Chapter 21

At a certain location, the horizontal component of the earth's magnetic field is \(2.5 \times 10^{-5} \mathrm{~T}\), due north. A proton moves eastward with just the right speed, so the magnetic force on it balances its weight. Find the speed of the proton.

A charge of \(-8.3 \mu \mathrm{C}\) is traveling at a speed of \(7.4 \times 10^{6} \mathrm{~m} / \mathrm{s}\) in a region of space where there is a magnetic field. The angle between the velocity of the charge and the field is \(52^{\circ}\). A force of magnitude \(5.4 \times 10^{-3} \mathrm{~N}\) acts on the charge. What is the magnitude of the magnetic field?

In a television set, electrons are accelerated from rest through a potential difference of 19 \(\mathrm{kV}\). The electrons then pass through a 0.28 - T magnetic field that deflects them to the appropriate spot on the screen. Find the magnitude of the maximum magnetic force that an electron can experience.

When a charged particle moves at an angle of \(25^{\circ}\) with respect to a magnetic field, it experiences a magnetic force of magnitude \(F\). At what angle (less than \(90^{\circ}\) ) with respect to this field will this particle, moving at the same speed, experience a magnetic force of magnitude \(2 F ?\)

A charge is moving perpendicular to a magnetic field and experiences a force whose magnitude is \(2.7 \times 10^{-3} \mathrm{~N}\). If this same charge were to move at the same speed and the angle between its velocity and the same magnetic field were \(38^{\circ},\) what would be the magnitude of the magnetic force that the charge would experience?

An electron is moving through a magnetic field whose magnitude is \(8.70 \times 10^{-4} \mathrm{~T}\). The electron experiences only a magnetic force and has an acceleration of magnitude \(3.50 \times 10^{14} \mathrm{~m} / \mathrm{s}^{2}\). At a certain instant, it has a speed of \(6.80 \times 10^{6} \mathrm{~m} / \mathrm{s}\). Determine the angle \(\theta\) (less than \(90^{\circ}\) ) between the electron's velocity and the magnetic field.

One component of a magnetic field has a magnitude of \(0.048 \mathrm{~T}\) and points along the \(+x\) axis, while the other component has a magnitude of \(0.065 \mathrm{~T}\) and points along the \(-y\) axis. A particle carrying a charge of \(+2.0 \times 10^{-5} \mathrm{C}\) is moving along the \(+z\) axis at a speed of \(4.2 \times 10^{3} \mathrm{~m} / \mathrm{s}\). (a) Find the magnitude of the net magnetic force that acts on the particle. (b) Determine the angle that the net force makes with respect to the \(+x\) axis.

The electrons in the beam of a television tube have a kinetic energy of \(2.40 \times 10^{-15} \mathrm{~J}\). Initially, the electrons move horizontally from west to east. The vertical component of the earth's magnetic field points down, toward the surface of the earth, and has a magnitude of \(2.00 \times 10^{-5} \mathrm{~T}\). (a) In what direction are the electrons deflected by this field component? (b) What is the acceleration of an electron in part (a)?

A magnetic field has a magnitude of \(1.2 \times 10^{-3} \mathrm{~T}\), and an electric field has a magnitude of \(4.6 \times 10^{3} \mathrm{~N} / \mathrm{C}\). Both fields point in the same direction. A positive \(1.8-\mu \mathrm{C}\) charge moves at a speed of \(3.1 \times 10^{6} \mathrm{~m} / \mathrm{s}\) in a direction that is perpendicular to both fields. Determine the magnitude of the net force that acts on the charge.

In the operating room, anesthesiologists use mass spectrometers to monitor the respiratory gases of patients undergoing surgery. One gas that is often monitored is the anesthetic isoflurane (molecular mass \(\left.=3.06 \times 10^{-25} \mathrm{~kg}\right)\). In a spectrometer, a singly ionized molecule of isoflurane (charge \(=+e\) ) moves at a speed of \(7.2 \times 10^{3} \mathrm{~m} / \mathrm{s}\) on a circular path that has a radius of \(0.10 \mathrm{~m}\). What is the magnitude of the magnetic field that the spectrometer uses?

