In Figure P19.2, assume that in each case the velocity

An electron gun res electrons into a magnetic eld directed straight downward. Find the direction of the force exerted by the eld on an electron for each of the following directions of the electrons velocity: (a) horizontal and due north; (b) horizontal and 30west of north; (c)due north, but at 30 below the horizontal; (d) straight upward. (Remember that an electron has a negative charge.)

a) Find the direction of the force on a proton (a positively charged particle) moving through the magnetic elds in Figure P19.2 (page 652), as shown. (b) Repeat part (a), assuming the moving particle is an electron.

Find the direction of the magnetic eld acting on the positively charged particle moving in the various situations shown in Figure P19.3 (page 652) if the direction of the magnetic force acting on it is as indicated.

Determine the initial direction of the deection of charged particles as they enter the magnetic elds, as shown in Figure P19.4 (page 652).

At the equator, near the surface of Earth, the magnetic eld is approximately 50.0  T northward, and the electric eld is about 100 N/C downward in fair weather. Find the gravitational, electric, and magnetic forces on an electron with an instantaneous velocity of 6.00  106 m/s directed to the east in this environment.

The magnetic eld of the Earth at a certain location is directed vertically downward and has a magnitude of 50.0  T. A proton is moving horizontally toward the west in this eld with a speed of 6.20  106 m/s. What are the direction and magnitude of the magnetic force the eld exerts on the proton?

What velocity would a proton need to circle Earth 1 000 km above the magnetic equator, where Earths magnetic eld is directed horizontally north and has a magnitude of 4.00  108 T?

An electron is accelerated through 2 400 V from rest and then enters a region where there is a uniform 1.70-T magnetic eld. What are (a) the maximum and (b) the minimum magnitudes of the magnetic force acting on this electron?

A proton moves perpendicularly to a uniform magnetic eld at 1.0  107 m/s and exhibits an acceleration of 2.0  1013 m/s2 in the x-direction when its velocity is in the z-direction. Determine the magnitude and direction of the eld.

Sodium ions (Na ) move at 0.851m/s through a bloodstream in the arm of a person standing near a large magnet. The magnetic eld has a strength of 0.254T and makes an angle of 51.0 with the motion of the sodium ions. The arm contains 100cm3 of blood with 3.00  1020 Na ions per cubic centimeter. If no other ions were present in the arm, what would be the magnetic force on the arm?

A current I  15 A is directed along the positive x-axis and perpendicular to a magnetic eld. A magnetic force per unit length of 0.12 N/m acts on the conductor in the negative y-direction. Calculate the magnitude and direction of the magnetic eld in the region through which the current passes.

In Figure P19.2, assume that in each case the velocity vector shown is replaced with a wire carrying a current in the direction of the velocity vector. For each case, nd the direction of the magnetic force acting on the wire.

In Figure P19.3, assume that in each case the velocity vector shown is replaced with a wire carrying a current in the direction of the velocity vector. For each case, nd the direction of the magnetic eld that will produce the magnetic force shown.

A wire having a mass per unit length of 0.500 g/cm carries a 2.00-A current horizontally to the south. What are the direction and magnitude of the minimum magnetic eld needed to lift this wire vertically upward?

A wire carries a current of 10.0 A in a direction that makes an angle of 30.0 with the direction of a magnetic eld of strength 0.300 T. Find the magnetic force on a 5.00-m length of the wire.

At a certain location, Earth has a magnetic eld of 0.60  104 T, pointing 75 below the horizontal in a northsouth plane. A 10.0-m-long straight wire carries a 15-A current. (a) If the current is directed horizontally toward the east, what are the magnitude and direction of the magnetic force on the wire? (b) What are the magnitude and direction of the force if the current is directed vertically upward?

A wire with a mass of 1.00 g/cm is placed on a horizontal surface with a coefcient of friction of 0.200. The wire carries a current of 1.50 A eastward and moves horizontally to the north. What are the magnitude and the direction of the smallest vertical magnetic eld that enables the wire to move in this fashion?

A conductor suspended by two exible wires as shown in Figure P19.18 has a mass per unit length of 0.040 0 kg/m. What current must exist in the conductor for the tension in the supporting wires to be zero when the magnetic eld is 3.60 T into the page? What is the required direction for the current?

An unusual message delivery system is pictured in Figure P19.19. A 15-cm length of conductor that is free to move is held in place between two thin conductors. When a 5.0-A current is directed as shown in the gure, the wire segment moves upward at a constant velocity. If the mass of the wire is 15g, nd the magnitude and direction of the minimum magnetic eld that is required to move the wire. (The wire slides without friction on the two vertical conductors.)

