetic field \(B\) at the center of a circular coil of wire carrying a current \(I\) (as in Fig. 20-9) is \(B=\frac{\mu_{0} N I}{2 r},\) where \(N\) is the number of loops in the coil and \(r\) is its radius. Imagine a simple model in which the Earth's magnetic field of about \(1 \mathrm{G}\left(=1 \times 10^{-4} \mathrm{~T}\right)\) near the poles is produced by a single current loop around the equator. Roughly estimate the current this loop would carry. Equation Transcription: Text Transcription: B I B=mu_0 NI/2r N 1 G (=1 times 10^-4 T)

A proton follows a spiral path through a gas in a uniform magnetic field of 0.010 T, perpendicular to the plane of the spiral, as shown in Fig. 20–66. In two successive loops, at points P and Q, the radii are 10.0 mm and 8.5 mm, respectively. Calculate the change in the kinetic energy of the proton as it travels from P to Q.

Two long straight aluminum wires, each of diameter 0.42 mm, carry the same current but in opposite directions. They are suspended by 0.50-m-long strings as shown in Fig. 20–67. If the suspension strings make an angle of 3.0° with the vertical and are hanging freely, what is the current in the wires?

An electron enters a uniform magnetic field \(B=0.23 \mathrm{~T}\) at a \(45^{\circ}\) angle to \(\overrightarrow{\mathbf{B}}\). Determine the radius \(r\) and pitch \(p\) (distance between loops) of the electron's helical path assuming its speed is \(3.0 \times 10^{6} \mathrm{~m} / \mathrm{s}\). Equation Transcription: Text Transcription: B=0.23 T 45^circ B r p 3.0 times 10^6 m/s

A motor run by a \(9.0-\mathrm{V}\) battery has a 20 -turn square coil with sides of length \(5.0 \mathrm{~cm}\) and total resistance \(28 \Omega\). When spinning, the magnetic field felt by the wire in the coil is \(0.020 \mathrm{~T}\). What is the maximum torque on the motor? Equation Transcription: Text Transcription: 9.0-V 28 Omega 0.020 T

Electrons are accelerated horizontally by 2.2 kV. They then pass through a uniform magnetic field B for a distance of 3.8 cm, which deflects them upward so they reach the top of a screen 22 cm away, 11 cm above the center. Estimate the value of B.

A 175-g model airplane charged to 18.0 mC and traveling at 3.4 m/s passes within 8.6 cm of a wire, nearly parallel to its path, carrying a 25-A current. What acceleration (in g’s) does this interaction give the airplane?

A uniform conducting rod of length \(\ell\) and mass \(m\) sits atop a fulcrum, which is placed a distance \(\ell / 4\) from the rod's left-hand end and is immersed in a uniform magnetic field of magnitude \(B\) directed into the page (Fig. 20-69). An object whose mass \(M\) is 6.0 times greater than the rod's mass is hung from the rod's left-hand end. What current (direction and magnitude) should flow through the rod in order for it to be "balanced" (i.e., be at rest horizontally) on the fulcrum? (Flexible connecting wires which exert negligible force on the rod are not shown.) Equation Transcription: ? ?/4 Text Transcription: ell m ell/4 B M

Suppose the Earth's magnetic field at the equator has magnitude \(0.50 \times 10^{-4} \mathrm{~T}\) and a northerly direction at all points. Estimate the speed a singly ionized uranium ion \((m=238 \mathrm{u}, q=+e)\) would need to circle the Earth \(6.0 \mathrm{~km}\) above the equator. Can you ignore gravity? [Ignore relativity.] Equation Transcription: Text Transcription: 0.50 times 10^-4 T (m=238 u, q=+e) 6.0 km

A particle with charge \(q\) and momentum \(p\), initially moving along the \(x\) axis, enters a region where a uniform magnetic field \(B_{0}\) extends over a width \(x=\ell\) as shown in Fig. 20-70. The particle is deflected a distance \(d\) in the \(+y\) direction as it traverses the field. Determine (a) whether \(q\) is positive or negative, and (b) the magnitude of its momentum \(p\) in terms of \(q, B_{0}, \ell\), and \(d\). Equation Transcription: ? ?, Text Transcription: q p x B0 x=ell d +y q p q,B0,ell, d

A bolt of lightning strikes a metal flag pole, one end of which is anchored in the ground. Estimate the force the Earth’s magnetic field can exert on the flag pole while the lightning-induced current flows. See Example 18–10.

A sort of “projectile launcher” is shown in Fig. 20–71. A large current moves in a closed loop composed of fixed rails, a power supply, and a very light, almost frictionless bar (pale green) touching the rails. A magnetic field is perpendicular to the plane of the circuit. If the bar has a length \(\ell = 28\ cm\), a mass of 1.5 g, and is placed in a field of 1.7 T, what constant current flow is needed to accelerate the bar from rest to 28 m/s in a distance of 1.0 m? In what direction must the magnetic field point?

The cyclotron (Fig. 2072) is a device used to accelerate elementary particles such as protons to high speeds. Particles starting at point A with some initial velocity travel in semicircular orbits in the magnetic field B. The particles are accelerated to higher speeds each time they pass through the gap between the metal dees, where there is an electric field E. (There is no electric field inside the hollow metal dees where the electrons move in circular paths.) The electric field changes direction each half- cycle, owing to an ac voltage so that the particles are increased in speed at each passage through the gap. (a) Show that the frequency f of the voltage must be where q is the charge on the particles and m their mass. (b) Show that the kinetic energy of the particles increases by each revolution, assuming that the gap is small. (c) If the radius of the cyclotron is 2.0 m and the magnetic field strength is 0.50 T, what will be the maximum kinetic energy of accelerated protons in MeV?

The cyclotron (Fig. 2072) is a device used to accelerate elementary particles such as protons to high speeds. Particles starting at point A with some initial velocity travel in semicircular orbits in the magnetic field B. The particles are accelerated to higher speeds each time they pass through the gap between the metal dees, where there is an electric field E. (There is no electric field inside the hollow metal dees where the electrons move in circular paths.) The electric field changes direction each half- cycle, owing to an ac voltage so that the particles are increased in speed at each passage through the gap. (a) Show that the frequency f of the voltage must be where q is the charge on the particles and m their mass. (b) Show that the kinetic energy of the particles increases by each revolution, assuming that the gap is small. (c) If the radius of the cyclotron is 2.0 m and the magnetic field strength is 0.50 T, what will be the maximum kinetic energy of accelerated protons in MeV?

In Fig. 2073 the top wire is 1.00-mm-diameter copper wire and is suspended in air due to the two magnetic forces from the bottom two wires. The current flow through the two bottom wires is 75 A in each. Calculate the required current flow in the suspended wire (M).

You want to get an idea of the magnitude of magnetic fields produced by overhead power lines. You estimate that a transmission wire is about 13 m above the ground. The local power company tells you that the lines operate at 240 kV and provide a maximum power of 46 MW. Estimate the magnetic field you might experience walking under one such power line, and compare to the Earths field.

Two long parallel wires 8.20 cm apart carry 19.2-A currents in the same direction. Determine the magnetic field vector at a point P, 12.0 cm from one wire and 13.0 cm from the other (Fig. 20–74). [Hint: Use the law of cosines; see Appendix A or inside rear cover.]

Problem 1COQ Which of the following can experience a force when placed in the magnetic field of a magnet? (a) An electric charge at rest. (c) An electric current in a wire. (b) An electric charge moving. (d) Another magnet.

Problem 1MCQ Indicate which of the following will produce a magnetic field: (a) A magnet. (b) The Earth. (c) An electric charge at rest. (d) A moving electric charge. (e) An electric current. (f ) The voltage of a battery not connected to anything. (g) An ordinary piece of iron. (h) A piece of any metal.

Problem 1P (I) (a) What is the force per meter of length on a straight wire carrying a 6.40-A current when perpendicular to a 0.90-T uniform magnetic field? (b) What if the angle between the wire and field is 35.0°?

Problem 1Q A compass needle is not always balanced parallel to the Earth’s surface, but one end may dip downward. Explain.

Problem 1SL How many magnetic force equations are there in Chapter 20? List each one and explain when it applies. For each magnetic force equation, show how the units work out to give force in newtons.

A current in a wire points into the page as shown at the right. In which direction is the magnetic field at point A (choose below)?

How much current is flowing in a wire 4.80 m long if the maximum force on it is 0.625 N when placed in a uniform 0.0800-T field?

Problem 2Q Explain why the Earth’s “north pole” is really a magnetic south pole. Indicate how north and south magnetic poles were defined and how we can tell experimentally that the north pole is really a south magnetic pole.