Suppose that an ion source in a mass spectrometer produces doubly ionized gold ions \(\left(\mathrm{Au}^{2+}\right),\) each with a mass of \(3.27 \times 10^{-25} \mathrm{~kg} .\) The ions are accelerated from rest through a potential difference of \(1.00 \mathrm{kV}\). Then, a 0.500 -T magnetic field causes the ions to follow a circular path. Determine the radius of the path.

An \(\alpha\) -particle has a charge of \(+2 e\) and a mass of \(6.64 \times 10^{-27} \mathrm{~kg} .\) It is accelerated from rest through a potential difference that has a value of \(1.20 \times 10^{6} \mathrm{~V}\) and then enters a uniform magnetic field whose magnitude is \(2.20 \mathrm{~T}\). The \(\alpha\) -particle moves perpendicular to the magnetic field at all times. What is (a) the speed of the \(\alpha\) -particle, (b) the magnitude of the magnetic force on it, and (c) the radius of its circular path?

Two isotopes of carbon, carbon- 12 and carbon- \(13,\) have masses of \(19.93 \times 10^{-27} \mathrm{~kg}\) and \(21.59 \times 10^{-27} \mathrm{~kg},\) respectively. These two isotopes are singly ionized \((+e)\) and each is given a speed of \(6.667 \times 10^{5} \mathrm{~m} / \mathrm{s}\). The ions then enter the bending region of a mass spectrometer where the magnetic field is \(0.8500 \mathrm{~T}\). Determine the spatial separation between the two isotopes after they have traveled through a half-circle.

The ion source in a mass spectrometer produces both singly and doubly ionized species, \(\mathrm{X}^{+}\) and \(\mathrm{X}^{2+}\). The difference in mass between these species is too small to be detected. Both species are accelerated through the same electric potential difference, and both experience the same magnetic field, which causes them to move on circular paths. The radius of the path for the species \(\mathrm{X}^{+}\) is \(r_{1}\), while the radius for species \(\mathrm{X}^{2+}\) is \(r_{2} .\) Find the ratio \(r_{1} / r_{2}\) of the radii.

A positively charged particle of mass \(7.2 \times 10^{-8} \mathrm{~kg}\) is traveling due east with a speed of \(85 \mathrm{~m} / \mathrm{s}\) and enters a 0.31 -T uniform magnetic field. The particle moves through onequarter of a circle in a time of \(2.2 \times 10^{-3} \mathrm{~s}\), at which time it leaves the field heading due south. All during the motion the particle moves perpendicular to the magnetic field. (a) What is the magnitude of the magnetic force acting on the particle? (b) Determine the magnitude of its charge.

A particle of mass \(6.0 \times 10^{-8} \mathrm{~kg}\) and charge \(+7.2 \mu \mathrm{C}\) is traveling due east. It enters perpendicularly a magnetic field whose magnitude is \(3.0 \mathrm{~T}\). After entering the field, the particle completes one-half of a circle and exits the field traveling due west. How much time does the particle spend traveling in the magnetic field?

A \(45-\mathrm{m}\) length of wire is stretched horizontally between two vertical posts. The wire carries a current of \(75 \mathrm{~A}\) and experiences a magnetic force of \(0.15 \mathrm{~N}\). Find the magnitude of the earth's magnetic field at the location of the wire, assuming the field makes an angle of \(60.0^{\circ}\) with respect to the wire.

A wire of length \(0.655 \mathrm{~m}\) carries a current of \(21.0 \mathrm{~A}\). In the presence of a \(0.470-\mathrm{T}\) magnetic field, the wire experiences a force of \(5.46 \mathrm{~N}\). What is the angle (less than \(90^{\circ}\) ) between the wire and the magnetic field?

At New York City, the earth's magnetic field has a vertical component of \(5.2 \times 10^{-5} \mathrm{~T}\) that points downward (perpendicular to the ground) and a horizontal component of \(1.8 \times 10^{-5} \mathrm{~T}\) that points toward geographic north (parallel to the ground). What is the magnitude and direction of the magnetic force on a 6.0 -m long, straight wire that carries a current of 28 A perpendicularly into the ground?