A wire 2.80 m in length carries a current of 5.00 A in a region where a uniform magnetic eld has a magnitude of 0.390 T. Calculate the magnitude of the magnetic force on the wire, assuming the angle between the magnetic eld and the current is (a) 60.0, (b) 90.0, (c) 120.

In Figure P19.21, the cube is 40.0cm on each edge. Four straight segments of wireab, bc, cd, and daform a closed loop that carries a current I  5.00A in the direction shown. A uniform magnetic eld of magnitude B  0.0200T is in the positive y-direction. Determine the magnitude and direction of the magnetic force on each segment.

A current of 17.0 mA is maintained in a single circular loop with a circumference of 2.00 m. A magnetic eld of 0.800 T is directed parallel to the plane of the loop. What is the magnitude of the torque exerted by the magnetic eld on the loop?

An eight-turn coil encloses an elliptical area having a major axis of 40.0 cm and a minor axis of 30.0 cm (Fig. P19.23). The coil lies in the plane of the page and has a 6.00-A current owing clockwise around it. If the coil is in a uniform magnetic eld of 2.00  104 T directed toward the left of the page, what is the magnitude of the torque on the coil? [Hint: The area of an ellipse is A   ab, where a and b are, respectively, the semimajor and semiminor axes of the ellipse.]

A rectangular loop consists of 100 closely wrapped turns and has dimensions 0.40 m by 0.30 m. The loop is hinged along the y-axis, and the plane of the coil makes an angle of 30.0 with the x-axis (Fig. P19.24). What is the magnitude of the torque exerted on the loop by a uniform magnetic eld of 0.80 T directed along the x-axis when the current in the windings has a value of 1.2 A in the direction shown? What is the expected direction of rotation of the loop?

A long piece of wire with a mass of 0.100 kg and a total length of 4.00 m is used to make a square coil with a side of 0.100 m. The coil is hinged along a horizontal side, carries a 3.40-A current, and is placed in a vertical magnetic eld with a magnitude of 0.010 0 T. (a) Determine the angle that the plane of the coil makes with the vertical when the coil is in equilibrium. (b) Find the torque acting on the coil due to the magnetic force at equilibrium.

A copper wire is 8.00 m long and has a cross-sectional area of 1.00  104 m2. The wire forms a one-turn loop in the shape of square and is then connected to a battery that applies a potential difference of 0.100 V. If the loop is placed in a uniform magnetic eld of magnitude 0.400 T, what is the maximum torque that can act on it? The resistivity of copper is 1.70  108 m.

A proton moving freely in a circular path perpendicular to a constant magnetic eld takes 1.00  s to complete one revolution. Determine the magnitude of the magnetic eld.

A cosmic-ray proton in interstellar space has an energy of 10.0 MeV and executes a circular orbit having a radius equal to that of Mercurys orbit around the Sun, which is 5.80  1010 m. What is the magnetic eld in that region of space?

Figure P19.29a is a diagram of a device called a velocity selector, in which particles of a specic velocity pass through undeected while those with greater or lesser velocities are deected either upwards or downwards. An electric eld is directed perpendicular to a magnetic eld, producing an electric force and a magnetic force on the charged particle that can be equal in magnitude and opposite in direction (Fig. P19.29b) and hence cancel. Show that particles with a speed of v  E/B will pass through the velocity selector undeected.

Consider the mass spectrometer shown schematically in Figure P19.30. The electric eld between the plates of the velocity selector is 950 V/m, and the magnetic elds in both the velocity selector and the deection chamber have magnitudes of 0.930 T. Calculate the radius of the path in the system for a singly charged ion with mass m  2.18  1026 kg. [Hint: See Problem 29.]

A singly charged positive ion has a mass of 2.50  1026 kg. After being accelerated through a potential difference of 250 V, the ion enters a magnetic eld of 0.500 T, in a direction perpendicular to the eld. Calculate the radius of the path of the ion in the eld.

A mass spectrometer is used to examine the isotopes of uranium. Ions in the beam emerge from the velocity selector at a speed of 3.00  105 m/s and enter a uniform magnetic eld of 0.600 T directed perpendicularly to the velocity of the ions. What is the distance between the impact points formed on the photographic plate by singly charged ions of 235U and 238U?

A proton is at rest at the plane vertical boundary of a region containing a uniform vertical magnetic eld B. An alpha particle moving horizontally makes a head-on elastic collision with the proton. Immediately after the collision, both particles enter the magnetic eld, moving perpendicular to the direction of the eld. The radius of the protons trajectory is R. Find the radius of the alpha particles trajectory. The mass of the alpha particle is four times that of the proton, and its charge is twice that of the proton.

In each of parts (a), (b), and (c) of Figure P19.34, nd the direction of the current in the wire that would produce a magnetic eld directed as shown.