Problem 2SL An electron is moving north at a constant speed of 3.0 X 104 m/s. (a) In what direction should an electric field point if the electron is to be accelerated to the east? (b) In what direction should a magnetic field point if the electron is to be accelerated to the west? (c) If the electric field of part a has a strength of 330 V/m, what magnetic field (magnitude and direction) will produce zero net force on the electron? (d) If the electron in part c is moving faster than 3.0 X 104 m/s, in which direction will it be accelerated? What if it is moving slower than 3.0 X 104 m/s? (e) Now consider electrons that move perpendicular to both a magnetic field and to an electric field, which are perpendicular to each other. If only electrons with speeds of 5.0 X 104 m/s go straight through undeflected, what is the ratio of the magnitudes of electric field to magnetic field? Without knowing the value of the electric field, can you know the value of the magnetic field?

Problem 3MCQ In which direction (see above) is the magnetic field at point B?

Problem 3P (I) A 240-m length of wire stretches between two towers and carries a 120-A current. Determine the magnitude of the force on the wire due to the Earth’s magnetic field of 5.0 X 10-5 which makes an angle of 68° with the wire.

Problem 3Q In what direction are the magnetic field lines surrounding a straight wire carrying a current that is moving directly away from you? Explain.

() A particle of charge q moves in a circular path of radius r in a uniform magnetic field \(\vec{B}\).If the magnitude of the magnetic field is double, and the kinetic energy of the particle is the same, how does the angular momentum of the particle differ? (b) Show that the magnetic dipole moment M (Section 20–9) of an electron orbiting the proton nucleus of a hydrogen atom is related to the orbital angular momentum L of the electron by \(M=\frac{e}{2 m} L\) Equation transcription: Text transcription: vec{B} M=frac{e}{2 m} L

Problem 4MCQ When a charged particle moves parallel to the direction of a magnetic field, the particle travels in a (a) straight line. (c) helical path. (b) circular path. (d) hysteresis loop.

Problem 4P (I) A 2.6-m length of horizontal wire carries a 4.5-A current toward the south. The dip angle of the Earth’s magnetic field makes an angle of 41° to the wire. Estimate the magnitude of the magnetic force on the wire due to the Earth’s magnetic field of 5.5 X 10-5 T.

Problem 4Q A horseshoe magnet is held vertically with the north pole on the left and south pole on the right. A wire passing between the poles, equidistant from them, carries a current directly away from you. In what direction is the force on the wire? Explain.

(a) Two long parallel wires, each 2.0 mm in diameter and 9.00 cm apart, carry equal 1.0-A currents in the same direction, Fig. 20–75. Determine \(\vec{B}\) along the x axis between the wires as a function of x. (b) Graph B vs. x from x = 1.0 mm to x =89.0 mm. \(d=9.00 \mathrm{~cm}\) Equation transcription: Text transcription: vec{B} d=9.00{~cm}

Problem 5MCQ As a proton moves through space, it creates (a) an electric field only. (b) a magnetic field only. (c) both an electric field and magnetic field. (d) nothing; the electric field and magnetic fields cancel each other out.

Problem 5P (I) The magnetic force per meter on a wire is measured to be only 45% of its maximum possible value. What is the angle between the wire and the magnetic field?

Problem 5Q Will a magnet attract any metallic object, such as those made of aluminum or copper? (Try it and see.) Why is this so?

The force on a moving particle in a magnetic field is the idea behind electromagnetic pumping. It can be used to pump metallic fluids (such as sodium) and to pump blood in artificial heart machines. A basic design is shown in Fig. 20–76. For blood, an electric field is applied perpendicular to a blood vessel and to the magnetic field. Explain in detail how ions in the blood are caused to move. Do positive and negative ions feel a force in the same direction?

Problem 6MCQ Which statements about the force on a charged particle placed in a magnetic field are true? (a) A magnetic force is exerted only if the particle is moving. (b) The force is a maximum if the particle is moving in the direction of the field. (c) The force causes the particle to gain kinetic energy. (d) The direction of the force is along the magnetic field. (e) A magnetic field always exerts a force on a charged particle.

Problem 6P (II) The force on a wire carrying 6.45 A is a maximum of 1.28 N when placed between the pole faces of a magnet. If the pole faces are 55.5 cm in diameter, what is the approximate strength of the magnetic field?

Problem 6Q Two iron bars attract each other no matter which ends are placed close together. Are both magnets? Explain.

Problem 7MCQ Which of the following statements is false? The magnetic field of a current-carrying wire (a) is directed circularly around the wire. (b) decreases inversely with the distance from the wire. (c) exists only if the current in the wire is changing. (d) depends on the magnitude of the current.

Problem 7P (II) The force on a wire is a maximum of 8.50 X 10-2 N when placed between the pole faces of a magnet. The current flows horizontally to the right and the magnetic field is vertical. The wire is observed to “jump” toward the observer when the current is turned on. (a) What type of magnetic pole is the top pole face? (b) If the pole faces have a diameter of 10.0 cm, estimate the current in the wire if the field is 0.220 T. (c) If the wire is tipped so that it makes an angle of 10.0° with the horizontal, what force will it now feel? [Hint: What length of wire will now be in the field?]

Problem 7Q The magnetic field due to current in wires in your home can affect a compass. Discuss the effect in terms of currents, including if they are ac or dc.

Problem 8MCQ A wire carries a current directly away from you. Which way do the magnetic field lines produced by this wire point? (a) They point parallel to the wire in the direction of the current. (b) They point parallel to the wire opposite the direction of the current. (c) They point toward the wire. (d) They point away from the wire. (e) They make circles around the wire.

(II) Suppose a straight 1.00-mm-diameter copper wire could just “float” horizontally in air because of the force due to the Earth’s magnetic field \(\vec{B}\), which is horizontal, perpendicular to the wire, and of magnitude \(5.0 \times 10^{-5} T\). What current would the wire carry? Does the answer seem feasible? Explain briefly. Equation transcription Text transcription: vec{B} 5.0 times 10^{-5} T

Problem 8Q If a negatively charged particle enters a region of uniform magnetic field which is perpendicular to the particle’s velocity, will the kinetic energy of the particle increase, decrease, or stay the same? Explain your answer. (Neglect gravity and assume there is no electric field.)

A proton enters a uniform magnetic field that is perpendicular to the proton’s velocity (Fig. 20–51). What happens to the kinetic energy of the proton? () It increases. (b) It decreases. (c) It stays the same. (d) It depends on the velocity direction. (e) It depends on the B field direction.

Problem 9P (I) Determine the magnitude and direction of the force on an electron traveling 7.75 X 105 m/s horizontally to the east in a vertically upward magnetic field of strength 0.45 T.

In Fig. 20–47, charged particles move in the vicinity of a current-carrying wire. For each charged particle, the arrow indicates the initial direction of motion of the particle, and the + or - indicates the sign of the charge. For each of the particles, indicate the direction of the magnetic force due to the magnetic field produced by the wire. Explain.

Problem 10MCQ For a charged particle, a constant magnetic field can be used to change (a) only the direction of the particle’s velocity. (b) only the magnitude of the particle’s velocity. (c) both the magnitude and direction of the particle’s velocity. (d) None of the above.

Problem 10P (I) An electron is projected vertically upward with a speed Of 1.70 X 106 into a uniform magnetic field of 0.640 T that is directed horizontally away from the observer. Describe the electron’s path in this field.

Three particles, a, b, and c, enter a magnetic field and follow paths as shown in Fig. 20–48. What can you say about the charge on each particle? Explain.

Which of the following statements about the force on a charged particle due to a magnetic field are not valid? (a) It depends on the particle’s charge. (b) It depends on the particle’s velocity. (c) It depends on the strength of the external magnetic field. (d) It acts at right angles to the direction of the particle’s motion. (e) None of the above; all of these statements are valid.

(I) Alpha particles (charge q = +2e, mass \(m = 6.6 \times 10^{-27}\ kg\) move at \(1.6 \times 10^6 m/s\). What magnetic field strength would be required to bend them into a circular path of radius r = 0.14 m?

Problem 11Q Can an iron rod attract a magnet? Can a magnet attract an iron rod?What must you consider to answer these questions?

Problem 12MCQ Two parallel wires are vertical. The one on the left carries a 10-A current upward. The other carries 5-A current downward. Compare the magnitude of the force that each wire exerts on the other. (a) The wire on the left carries twice as much current, so it exerts twice the force on the right wire as the right one exerts on the left one. (b) The wire on the left exerts a smaller force. It creates a magnetic field twice that due to the wire on the right; and therefore has less energy to cause a force on the wire on the right. (c) The two wires exert the same force on each other. (d) Not enough information; we need the length of the wire.