A square coil of wire containing a single turn is placed in a uniform 0.25 -T magnetic field, as the drawing shows. Each side has a length of \(0.32 \mathrm{~m}\), and the current in the coil is 12 A. Determine the magnitude of the magnetic force on each of the four sides.

A wire carries a current of \(0.66 \mathrm{~A}\). This wire makes an angle of \(58^{\circ}\) with respect to a magnetic field of magnitude \(4.7 \times 10^{-5} \mathrm{~T}\). The wire experiences a magnetic force of magnitude \(7.1 \times 10^{-5} \mathrm{~N}\). What is the length of the wire?

A wire carries a current of 0.66 A. This wire makes an angle of \(58^{\circ}\) with respect to a magnetic field of magnitude \(4.7 \times 10^{-5} \mathrm{~T}\). The wire experiences a magnetic force of magnitude \(7.1 \times 10^{-5} \mathrm{~N}\). What is the length of the wire?

The \(x, y,\) and \(z\) components of a magnetic field are \(B_{x}=0.10 \mathrm{~T}, B_{y}=0.15 \mathrm{~T},\) and \(B_{z}=0.17 \mathrm{~T}\). A \(25-\mathrm{cm}\) wire is oriented along the \(z\) axis and carries a current of \(4.3 \mathrm{~A}\) What is the magnitude of the magnetic force that acts on this wire?

A copper rod of length \(0.85 \mathrm{~m}\) is lying on a frictionless table (see the drawing). Each end of the rod is attached to a fixed wire by an unstretched spring that has a spring constant of \(k=75 \mathrm{~N} / \mathrm{m}\). A magnetic field with a strength of \(0.16 \mathrm{~T}\) is oriented perpendicular to the surface of the table. (a) What must be the direction of the current in the copper rod that causes the springs to stretch? (b) If the current is \(12 \mathrm{~A}\), by how much does each spring stretch?

Consult Interactive Solution \(21.3421 .3\) at to explore a model for solving this problem. The drawing shows a thin, uniform rod, which has a length of \(0.45 \mathrm{~m}\) and a mass of \(0.094 \mathrm{~kg}\). This rod lies in the plane of the paper and is attached to the floor by a hinge at point \(P\). A uniform magnetic field of \(0.36 \mathrm{~T}\) is directed perpendicularly into the plane of the paper. There is a current \(I=4.1 \mathrm{~A}\) in the rod, which does not rotate clockwise or counterclockwise. Find the angle \(\theta\). (Hint: The magnetic force may be taken to act at the center of gravity.)

The two conducting rails in the drawing are tilted upward so they each make an angle of \(30.0^{\circ}\) with respect to the ground. The vertical magnetic field has a magnitude of \(0.050 \mathrm{~T}\) The \(0.20-\mathrm{kg}\) aluminum rod (length \(=1.6 \mathrm{~m}\) ) slides without friction down the rails at a constant velocity. How much current flows through the bar?

A wire has a length of \(7.00 \times 10^{-2} \mathrm{~m}\) and is used to make a circular coil of one turn. There is a current of \(4.30 \mathrm{~A}\) in the wire. In the presence of a 2.50-T magnetic field, what is the maximum torque that this coil can experience?

The 1200 -turn coil in a dc motor has an area per turn of \(1.1 \times 10^{-2} \mathrm{~m}^{2}\). The design for the motor specifies that the magnitude of the maximum torque is \(5.8 \mathrm{~N} \cdot \mathrm{m}\) when the coil is placed in a 0.20 -T magnetic field. What is the current in the coil?

Two coils have the same number of circular turns and carry the same current. Each rotates in a magnetic field as in Figure \(21-21\). Coil 1 has a radius of \(5.0 \mathrm{~cm}\) and rotates in a 0.18 -T field. Coil 2 rotates in a 0.42 - T field. Each coil experiences the same maximum torque. What is the radius (in \(\mathrm{cm}\) ) of coil \(2 ?\)

A coil carries a current and experiences a torque due to a magnetic field. The value of the torque is \(80.0 \%\) of the maximum possible torque. (a) What is the smallest angle between the magnetic field and the normal to the plane of the coil? (b) Make a drawing, showing how this coil would be oriented relative to the magnetic field. Be sure to include the angle in the drawing.