A lightning bolt may carry a current of 1.00  104 A for a short time. What is the resulting magnetic eld 100 m from the bolt? Suppose that the bolt extends far above and below the point of observation.

In 1962, measurements of the magnetic eld of a large tornado were made at the Geophysical Observatory in Tulsa, Oklahoma. If the magnitude of the tornados eld was B  1.50  108 T pointing north when the tornado was 9.00 km east of the observatory, what current was carried up or down the funnel of the tornado? Model the vortex as a long, straight wire carrying a current.

A cardiac pacemaker can be affected by a static magnetic eld as small as 1.7 mT. How close can a pacemaker wearer come to a long, straight wire carrying 20 A?

The two wires shown in Figure P19.38 carry currents of 5.00 A in opposite directions and are separated by 10.0 cm. Find the direction and magnitude of the net magnetic eld (a) at a point midway between the wires; (b) at point P1, 10.0 cm to the right of the wire on the right, and (c) at point P2, 20.0 cm to the left of the wire on the left.

Four long, parallel conductors carry equal currents of I  5.00 A. Figure P19.39 is an end view of the conductors. The direction of the current is into the page at points A and B (indicated by the crosses) and out of the page at C and D (indicated by the dots). Calculate the magnitude and direction of the magnetic eld at point P, located at the center of the square with edge of length 0.200 m.

The two wires in Figure P19.40 carry currents of 3.00 A and 5.00 A in the direction indicated. (a) Find the direction and magnitude of the magnetic eld at a point midway between the wires. (b) Find the magnitude and direction of the magnetic eld at point P, located 20.0 cm above the wire carrying the 5.00-A current

A wire carries a 7.00-A current along the x-axis, and another wire carries a 6.00-A current along the y-axis, as shown in Figure P19.41. What is the magnetic eld at point P, located at x  4.00 m, y  3.00 m?

A long, straight wire lies on a horizontal table and carries a current of 1.20  A. In a vacuum, a proton moves parallel to the wire (opposite the direction of the current) with a constant velocity of 2.30  104 m/s at a constant distance d above the wire. Determine the value of d. (You may ignore the magnetic eld due to Earth.)

The magnetic eld 40.0 cm away from a long, straight wire carrying current 2.00 A is 1.00  T. (a) At what distance is it 0.100  T? (b) At one instant, the two conductors in a long household extension cord carry equal 2.00-A currents in opposite directions. The two wires are 3.00 mm apart. Find the magnetic eld 40.0 cm away from the middle of the straight cord, in the plane of the two wires. (c) At what distance is it one-tenth as large? (d) The center wire in a coaxial cable carries current 2.00 A in one direction, and the sheath around it carries current 2.00 A in the opposite direction. What magnetic eld does the cable create at points outside?

Two parallel wires are 10.0 cm apart, and each carries a current of 10.0 A. (a) If the currents are in the same direction, nd the force per unit length exerted on one of the wires by the other. Are the wires attracted to or repelled by each other? (b) Repeat the problem with the currents in opposite directions.

A wire with a weight per unit length of 0.080 N/m is suspended directly above a second wire. The top wire carries a current of 30.0 A and the bottom wire carries a current of 60.0 A. Find the distance of separation between the wires so that the top wire will be held in place by magnetic repulsion.

In Figure P19.46, the current in the long, straight wire is I1  5.00 A, and the wire lies in the plane of the rectangular loop, which carries 10.0 A. The dimensions shown are c  0.100 m, a  0.150 m, and   0.450 m. Find the magnitude and direction of the net force exerted by the magnetic eld due to the straight wire on the loop.

What current is required in the windings of a long solenoid that has 1 000 turns uniformly distributed over a length of 0.400 m in order to produce a magnetic eld of magnitude 1.00  104 T at the center of the solenoid?

It is desired to construct a solenoid that will have a resistance of 5.00  (at 20C) and produce a magnetic eld of 4.00  102 T at its center when it carries a current of 4.00 A. The solenoid is to be constructed from copper wire having a diameter of 0.500 mm. If the radius of the solenoid is to be 1.00 cm, determine (a) the number of turns of wire needed and (b) the length the solenoid should have.

A single-turn square loop of wire 2.00cm on a side carries a counterclockwise current of 0.200 A. The loop is inside a solenoid, with the plane of the loop perpendicular to the magnetic eld of the solenoid. The solenoid has 30 turns per centimeter and carries a counterclockwise current of 15.0 A. Find the force on each side of the loop and the torque acting on the loop.