(I) Find the direction of the force on a negative charge for each diagram shown in Fig. 20–52, where (green) is the velocity of the charge and \(\vec{B}\) (blue) is the direction of the magnetic field. ( ? means the vector points inward. means it points outward, toward you.) \(\vec{B}\) \(\vec{v}\) Equation transcription: Text transcription: vec{B} vec{v}

A positively charged particle in a nonuniform magnetic field follows the trajectory shown in Fig. 20–49. Indicate the direction of the magnetic field at points near the path, assuming the path is always in the plane of the page, and indicate the relative magnitudes of the field in each region. Explain your answers.

(I) Determine the direction of \(\vec{B}\) for each case in Fig. 20–53, where \(\vec{F}\) represents the maximum magnetic force on a positively charged particle moving with velocity \(\vec{v}\). \(\vec{F}\) \(\vec{B}\) \(\vec{v}\) Equation transcription: Text transcription: vec{B} vec{F} vec{v}

Problem 13Q Explain why a strong magnet held near a CRT television screen (Section 17–11) causes the picture to become distorted. Also, explain why the picture sometimes goes completely black where the field is the strongest. [But don’t risk damage to your TV by trying this.]

(II) What is the velocity of a beam of electrons that goes undeflected when moving perpendicular to an electric and to a magnetic field. \(\vec{E}\) and \(\vec{B}\) are also perpendicular to each other and have magnitudes \(7.7 x 10^{-3} T\), and respectively. What is the radius of the electron orbit if the electric field is turned off? Equation transcription: Text transcription: vec{B} vec{E} 7.7 x 10^{-3} T

Problem 14Q Suppose you have three iron rods, two of which are magnetized but the third is not. How would you determine which two are the magnets without using any additional objects?

Problem 15P (II) A helium ion (Q =+2e) whose mass is 6.6 C 10-27 kg is accelerated by a voltage of 3700 V. (a)What is its speed? (b) What will be its radius of curvature if it moves in a plane perpendicular to a uniform 0.340-T field? (c) What is its period of revolution?

Problem 15Q Can you set a resting electron into motion with a magnetic field? With an electric field? Explain.

Problem 16P (II) For a particle of mass m and charge q moving in a circular path in a magnetic field B, (a) show that its kinetic energy is proportional to r2 the square of the radius of curvature of its path. (b) Show that its angular momentum is L =q Br2 around the center of the circle.

Problem 16Q A charged particle is moving in a circle under the influence of a uniform magnetic field. If an electric field that points in the same direction as the magnetic field is turned on, describe the path the charged particle will take.

(II) A 1.5-MeV (kinetic energy) proton enters a 0.30-T field, in a plane perpendicular to the field. What is the radius of its path? See Section 17–4.

Problem 17Q A charged particle moves in a straight line through a particular region of space. Could there be a nonzero magnetic field in this region? If so, give two possible situations.

Problem 18P (II) An electron experiences the greatest force as it travels 2.8 X 106 m/s in a magnetic field when it is moving north-ward. 6.2 X 10-13 N. The force is vertically upward and of magnitude What is the magnitude and direction of the magnetic field?

Problem 18Q If a moving charged particle is deflected sideways in some region of space, can we conclude, for certain, that in that region? Explain.

Problem 19P (II) A proton and an electron have the same kinetic energy upon entering a region of constant magnetic field. What is the ratio of the radii of their circular paths?

Problem 19Q Two insulated long wires carrying equal currents I cross at right angles to each other. Describe the magnetic force one exerts on the other.

Does the Earth’s magnetic field have a greater magnitude near the poles or near the equator? [How can you tell using the field lines in Fig. 20–5?]

Problem 20EB A straight wire carries a current directly toward you. In what direction are the magnetic field lines surrounding the wire?

Problem 20ED A straight power line carries 30 A and is perpendicular to the Earth’s magnetic field of 0.50 X 10–4 T.What magnitude force is exerted on 100m of this power line?

Problem 20EC A wire carrying current I is perpendicular to a magnetic field of strength B. Assuming a fixed length of wire, which of the following changes will result in decreasing the force on the wire by a factor of 2? (a) Decrease the angle from 90° to 45°; (b) decrease the angle from 90° to 30°; (c) decrease the current in the wire to I/2; (d) decrease the magnetic field to B/2; (e) none of these will do it.

Return to the Chapter-Opening Question, page 560, and answer it again now. Try to explain why you may have answered differently the first time.

What is the sign of the charge in Fig. 20–19? How would you modify the drawing if the charge had the opposite sign?

Suppose both \(I_{1}\) and \(I_{2}\) point into the page in Fig. 20–24. What then is the field B midway between the wires? Equation transcription: Text transcription: I{1} I{2}

(III) A proton (mass \(m_{p}\) ), a deuteron \(\left(m=2 m_{p}, Q=e\right)\), and an alpha particle \(\left(m=4 m_{p}, Q=2 e\right)\) are accelerated by the same potential difference and then enter a uniform magnetic field , where they move in circular paths perpendicular to . Determine the radius of the paths for the deuteron and alpha particle in terms of that for the proton. Equation transcription: Text transcription: M{p} (m=2 m{p}, Q=e) (m=4 m{p}, Q=2 e) vec{B}

Problem 20Q A horizontal current-carrying wire, free to move in Earth’s gravitational field, is suspended directly above a parallel, current-carrying wire. (a) In what direction is the current in the lower wire? (b) Can the lower wire be held in stable equilibrium due to the magnetic force of the upper wire? Explain.

Problem 21P (III) A 3.40-g bullet moves with a speed of 155 m/s perpendicular to the Earth’s magnetic field of 5.00 X 10-5 T. If the bullet possesses a net charge of 18.5 X 10-9 C, by what distance will it be deflected from its path due to the Earth’s magnetic field after it has traveled 1.50 km?

Problem 21Q What would be the effect on B inside a long solenoid if (a) the diameter of all the loops was doubled, (b) the spacing between loops was doubled, or (c) the solenoid’s length was doubled along with a doubling in the total number of loops?

Problem 22P (III) A Hall probe, consisting of a thin rectangular slab of current-carrying material, is calibrated by placing it in a known magnetic field of magnitude 0.10 T. When the field is oriented normal to the slab’s rectangular face, a Hall emf of 12 mV is measured across the slab’s width. The probe is then placed in a magnetic field of unknown magnitude B, and a Hall emf of 63 mV is measured. Determine B assuming that the angle ? between the unknown field and the plane of the slab’s rectangular face is (a) ? =90°, and (b) ? =60°

A type of magnetic switch similar to a solenoid is a relay (Fig. 20–50). A relay is an electromagnet (the iron rod inside the coil does not move) which, when activated, attracts a strip of iron on a pivot. Design a relay to close an electrical switch. A relay is used when you need to switch on a circuit carrying a very large current but do not want that large current flowing through the main switch. For example, a car’s starter switch is connected to a relay so that the large current needed for the starter doesn’t pass to the dashboard switch.

Problem 23P (III) The Hall effect can be used to measure blood flow rate because the blood contains ions that constitute an electric current. (a) Does the sign of the ions influence the emf? Explain. (b) Determine the flow velocity in an artery 3.3mm in diameter if the measured emf across the width of the artery is 0.13 mV and B is 0.070 T. (In actual practice, an alternating magnetic field is used.)

Two ions have the same mass, but one is singly ionized and the other is doubly ionized. How will their positions on the film of a mass spectrometer (Fig. 20–41) differ? Explain.

(III) A long copper strip wide and thick is placed in a 1.2-T magnetic field as in Fig. 20-21a. When a steady current of 15 A passes through it, the Hall emf is measured to be \(1.02 \mu V\). Determine the drift velocity of the electrons and the density of free (conducting) electrons (number per unit volume) in the copper. [Hint: See also Section Equation transcription: Text transcription: 1.02 mu V

Problem 24Q Why will either pole of a magnet attract an unmagnetized piece of iron?

Problem 25P (I) Jumper cables used to start a stalled vehicle often carry a 65-A current. How strong is the magnetic field 4.5 cm from one cable? Compare to the Earth’s magnetic field (5.0 X 10-5 T).

Problem 25Q An unmagnetized nail will not attract an unmagnetized paper clip. However, if one end of the nail is in contact with a magnet, the other end will attract a paper clip. Explain.