Two pieces of the same wire have the same length. From one piece, a square coil containing a single loop is made. From the other, a circular coil containing a single loop is made. The coils carry different currents. When placed in the same magnetic field with the same orientation, they experience the same torque. What is the ratio \(I_{\text {square }} / I_{\text {circle }}\) of the current in the square coil to that in the circular coil?

Consult Interactive Solution \(\underline{21.43} 21.43\) at to see how this problem can be solved. The coil in Figure \(21-22 a\) contains 410 turns and has an area per turn of \(3.1 \times 10^{-3} \mathrm{~m}^{2}\). The magnetic field is \(0.23 \mathrm{~T},\) and the current in the coil is \(0.26 \mathrm{~A} .\) A brake shoe is pressed perpendicularly against the shaft to keep the coil from turning. The coefficient of static friction between the shaft and the brake shoe is \(0.76 .\) The radius of the shaft is \(0.012 \mathrm{~m}\). What is the magnitude of the minimum normal force that the brake shoe exerts on the shaft?

A square coil and a rectangular coil are each made from the same length of wire. Each contains a single turn. The long sides of the rectangle are twice as long as the short sides. Find the ratio \(\tau\) square \(/ \tau_{\text {rectangle}}\) of the maximum torques that these coils experience in the same magnetic field when they contain the same current.

In the model of the hydrogen atom due to Niels Bohr, the electron moves around the proton at a speed of \(2.2 \times 10^{6} \mathrm{~m} / \mathrm{s}\) in a circle of radius \(5.3 \times 10^{-11} \mathrm{~m}\). Considering the orbiting electron to be a small current loop, determine the magnetic moment associated with this motion. (Hint: The electron travels around the circle in a time equal to the period of the motion.)

What must be the radius of a circular loop of wire so the magnetic field at its center is \(1.8 \times 10^{-4} \mathrm{~T}\) when the loop carries a current of \(12 \mathrm{~A} ?\)

A long, straight wire carries a current of 48 A. The magnetic field produced by this current at a certain point is \(8.0 \times 10^{-5} \mathrm{~T}\). How far is the point from the wire?

The magnetic field produced by the solenoid in a magnetic resonance imaging (MRI) system designed for measurements on whole human bodies has a field strength of \(7.0 \mathrm{~T},\) and the current in the solenoid is \(2.0 \times 10^{2} \mathrm{~A} .\) What is the number of turns per meter of length of the solenoid? Note that the solenoid used to produce the magnetic field in this type of system has a length that is not very long compared to its diameter. Because of this and other design considerations, your answer will be only an approximation.

In a lightning bolt, \(15 \mathrm{C}\) of charge flows during a time of \(1.5 \times 10^{-3} \mathrm{~s}\). Assuming that the lightning bolt can be represented as a long, straight line of current, what is the magnitude of the magnetic field at a distance of \(25 \mathrm{~m}\) from the bolt?

Multiple-Concept Example 7 discusses how problems like this one can be solved. A \(+6.00 \mu \mathrm{C}\) charge is moving with a speed of \(7.50 \times 10^{4} \mathrm{~m} / \mathrm{s}\) parallel to a very long, straight wire. The wire is \(5.00 \mathrm{~cm}\) from the charge and carries a current of \(67.0 \mathrm{~A}\) in a direction opposite to that of the moving charge. Find the magnitude and direction of the force on the charge.

Two circular loops of wire, each containing a single turn, have the same radius of \(4.0 \mathrm{~cm}\) and a common center. The planes of the loops are perpendicular. Each carries a current of 1.7 A. What is the magnitude of the net magnetic field at the common center?

A very long, straight wire carries a current of \(0.12 \mathrm{~A}\). This wire is tangent to a singleturn, circular wire loop that also carries a current. The directions of the currents are such that the net magnetic field at the center of the loop is zero. Both wires are insulated and have diameters that can be neglected. How much current is there in the loop?