(I) If an electric wire is allowed to produce a magnetic field no larger than that of the Earth \(0.50 \times 10^{-4} \ \mathrm t\) at a distance of 12 cm from the wire, what is the maximum current the wire can carry?

Problem 27P (I) Determine the magnitude and direction of the force between two parallel wires 25 m long and 4.0 cm apart, each carrying 25 A in the same direction.

Problem 28P (I) A vertical straight wire carrying an upward 28-A current exerts an attractive force per unit length of 7.8 X 10-4 N/m on a second parallel wire 9.0 cm away. What current (magnitude and direction) flows in the second wire?

(II) In Fig. 20-54, a long straight wire carries current out of the page toward you. Indicate, with appropriate arrows, the direction and (relative) magnitude of \(\vec{B}\) at each of the points , and in the plane of the page. Equation transcription: Text transcription: vec{B}

Problem 30P (II) An experiment on the Earth’s magnetic field is being carried out 1.00 m from an electric cable. What is the maximum allowable current in the cable if the experiment is to be accurate to ± 3.0%?

(II) A rectangular loop of wire is placed next to a straight wire, as shown in Fig. 20–55. There is a current of 3.5 A in both wires. Determine the magnitude and direction of the net force on the loop. \(3.5 A\) \(3.5 A\) \(3.0 \mathrm{~cm}\) \(10.0 \mathrm{~cm}\) Equation transcription: Text transcription: 3.5 A 3.5 A 3.0{~cm} 10.0{~cm}

Problem 32P (II) A horizontal compass is placed 18 cm due south from a straight vertical wire carrying a 48-A current downward. In what direction does the compass needle point at this location? Assume the horizontal component of the Earth’s field at this point is 0.45 X 10-4 T and the magnetic declination is 0°

Problem 33P (II) A long horizontal wire carries 24.0 A of current due north. What is the net magnetic field 20.0 cm due west of the wire if the Earth’s field there points downward, 44° below the horizontal, and has magnitude 5.0 X 10-5 T?

Problem 34P (II) A straight stream of protons passes a given point in space at a rate of 2.5 X 109 protons/s. What magnetic field do they produce 1.5 m from the beam?

Problem 35P (II) Determine the magnetic field midway between two long straight wires 2.0 cm apart in terms of the current I in one when the other carries 25 A. Assume these currents are (a) in the same direction, and (b) in opposite directions.

Problem 36P (II) Two straight parallel wires are separated by 7.0 cm. There is a 2.0-A current flowing in the first wire. If the magnetic field strength is found to be zero between the two wires at a distance of 2.2 cm from the first wire, what is the magnitude and direction of the current in the second wire?

(II) Two long straight wires each carry a current I out of the page toward the viewer, Fig. 20–56. Indicate, with appropriate arrows, the direction of \(\vec {\mathrm B}\) at each of the points 1 to 6 in the plane of the page. State if the field is zero at any of the points.

(II) A power line carries a current of 95 A west along the tops of 8.5-m-high poles. (a) What is the magnitude and direction of the magnetic field produced by this wire at the ground directly below? How does this compare with the Earth’s magnetic field of about \(\frac{1}{2} G\)? (b) Where would the wire’s magnetic field cancel the Earth’s field? Equation transcription: Text transcription: frac{1}{2} G

Problem 39P (II) A compass needle points 17° E of N outdoors. However, when it is placed 12.0 cm to the east of a vertical wire inside a building, it points 32° E of N. What is the magnitude and direction of the current in the wire? The Earth’s field there is 0.50 X 10–4 T and is horizontal.

(II) A long pair of insulated wires serves to conduct 24.5 A of dc current to and from an instrument. If the wires are of negligible diameter but are 2.8 mm apart, what is the magnetic field 10.0 cm from their midpoint, in their plane (Fig.20–57)? Compare to the magnetic field of the Earth. \(10.0 \mathrm{~cm}\) \(2.8 \mathrm{~cm}\) Equation transcription: Text transcription: 10.0{~cm} 2.8{~cm}

Problem 41P (II) A third wire is placed in the plane of the two wires shown in Fig. 20–57 parallel and just to the right. If it carries 25.0 A upward, what force per meter of length does it exert on each of the other two wires? Assume it is 2.8mm from the nearest wire, center to center.

(III) Two long thin parallel wires 13.0 cm apart carry 28-A currents in the same direction. Determine the magnetic field vector at a point 10.0 cm from one wire and 6.0 cm from the other (Fig. 20–58). \(6.0 \mathrm{~cm}\) \(13.0 \mathrm{~cm}\) \(10.0 \mathrm{~cm}\) Equation transcription: Text transcription: 6.0{~cm} 13.0{~cm} 10.0{~cm}

(III) Two long wires are oriented so that they are perpendicular to each other. At their closest, they are 20.0 cm apart (Fig. 20–59). What is the magnitude of the magnetic field at a point midway between them if the top one carries a current of 20.0 A and the bottom one carries 12.0 A? \(I_{T}=20.0 A\) \(10.0 \mathrm{~cm}\) \(I_{B}=12.0 \mathrm{~A}\) Equation transcription: Text transcription: I{T}=20.0 A 10.0{~cm} I{B}=12.0{~A}

Problem 44P (I) A thin 12-cm-long solenoid has a total of 460 turns of wire and carries a current of 2.0 A. Calculate the field inside the solenoid near the center.

Problem 45P (I) A 30.0-cm-long solenoid 1.25 cm in diameter is to produce a field of 4.65 mT at its center. How much current should the solenoid carry if it has 935 turns of the wire?

Problem 46P (I) A 42-cm-long solenoid, 1.8 cm in diameter, is to produce a 0.030-T magnetic field at its center. If the maximum current is 4.5 A, how many turns must the solenoid have?

Problem 47P (II) A 550-turn horizontal solenoid is 15 cm long. The current in its coils is 38 A. A straight wire cuts through the center of the solenoid, along a 3.0-cm diameter. This wire carries a 22-A current downward (and is connected by other wires that don’t concern us). What is the force on this wire assuming the solenoid’s magnetic field points due east?

Problem 48P (III) You have 1.0 kg of copper and want to make a practical solenoid that produces the greatest possible magnetic field for a given voltage. Should you make your copper wire long and thin, short and fat, or something else? Consider other variables, such as solenoid diameter, length, and so on. Explain your reasoning.

(III) A toroid is a solenoid in the shape of a donut (Fig. 20–60). Use Ampère’s law along the circular paths, shown dashed in Fig. 20–60a, to determine that the magnetic field () inside the toroid is B = \(\mu_{a} N I \backslash 2 \pi R\), where N is the total number of turns, and (b) outside the toroid is B = 0. (c) Is the field inside a toroid uniform like a solenoid’s? If not, how does it vary? Equation transcription: Text transcription: mu{a} N I \backslash 2 pi R

(III) () Use Ampère’s law to show that the magnetic field between the conductors of a coaxial cable (Fig. 20–61) is \(B=\mu_{a} \backslash 2 \pi r\) if r (distance from center) is greater than the radius of the inner wire and less than the radius of the outer cylindrical braid (= ground). (b) Show that B = 0 outside the coaxial cable. Equation transcription: Text transcription: B=mu{a} backslash 2 pi r

Problem 51P (I) A single square loop of wire 22.0 cm on a side is placed with its face parallel to the magnetic field as in Fig. 20–34b. When 5.70 A flows in the coil, the torque on it is 0.325 m.N. What is the magnetic field strength?

Problem 52P (I) If the current to a motor drops by 12%, by what factor does the output torque change?

Problem 53P (I) A galvanometer needle deflects full scale for a 53.0 µA current. What current will give full-scale deflection if the magnetic field weakens to 0.760 of its original value?

Problem 54P (II) A circular coil 12.0 cm in diameter and containing nine loops lies flat on the ground. The Earth’s magnetic field at this location has magnitude 5.50 X 10-5 T and points into the Earth at an angle of 56.0° below a line pointing due north. If a 7.20-A clockwise current passes through the coil, (a) determine the torque on the coil, and (b) which edge of the coil rises up: north, east, south, or west?

Problem 55P (I) Protons move in a circle of radius 6.10 cm in a 0.566-T magnetic field. What value of electric field could make their paths straight? In what direction must the electric field point?

(II) Suppose the electric field between the electric plates in the mass spectrometer of Fig. 20–41 is \(2.88 \times 10^{4} \mathrm{~V} / \mathrm{m}\) and the magnetic fields are \(B=B^{\prime}=0.68 T\). The source contains carbon isotopes of mass numbers 12, 13, and 14 from a long-dead piece of a tree. (To estimate masses of the atoms, multiply by \(1.67 \times 10^{-27} \mathrm{~kg}\).) How far apart are the lines formed by the singly charged ions of each type on the photographic film? What if the ions were doubly charged? Equation transcription: Text transcription: 2.88 times 10^{4}{~V} /{m} B=B^{prime}=0.68 T 1.67 times 10^{-27}{~kg}

Problem 56P (I) Protons move in a circle of radius 6.10 cm in a 0.566-T magnetic field. What value of electric field could make their paths straight? In what direction must the electric field point?

Problem 58P (II) One form of mass spectrometer accelerates ions by a voltage V before they enter a magnetic field B. The ions are assumed to start from rest. Show that the mass of an ion m =qB2R2/2V, is where R is the radius of the ions’ path in the magnetic field and q is their charge.

Problem 59P (II) An unknown particle moves in a straight line through crossed electric and magnetic fields with E = 1.5 KV/m and B =0.034T. If the electric field is turned off, the particle moves in a circular path of radius r =2.7 cm. What might the particle be?

(III) A mass spectrometer is monitoring air pollutants. It is difficult, however, to separate molecules of nearly equal mass such as CO (28.0106 u) and \(N_{2}\) (28.0134 u). How large a radius of curvature must a spectrometer have (Fig. 20–41) if these two molecules are to be separated on the film by 0.50 mm? Equation transcription: Text transcription: N{2}

Problem 61P (I) A long thin iron-core solenoid has 380 loops of wire per meter, and a 350-mA current flows through the wire. If the permeability of the iron is 3000 µ0, what is the total field B inside the solenoid?

Problem 62P (II) An iron-core solenoid is 38 cm long and 1.8 cm in diameter, and has 780 turns of wire. The magnetic field inside the solenoid is 2.2 T when 48 A flows in the wire. What is the permeability at this high field strength?

Problem 63P (II) The following are some values of B and B0 for a piece of iron as it is being magnetized (note different units): B0 (10-4 T) 0.0 0.13 0.25 0.50 0.63 0.78 1.0 1.3 B(T) 0.0 0.0042 0.010 0.028 0.043 0.095 0.45 0.67 B0 (10-4 T) 1.9 2.5 6.3 13.0 130 1300 10,000 B(T) 1.01 1.18 1.44 1.58 1.72 2.26 3.15 Determine the magnetic permeability for each value and plot a graph of µ versus B0 .

Problem 64GP Two long straight parallel wires are 15 cm apart. Wire A carries 2.0-A current. Wire B’s current is 4.0 A in the same direction. (a) Determine the magnetic field magnitude due to wire A at the position of wire B. (b) Determine the magnetic field due to wire B at the position of wire A. (c) Are these two magnetic fields equal and opposite? Why or why not? (d) Determine the force on wire A due to wire B, and the force on wire B due to wire A. Are these two forces equal and opposite? Why or why not?

Problem 65GP Protons with momentum 4.8*10-21 kg.m/s are magnetically steered clockwise in a circular path 2.2 m in diameter. Determine the magnitude and direction of the field in the magnets surrounding the beam pipe.

A small but rigid wire carrying a 5.0-A current (Fig. 20–62) is placed inside a solenoid. The solenoid is 15.0 cm long and has 700 loops of wire, and the current in each loop is 7.0 A. What is the net force on the ´-shaped wire? FIGURE 20–62 Problem 66.

The power cable for an electric trolley (Fig. 20–63) carries a horizontal current of 330 A toward the east. The Earth’s magnetic field has a strength \(5.0 \times 10^{-5} T\) and makes an angle of dip of 22° at this location. Calculate the magnitude and direction of the magnetic force on an 18-m length of this cable. \(I=330 A\) FIGURE 20–63 Problem 67. Equation transcription: Text transcription: 5.0 times 10^{-5} T I=330 A

Problem 68GP A particle of charge q moves in a circular path of radius r perpendicular to a uniform magnetic field B. Determine its linear momentum in terms of the quantities given.

Problem 69GP An airplane has acquired a net charge of 1280 µC. If the Earth’s magnetic field of 5.0 x 10-5 T is perpendicular to the airplane’s velocity of magnitude 120 m/s, determine the force on the airplane.

Problem 70GP A 32-cm-long solenoid, 1.8 cm in diameter, is to produce a 0.050-T magnetic field at its center. If the maximum current is 6.4 A, how many turns must the solenoid have?

Problem 71GP Near the equator, the Earth’s magnetic field points almost horizontally to the north and has magnitude B=0.50*10-4 T. What should be the magnitude and direction for the velocity of an electron if its weight is to be exactly balanced by the magnetic force?

Problem 72GP A doubly charged helium atom, whose mass is 6.6*10-27 kg, is accelerated by a voltage of 3200 V. (a) What will be its radius of curvature in a uniform 0.240-T field? (b) What is its period of revolution?

Four very long straight parallel wires, located at the corners of a square of side , carry equal currents \(I_{0}\) perpendicular to the page as shown in Fig. 20–64. Determine the magnitude and direction \(\vec{B}\) of at the center C of the square. FIGURE 20–64 Problem 73. Equation transcription: Text transcription: I{0} vec{B}

Problem 74GP (a) What value of magnetic field would make a beam of electrons, traveling to the west at a speed of 4.8*106 m/s, go undeflected through a region where there is a uniform electric field of 12000 V/m pointing south? (b) What is the direction of the magnetic field if it is perpendicular to the electric field? (c) What is the frequency of the circular orbit of the electrons if the electric field is turned off?

Magnetic fields are very useful in particle accelerators for “beam steering”; that is, the magnetic fields can be used to change the direction of the beam of charged particles without altering their speed (Fig. 20–65). Show how this could work with a beam of protons. What happens to protons that are not moving with the speed for which the magnetic field was designed? If the field extends over a region 5.0 cm wide and has a magnitude of 0.41 T, by approximately what angle will a beam of protons traveling at \(2.5 \times 10^{6} \mathrm{~m} / \mathrm{s}\) be bent? Evacuated tubes, inside of which the protons move with velocity indicated by the green arrows FIGURE 20–65 Problem 75. Equation transcription: Text transcription: 2.5 times 10^{6}{~m} /{s}

The magnetic field at the center of a circular coil of wire carrying a current (as in Fig. is \(B=\frac{\mu_{0} N L}{2 r}\) where is the number of loops in the coil and is its radius. Imagine a simple model in which the Earth's magnetic field of about \(1 G\left(=1 \times 10^{-4} T\right)\) near the poles is produced by a single current loop around the equator. Roughly estimate the current this loop would carry. Equation transcription: Text transcription: B=frac{\mu{0} N L}{2 r} 1 G(=1 times 10^{-4} T)

A proton follows a spiral path through a gas in a uniform magnetic field of \(0.010 \mathrm{~T}\), perpendicular to the plane of the spiral, as shown in Fig. 20–66. In two successive loops, at points P and Q, the radii are 10.0 mm and 8.5 mm, respectively. Calculate the change in the kinetic energy of the proton as it travels from P to Q. FIGURE 20–66 Problem 77 Equation transcription: Text transcription: .0.010{~T}

Two long straight aluminum wires, each of diameter 0.42 mm, carry the same current but in opposite directions. They are suspended by 0.50-m-long strings as shown in Fig. 20–67. If the suspension strings make an angle of \(3.0^{0}\) with the vertical and are hanging freely, what is the current in the wires? FIGURE 20–67 Problem 78. Equation transcription: Text transcription: 3.0^{0}

Problem 79GP An electron enters a uniform magnetic field B = 0.23 T at a 45° angle to Determine the radius r and pitch p (distance between loops) of the electron’s helical path assuming its speed is 3.0 x 106 m/s. See Fig. 20–68.

Problem 80GP A motor run by a 9.0-V battery has a 20-turn square coil with sides of length 5.0 cm and total resistance 28 ?. When spinning, the magnetic field felt by the wire in the coil is 0.020 T. What is the maximum torque on the motor?

Problem 81GP Electrons are accelerated horizontally by 2.2 kV. They then pass through a uniform magnetic field B for a distance of 3.8 cm, which deflects them upward so they reach the top of a screen 22 cm away, 11 cm above the center. Estimate the value of B.

Problem 82GP A 175-g model airplane charged to 18.0 mC and traveling at 3.4 m/s passes within 8.6 cm of a wire, nearly parallel to its path, carrying a 25-A current.What acceleration (in g’s) does this interaction give the airplane?