Review Interactive Solution \(\underline{21.55} 21.55\) at for one approach to this problem. Two circular coils are concentric and lie in the same plane. The inner coil contains 140 turns of wire, has a radius of \(0.015 \mathrm{~m}\), and carries a current of \(7.2 \mathrm{~A}\). The outer coil contains 180 turns and has a radius of \(0.023 \mathrm{~m}\). What must be the magnitude and direction (relative to the current in the inner coil) of the current in the outer coil, such that the net magnetic field at the common center of the two coils is zero?

Review Interactive Solution 21.5521 .55 at for one approach to this problem. Two circular coils are concentric and lie in the same plane. The inner coil contains 140 turns of wire, has a radius of \(0.015 \mathrm{~m}\), and carries a current of \(7.2 \mathrm{~A}\). The outer coil contains 180 turns and has a radius of \(0.023 \mathrm{~m}\). What must be the magnitude and direction (relative to the current in the inner coil) of the current in the outer coil, such that the net magnetic field at the common center of the two coils is zero?

Two parallel rods are each \(0.50 \mathrm{~m}\) in length. They are attached at their centers to either end of a spring (spring constant \(=150 \mathrm{~N} / \mathrm{m}\) ) that is initially neither stretched nor compressed. When 950 A of current is in each rod in the same direction, the spring is observed to be compressed by \(2.0 \mathrm{~cm} .\) Treat the rods as long, straight wires and find the separation between them when the current is present.

A piece of copper wire has a resistance per unit length of \(5.90 \times 10^{-3} \Omega / \mathrm{m}\). The wire is wound into a thin, flat coil of many turns that has a radius of \(0.140 \mathrm{~m}\). The ends of the wire are connected to a \(12.0-\mathrm{V}\) battery. Find the magnetic field strength at the center of the coil.

The drawing shows an end-on view of three wires. They are long, straight, and perpendicular to the plane of the paper. Their cross sections lie at the corners of a square. The currents in wires 1 and 2 are \(I_{1}=I_{2}=I\) and are directed into the paper. What is the direction of the current in wire \(3,\) and what is the ratio \(I_{3} / I,\) such that the net magnetic field at the empty corner is zero?

Suppose a uniform magnetic field is everywhere perpen dicular to this page. The field points directly upward toward you. A circular path is drawn on the page. Use Ampère's law to show that there can be no net current passing through the circular surface.

Refer to Interactive Solution \(21.6221 .62\) at for help with problems like this one. A very long, hollow cylinder is formed by rolling up a thin sheet of copper. Electric charges flow along the copper sheet parallel to the axis of the cylinder. The arrangement is, in effect, a hollow tube of current \(I\). Use Ampère's law to show that the magnetic field (a) is \(\mu_{0} I /(2 \pi r)\) outside the cylinder at a distance \(r\) from the axis and (b) is zero at any point within the hollow interior of the cylinder. (Hint: For closed paths, use circles perpendicular to and centered on the axis of the cylinder.)

At for help with problems like this one. A very long, hollow cylinder is formed by rolling up a thin sheet of copper. Electric charges flow along the copper sheet parallel to the axis of the cylinder. The arrangement is, in effect, a hollow tube of current \(I\). Use Ampère's law to show that the magnetic field (a) is \(\mu_{0} I /(2 \pi r)\) outside the cylinder at a distance \(r\) from the axis and \((\mathrm{b})\) is zero at any point within the hollow interior of the cylinder. (Hint: For closed paths, use circles perpendicular to and centered on the axis of the cylinder.)

A long, cylindrical conductor is solid throughout and has a radius \(R\). Electric charges flow parallel to the axis of the cylinder and pass uniformly through the entire cross section. The arrangement is, in effect, a solid tube of current \(I_{0}\). The current per unit cross-sectional area (i.e., the current density) is \(I_{0} /\left(\pi R^{2}\right)\). Use Ampère's law to show that the magnetic field inside the conductor at a distance \(r\) from the axis is \(\mu_{0} l_{0} r /\left(2 \pi R^{2}\right) .\)