A uniform conducting rod of length and mass m sits atop a fulcrum, which is placed a distance \(I / 4\) from the rod’s left-hand end and is immersed in a uniform magnetic field of magnitude directed into the page (Fig. 20–69). An object whose mass is 6.0 times greater than the rod’s mass is hung from the rod’s left-hand end. What current (direction and magnitude) should flow through the rod in order for it to be “balanced” (i.e., be at rest horizontally) on the fulcrum? (Flexible connecting wires which exert negligible force on the rod are not shown.) FIGURE 20–69 Problem 83 Equation transcription: Text transcription: I / 4

Problem 84GP Suppose the Earth’s magnetic field at the equator has Magnitude 0.50 x 10-4 T and a northerly direction at all points. Estimate the speed a singly ionized uranium ion (m = 238, q = +e) would need to circle the Earth 6.0 km above the equator. Can you ignore gravity? [Ignore relativity.]

A particle with charge and momentum , initially moving along the axis, enters a region where a uniform magnetic field \(B_{0}\) extends over a width as shown in Fig. 20–70. The particle is deflected a distance in the direction as it traverses the field. Determine (a) whether is positive or negative, and (b) the magnitude of its momentum in terms of ,\(B_{0}\), and . FIGURE 20–70 Problem 85. Equation transcription: Text transcription: B{0}

Problem 86GP A bolt of lightning strikes a metal flag pole, one end of which is anchored in the ground. Estimate the force the Earth’s magnetic field can exert on the flag pole while the lightning-induced current flows. See Example 18–10.

A sort of “projectile launcher” is shown in Fig. 20–71. A large current moves in a closed loop composed of fixed rails, a power supply, and a very light, almost frictionless bar (pale green) touching the rails. A magnetic field is perpendicular to the plane of the circuit. If the bar has a length \(I=28 \mathrm{~cm}\), a mass of 1.5 g, and is placed in a field of 1.7 T, what constant current flow is needed to accelerate the bar from rest to \(28 \mathrm{~m} / \mathrm{s}\) in a distance of 1.0 m? In what direction must the magnetic field point FIGURE 20–71 Problem 87. Equation transcription: Text transcription: I=28{~cm} 28{~m} /{s}

The cyclotron (Fig. 20-72 ) is a device used to accelerate elementary particles such as protons to high speeds. Particles starting at point A with some initial velocity travel in semicircular orbits in the magnetic field B. The particles are accelerated to higher speeds each time they pass through the gap between the metal "dees," where there is an electric field E. (There is no electric field inside the hollow metal dees where the electrons move in circular paths.) The electric field changes direction each half-cycle, owing to an ac voltage \(V=V_{0} \sin 2 \pi f t\), so that the particles are increased in speed at each passage through the gap. (a) Show that the frequency \(f\) of the voltage must be \(f=B q / 2 \pi m\), where q is the charge on the particles and m their mass. (b) Show that the kinetic energy of the particles increases by \(2 q V_{0}\) each revolution, assuming that the gap is small. (c) If the radius of the cyclotron is 2.0 m and the magnetic field strength is 0.50 T, what will be the maximum kinetic energy of accelerated protons in MeV? FIGURE 20–72 A cyclotron. Problem 88. Equation Transcription: V=V0 sin? 2?ft f=Bq/2?m 2qV0 Text Transcription: V=V_0 sin? 2pi ft f=Bq/2pi m 2qV_0

Three long parallel wires are 3.8 cm from one another. (Looking along them, they are at three corners of an equilateral triangle.) The current in each wire is 8.00 A, but its direction in wire M is opposite to that in wires N and P (Fig. 20–73). (a) Determine the magnetic force per unit length on each wire due to the other two. (b) In Fig. 20–73, determine the magnitude and direction of the magnetic field at the midpoint of the line between wire M and wire N. FIGURE 20–73 Problems 89 and 90.

In Fig. 20–73 the top wire is 1.00-mm-diameter copper wire and is suspended in air due to the two magnetic forces from the bottom two wires. The current flow through the two bottom wires is 75 A in each. Calculate the required current flow in the suspended wire (M).

Problem 91GP You want to get an idea of the magnitude of magnetic fields produced by overhead power lines. You estimate that a transmission wire is about 13 m above the ground. The local power company tells you that the lines operate at 240 kV and provide a maximum power of 46 MW. Estimate the magnetic field you might experience walking under one such power line, and compare to the Earth’s field.

Two long parallel wires 8.20 cm apart carry 19.2-A currents in the same direction. Determine the magnetic field vector at a point P, 12.0 cm from one wire and 13.0 cm from the other (Fig. 20–74). [Hint: Use the law of cosines; see Appendix A or inside rear cover.] FIGURE 20–74 Problem 92.

(a) What is the force per meter of length on a straight wire carrying a 6.40-A current when perpendicular to a 0.90-T uniform magnetic field? (b) What if the angle between the wire and field is 35.0?

How much current is flowing in a wire 4.80 m long if the maximum force on it is 0.625 N when placed in a uniform 0.0800-T field?

A 240-m length of wire stretches between two towers and carries a 120-A current. Determine the magnitude of the force on the wire due to the Earths magnetic field of which makes an angle of 68 with the wire.

A 2.6-m length of horizontal wire carries a 4.5-A current toward the south. The dip angle of the Earths magnetic field makes an angle of 41 to the wire. Estimate the magnitude of the magnetic force on the wire due to the Earths magnetic field of .

The magnetic force per meter on a wire is measured to be only 45% of its maximum possible value. What is the angle between the wire and the magnetic field?

The force on a wire carrying 6.45 A is a maximum of 1.28 N when placed between the pole faces of a magnet. If the pole faces are 55.5 cm in diameter, what is the approximate strength of the magnetic field?

The force on a wire is a maximum of when placed between the pole faces of a magnet. The current flows horizontally to the right and the magnetic field is vertical. The wire is observed to jump toward the observer when the current is turned on. (a) What type of magnetic pole is the top pole face? (b) If the pole faces have a diameter of 10.0 cm, estimate the current in the wire if the field is 0.220 T. (c) If the wire is tipped so that it makes an angle of 10.0 with the horizontal, what force will it now feel? [Hint: What length of wire will now be in the field?]

Suppose a straight 1.00-mm-diameter copper wire could just float horizontally in air because of the force due to the Earths magnetic field which is horizontal, perpendicular to the wire, and of magnitude What current would the wire carry? Does the answer seem feasible? Explain briefly

Determine the magnitude and direction of the force on an electron traveling horizontally to the east in a vertically upward magnetic field of strength 0.45 T

An electron is projected vertically upward with a speed of into a uniform magnetic field of 0.640 T that is directed horizontally away from the observer. Describe the electrons path in this field.

Alpha particles (charge , mass ) move at What magnetic field strength would be required to bend them into a circular path of radius

Find the direction of the force on a negative charge for each diagram shown in Fig. 2052, where (green) is the velocity of the charge and (blue) is the direction of the magnetic field. ( means the vector points inward. means it points outward, toward you.)

Determine the direction of for each case in Fig. 2053, where represents the maximum magnetic force on a positively charged particle moving with velocity v B .

What is the velocity of a beam of electrons that goes undeflected when moving perpendicular to an electric and to a magnetic field. and are also perpendicular to each other and have magnitudes and respectively. What is the radius of the electron orbit if the electric field is turned off?

A helium ion whose mass is is accelerated by a voltage of 3700 V. (a) What is its speed? (b) What will be its radius of curvature if it moves in a plane perpendicular to a uniform 0.340-T field? (c) What is its period of revolution?

For a particle of mass m and charge q moving in a circular path in a magnetic field B, (a) show that its kinetic energy is proportional to the square of the radius of curvature of its path. (b) Show that its angular momentum is around the center of the circle.

A 1.5-MeV (kinetic energy) proton enters a 0.30-T field, in a plane perpendicular to the field. What is the radius of its path? See Section 174.

An electron experiences the greatest force as it travels in a magnetic field when it is moving northward. The force is vertically upward and of magnitude What is the magnitude and direction of the magnetic field

A proton and an electron have the same kinetic energy upon entering a region of constant magnetic field. What is the ratio of the radii of their circular paths?

A proton (mass ), a deuteron and an alpha particle are accelerated by the same potential difference V and then enter a uniform magnetic field where they move in circular paths perpendicular to Determine the radius of the paths for the deuteron and alpha particle in terms of that for the proton.

A 3.40-g bullet moves with a speed of perpendicular to the Earths magnetic field of If the bullet possesses a net charge of by what distance will it be deflected from its path due to the Earths magnetic field after it has traveled 1.50 km?

A Hall probe, consisting of a thin rectangular slab of current-carrying material, is calibrated by placing it in a known magnetic field of magnitude 0.10 T. When the field is oriented normal to the slabs rectangular face, a Hall emf of 12 mV is measured across the slabs width. The probe is then placed in a magnetic field of unknown magnitude B, and a Hall emf of 63 mV is measured. Determine B assuming that the angle between the unknown field and the plane of the slabs rectangular face is (a)

The Hall effect can be used to measure blood flow rate because the blood contains ions that constitute an electric current. (a) Does the sign of the ions influence the emf? Explain. (b) Determine the flow velocity in an artery 3.3 mm in diameter if the measured emf across the width of the artery is 0.13 mV and B is 0.070 T. (In actual practice, an alternating magnetic field is used.)

A long copper strip 1.8 cm wide and 1.0 mm thick is placed in a 1.2-T magnetic field as in Fig. 2021a. When a steady current of 15 A passes through it, the Hall emf is measured to be Determine (a) the drift velocity of the electrons and (b) the density of free (conducting) electrons (number per unit volume) in the copper. [Hint: See also Section 188.]

Jumper cables used to start a stalled vehicle often carry a 65-A current. How strong is the magnetic field 4.5 cm from one cable? Compare to the Earths magnetic field

If an electric wire is allowed to produce a magnetic field no larger than that of the Earth at a distance of 12 cm from the wire, what is the maximum current the wire can carry?

Determine the magnitude and direction of the force between two parallel wires 25 m long and 4.0 cm apart, each carrying 25 A in the same direction.

A vertical straight wire carrying an upward 28-A current exerts an attractive force per unit length of on a second parallel wire 9.0 cm away. What current (magnitude and direction) flows in the second wire?

In Fig. 2054, a long straight wire carries current I out of the page toward you. Indicate, with appropriate arrows, the direction and (relative) magnitude of at each of the points C, D, and E in the plane of the page.

An experiment on the Earths magnetic field is being carried out 1.00 m from an electric cable. What is the maximum allowable current in the cable if the experiment is to be accurate to

A rectangular loop of wire is placed next to a straight wire, as shown in Fig. 2055. There is a current of 3.5 A in both wires. Determine the magnitude and direction of the net force on the loop.

A horizontal compass is placed 18 cm due south from a straight vertical wire carrying a 48-A current downward. In what direction does the compass needle point at this location? Assume the horizontal component of the Earths field at this point is and the magnetic declination is 0

A long horizontal wire carries 24.0 A of current due north. What is the net magnetic field 20.0 cm due west of the wire if the Earths field there points downward, 44 below the horizontal, and has magnitude

A straight stream of protons passes a given point in space at a rate of What magnetic field do they produce 1.5 m from the beam?

Determine the magnetic field midway between two long straight wires 2.0 cm apart in terms of the current I in one when the other carries 25 A. Assume these currents are (a) in the same direction, and (b) in opposite directions

Two straight parallel wires are separated by 7.0 cm. There is a 2.0-A current flowing in the first wire. If the magnetic field strength is found to be zero between the two wires at a distance of 2.2 cm from the first wire, what is the magnitude and direction of the current in the second wire?

Two long straight wires each carry a current I out of the page toward the viewer, Fig. 2056. Indicate, with appropriate arrows, the direction of at each of the points 1 to 6 in the plane of the page. State if the field is zero at any of the points.

A power line carries a current of 95 A west along the tops of 8.5-m-high poles. (a) What is the magnitude and direction of the magnetic field produced by this wire at the ground directly below? How does this compare with the Earths magnetic field of about (b) Where would the wires magnetic field cancel the Earths field?

A compass needle points 17 E of N outdoors. However, when it is placed 12.0 cm to the east of a vertical wire inside a building, it points 32 E of N. What is the magnitude and direction of the current in the wire? The Earths field there is and is horizontal.

A long pair of insulated wires serves to conduct 24.5 A of dc current to and from an instrument. If the wires are of negligible diameter but are 2.8 mm apart, what is the magnetic field 10.0 cm from their midpoint, in their plane (Fig.2057)? Compare to the magnetic field of the Earth.

A third wire is placed in the plane of the two wires shown in Fig. 2057 parallel and just to the right. If it carries 25.0 A upward, what force per meter of length does it exert on each of the other two wires? Assume it is 2.8 mm from the nearest wire, center to center.

Two long thin parallel wires 13.0 cm apart carry 28-A currents in the same direction. Determine the magnetic field vector at a point 10.0 cm from one wire and 6.0 cm from the other (Fig. 2058

Two long wires are oriented so that they are perpendicular to each other. At their closest, they are 20.0 cm apart (Fig. 2059). What is the magnitude of the magnetic field at a point midway between them if the top one carries a current of 20.0 A and the bottom one carries 12.0 A?

A thin 12-cm-long solenoid has a total of 460 turns of wire and carries a current of 2.0 A. Calculate the field inside the solenoid near the center.

A 30.0-cm-long solenoid 1.25 cm in diameter is to produce a field of 4.65 mT at its center. How much current should the solenoid carry if it has 935 turns of the wire?

A 30.0-cm-long solenoid 1.25 cm in diameter is to produce a field of 4.65 mT at its center. How much current should the solenoid carry if it has 935 turns of the wire?

A 30.0-cm-long solenoid 1.25 cm in diameter is to produce a field of 4.65 mT at its center. How much current should the solenoid carry if it has 935 turns of the wire?

You have 1.0 kg of copper and want to make a practical solenoid that produces the greatest possible magnetic field for a given voltage. Should you make your copper wire long and thin, short and fat, or something else? Consider other variables, such as solenoid diameter, length, and so on. Explain your reasoning

A toroid is a solenoid in the shape of a donut (Fig. 2060). Use Ampres law along the circular paths, shown dashed in Fig. 2060a, to determine that the magnetic field (a) inside the toroid is where N is the total number of turns, and (b) outside the toroid is (c) Is the field inside a toroid uniform like a solenoids? If not, how does it vary?

Use Ampres law to show that the magnetic field between the conductors of a coaxial cable (Fig. 2061) is if r (distance from center) is greater than the radius of the inner wire and less than the radius of the outer cylindrical braid . (b) Show that outside the coaxial cable.

A single square loop of wire 22.0 cm on a side is placed with its face parallel to the magnetic field as in Fig. 2034b. When 5.70 A flows in the coil, the torque on it is What is the magnetic field strength?

If the current to a motor drops by 12%, by what factor does the output torque change?

A galvanometer needle deflects full scale for a current. What current will give full-scale deflection if the magnetic field weakens to 0.760 of its original value?

A circular coil 12.0 cm in diameter and containing nine loops lies flat on the ground. The Earths magnetic field at this location has magnitude and points into the Earth at an angle of 56.0 below a line pointing due north. If a 7.20-A clockwise current passes through the coil, (a) determine the torque on the coil, and (b) which edge of the coil rises up: north, east, south, or west?

Protons move in a circle of radius 6.10 cm in a 0.566-T magnetic field. What value of electric field could make their paths straight? In what direction must the electric field point?

In a mass spectrometer, germanium atoms have radii of curvature equal to 21.0, 21.6, 21.9, 22.2, and 22.8 cm. The largest radius corresponds to an atomic mass of 76 u. What are the atomic masses of the other isotopes?

Suppose the electric field between the electric plates in the mass spectrometer of Fig. 2041 is and the magnetic fields are The source contains carbon isotopes of mass numbers 12, 13, and 14 from a long-dead piece of a tree. (To estimate masses of the atoms, multiply by ) How far apart are the lines formed by the singly charged ions of each type on the photographic film? What if the ions were doubly charged?

One form of mass spectrometer accelerates ions by a voltage V before they enter a magnetic field B. The ions are assumed to start from rest. Show that the mass of an ion is where R is the radius of the ions path in the magnetic field and q is their charge.

An unknown particle moves in a straight line through crossed electric and magnetic fields with and If the electric field is turned off, the particle moves in a circular path of radius What might the particle be?

A mass spectrometer is monitoring air pollutants. It is difficult, however, to separate molecules of nearly equal mass such as CO (28.0106 u) and How large a radius of curvature must a spectrometer have (Fig. 2041) if these two molecules are to be separated on the film by 0.50 mm?

A long thin iron-core solenoid has 380 loops of wire per meter, and a 350-mA current flows through the wire. If the permeability of the iron is what is the total field B inside the solenoid

An iron-core solenoid is 38 cm long and 1.8 cm in diameter, and has 780 turns of wire. The magnetic field inside the solenoid is 2.2 T when 48 A flows in the wire. What is the permeability at this high field strength?

The following are some values of B and for a piece of iron as it is being magnetized (note different units):

Two long straight parallel wires are 15 cm apart. Wire A carries 2.0-A current. Wire Bs current is 4.0 A in the same direction. (a) Determine the magnetic field magnitude due to wire A at the position of wire B. (b) Determine the magnetic field due to wire B at the position of wire A. (c) Are these two magnetic fields equal and opposite? Why or why not? (d) Determine the force on wire A due to wire B, and the force on wire B due to wire A. Are these two forces equal and opposite? Why or why not?

Protons with momentum are magnetically steered clockwise in a circular path 2.2 m in diameter. Determine the magnitude and direction of the field in the magnets surrounding the beam pipe.

A small but rigid wire carrying a 5.0-A current (Fig. 2062) is placed inside a solenoid. The solenoid is 15.0 cm long and has 700 loops of wire, and the current in each loop is 7.0 A. What is the net force on the -shaped wire?

The power cable for an electric trolley (Fig. 2063) carries a horizontal current of 330 A toward the east. The Earths magnetic field has a strength and makes an angle of dip of 22 at this location. Calculate the magnitude and direction of the magnetic force on an 18-m length of this cable.

A particle of charge q moves in a circular path of radius r perpendicular to a uniform magnetic field B. Determine its linear momentum in terms of the quantities given.

An airplane has acquired a net charge of If the Earths magnetic field of is perpendicular to the airplanes velocity of magnitude determine the force on the airplane.

A 32-cm-long solenoid, 1.8 cm in diameter, is to produce a 0.050-T magnetic field at its center. If the maximum current is 6.4 A, how many turns must the solenoid have

Near the equator, the Earths magnetic field points almost horizontally to the north and has magnitude What should be the magnitude and direction for the velocity of an electron if its weight is to be exactly balanced by the magnetic force?

A doubly charged helium atom, whose mass is is accelerated by a voltage of 3200 V. (a) What will be its radius of curvature in a uniform 0.240-T field? (b) What is its period of revolution?

A doubly charged helium atom, whose mass is is accelerated by a voltage of 3200 V. (a) What will be its radius of curvature in a uniform 0.240-T field? (b) What is its period of revolution?

(a) What value of magnetic field would make a beam of electrons, traveling to the west at a speed of go undeflected through a region where there is a uniform electric field of pointing south? (b) What is the direction of the magnetic field if it is perpendicular to the electric field? (c) What is the frequency of the circular orbit of the electrons if the electric field is turned off?

Magnetic fields are very useful in particle accelerators for beam steering; that is, the magnetic fields can be used to change the direction of the beam of charged particles without altering their speed (Fig. 2065). Show how this could work with a beam of protons. What happens to protons that are not moving with the speed for which the magnetic field was designed? If the field extends over a region 5.0 cm wide and has a magnitude of 0.41 T, by approximately what angle will a beam of protons traveling at be bent?

The magnetic field B at the center of a circular coil of wire carrying a current I (as in Fig. 209) is where N is the number of loops in the coil and r is its radius. Imagine a simple model in which the Earths magnetic field of about 1 G near the poles is produced by a single current loop around the equator. Roughly estimate the current this loop would carry.

A proton follows a spiral path through a gas in a uniform magnetic field of 0.010 T, perpendicular to the plane of the spiral, as shown in Fig. 2066. In two successive loops, at points P and Q, the radii are 10.0 mm and 8.5 mm, respectively. Calculate the change in the kinetic energy of the proton as it travels from P to Q.

Two long straight aluminum wires, each of diameter 0.42 mm, carry the same current but in opposite directions. They are suspended by 0.50-m-long strings as shown in Fig. 2067. If the suspension strings make an angle of 3.0 with the vertical and are hanging freely, what is the current in the wires?

An electron enters a uniform magnetic field at a 45 angle to Determine the radius r and pitch p (distance between loops) of the electrons helical path assuming its speed is See Fig. 2068.

A motor run by a 9.0-V battery has a 20-turn square coil with sides of length 5.0 cm and total resistance When spinning, the magnetic field felt by the wire in the coil is 0.020 T. What is the maximum torque on the motor?

Electrons are accelerated horizontally by 2.2 kV. They then pass through a uniform magnetic field B for a distance of 3.8 cm, which deflects them upward so they reach the top of a screen 22 cm away, 11 cm above the center. Estimate the value of B.

A 175-g model airplane charged to 18.0 mC and traveling at 3.4 m/s passes within 8.6 cm of a wire, nearly parallel to its path, carrying a 25-A current.What acceleration (in g’s) does this interaction give the airplane?

A uniform conducting rod of length and mass m sits atop a fulcrum, which is placed a distance from the rods left-hand end and is immersed in a uniform magnetic field of magnitude B directed into the page (Fig. 2069). An object whose mass M is 6.0 times greater than the rods mass is hung from the rods left-hand end. What current (direction and magnitude) should flow through the rod in order for it to be balanced (i.e., be at rest horizontally) on the fulcrum? (Flexible connecting wires which exert negligible force on the rod are not shown.)

Suppose the Earth’s magnetic field at the equator has magnitude \(0.50 \times 10^{-4}\ T\) and a northerly direction at all points. Estimate the speed a singly ionized uranium ion (m = 238 u, q = +e) would need to circle the Earth 6.0 km above the equator. Can you ignore gravity? [Ignore relativity.]

A particle with charge q and momentum p, initially moving along the x axis, enters a region where a uniform magnetic field extends over a width as shown in Fig. 2070. The particle is deflected a distance d in the direction as it traverses the field. Determine (a) whether q is positive or negative, and (b) the magnitude of its momentum p in terms of B l, and d

A bolt of lightning strikes a metal flag pole, one end of which is anchored in the ground. Estimate the force the Earth’s magnetic field can exert on the flag pole while the lightning-induced current flows. See Example 18–10.

A sort of projectile launcher is shown in Fig. 2071. A large current moves in a closed loop composed of fixed rails, a power supply, and a very light, almost frictionless bar (pale green) touching the rails. A magnetic field is perpendicular to the plane of the circuit. If the bar has a length a mass of 1.5 g, and is placed in a field of 1.7 T, what constant current flow is needed to accelerate the bar from rest to in a distance of 1.0 m? In what direction must the magnetic field point?

The cyclotron (Fig. 2072) is a device used to accelerate elementary particles such as protons to high speeds. Particles starting at point A with some initial velocity travel in semicircular orbits in the magnetic field B. The particles are accelerated to higher speeds each time they pass through the gap between the metal dees, where there is an electric field E. (There is no electric field inside the hollow metal dees where the electrons move in circular paths.) The electric field changes direction each half-cycle, owing to an ac voltage so that the particles are increased in speed at each passage through the gap. (a) Show that the frequency f of the voltage must be where q is the charge on the particles and m their mass. (b) Show that the kinetic energy of the particles increases by each revolution, assuming that the gap is small. (c) If the radius of the cyclotron is 2.0 m and the magnetic field strength is 0.50 T, what will be the maximum kinetic energy of accelerated protons in MeV?

The cyclotron (Fig. 2072) is a device used to accelerate elementary particles such as protons to high speeds. Particles starting at point A with some initial velocity travel in semicircular orbits in the magnetic field B. The particles are accelerated to higher speeds each time they pass through the gap between the metal dees, where there is an electric field E. (There is no electric field inside the hollow metal dees where the electrons move in circular paths.) The electric field changes direction each half-cycle, owing to an ac voltage so that the particles are increased in speed at each passage through the gap. (a) Show that the frequency f of the voltage must be where q is the charge on the particles and m their mass. (b) Show that the kinetic energy of the particles increases by each revolution, assuming that the gap is small. (c) If the radius of the cyclotron is 2.0 m and the magnetic field strength is 0.50 T, what will be the maximum kinetic energy of accelerated protons in MeV?

In Fig. 20–73 the top wire is 1.00-mm-diameter copper wire and is suspended in air due to the two magnetic forces from the bottom two wires. The current flow through the two bottom wires is 75 A in each. Calculate the required current flow in the suspended wire (M).

You want to get an idea of the magnitude of magnetic fields produced by overhead power lines. You estimate that a transmission wire is about 13 m above the ground. The local power company tells you that the lines operate at 240 kV and provide a maximum power of 46 MW. Estimate the magnetic field you might experience walking under one such power line, and compare to the Earths field.

In Fig. 2073 the top wire is 1.00-mm-diameter copper wire and is suspended in air due to the two magnetic forces from the bottom two wires. The current flow through the two bottom wires is 75 A in each. Calculate the required current flow in the suspended wire (M).