An electron is moving at a speed of 1.0  104 m/s in a circular path of radius of 2.0 cm inside a solenoid. The magnetic eld of the solenoid is perpendicular to the plane of the electrons path. Find (a) the strength of the magnetic eld inside the solenoid and (b) the current in the solenoid if it has 25 turns per centimeter.

A circular coil consisting of a single loop of wire has a radius of 30.0 cm and carries a current of 25 A. It is placed in an external magnetic eld of 0.30 T. Find the torque on the wire when the plane of the coil makes an angle of 35 with the direction of the eld.

An electron enters a region of magnetic eld of magnitude 0.010 0 T, traveling perpendicular to the linear boundary of the region. The direction of the eld is perpendicular to the velocity of the electron. (a) Determine the time it takes for the electron to leave the eld-lled region, noting that its path is a semicircle. (b) Find the kinetic energy of the electron if the radius of its semicircular path is 2.00 cm.

Two long, straight wires cross each other at right angles, as shown in Figure P19.53. (a) Find the direction and magnitude of the magnetic eld at point P, which is in the same plane as the two wires. (b) Find the magnetic eld at a point 30.0 cm above the point of intersection (30.0 cm out of the page, toward you).

A 0.200-kg metal rod carrying a current of 10.0 A glides on two horizontal rails 0.500 m apart. What vertical magnetic eld is required to keep the rod moving at a constant speed if the coefcient of kinetic friction between the rod and rails is 0.100?

Two species of singly charged positive ions of masses 20.0  1027 kg and 23.4  1027 kg enter a magnetic eld at the same location with a speed of 1.00  105 m/s. If the strength of the eld is 0.200 T, and the ions move perpendicularly to the eld, nd their distance of separation after they complete one-half of their circular path.

Two parallel conductors carry currents in opposite directions, as shown in Figure P19.56. One conductor carries a current of 10.0 A. Point A is the midpoint between the wires, and point C is 5.00 cm to the right of the 10.0-A current. I is adjusted so that the magnetic eld at C is zero. Find (a) the value of the current I and (b) the value of the magnetic eld at A.

Using an electromagnetic owmeter (Fig. P19.57), a heart surgeon monitors the ow rate of blood through an artery. Electrodes A and B make contact with the outer surface of the blood vessel, which has interior diameter 3.00 mm. (a) For a magnetic eld magnitude of 0.040 0 T, a potential difference of 160  V appears between the electrodes. Calculate the speed of the blood. (b) Verify that electrode A is positive, as shown. Does the sign of the emf depend on whether the mobile ions in the blood are predominantly positively or negatively charged? Explain.

Two circular loops are parallel, coaxial, and almost in contact 1.00 mm apart (Fig. P19.58). Each loop is 10.0 cm in radius. The top loop carries a clockwise current of 140 A. The bottom loop carries a counterclockwise current of 140 A. (a) Calculate the magnetic force that the bottom loop exerts on the top loop. (b) The upper loop has a mass of 0.021 0 kg. Calculate its acceleration, assuming that the only forces acting on it are the force in part (a) and its weight. [Hint: The distance between the loops is small in comparison to their radius of curvature, so the loops may be treated as long, straight parallel wires.]

A 1.00-kg ball having net charge Q  5.00  C is thrown out of a window horizontally at a speed v  20.0 m/s. The window is at a height h  20.0 m above the ground. A uniform horizontal magnetic eld of magnitude B  0.010 0 T is perpendicular to the plane of the balls trajectory. Find the magnitude of the magnetic force acting on the ball just before it hits the ground. [Hint: Ignore magnetic forces in nding the balls nal velocity.]

The idea that static magnetic elds might have a therapeutic value has been around for centuries. A currently available rare-Earth magnet that is advertised to relieve joint pain is shown in Figure P19.60. It is 1.0 mm thick and has a eld strength of 5.0
102 T at the center of the at surface. The magnetic eld strength at points away from the center of the disk is inversely proportional to h3, where h is the distance from the midplane of the disk. How far from the surface of the disk will the eld strength be reduced to that of Earth (5.0
105 T)?

Two long parallel conductors carry currents I1  3.00 A and I2  3.00 A, both directed into the page in Figure P19.61. Determine the magnitude and direction of the resultant magnetic eld at P.

A uniform horizontal wire with a linear mass density of 0.50 g/m carries a 2.0-A current. It is placed in a constant magnetic eld with a strength of 4.0
103 T. The eld is horizontal and perpendicular to the wire. As the wire moves upward starting from rest, (a) what is its acceleration and (b) how long does it take to rise 50 cm? Neglect the magnetic eld of Earth.

Three long parallel conductors carry currents of I  2.0 A. Figure P19.63 is an end view of the conductors, with each current coming out of the page. Given that a  1.0 cm, determine the magnitude and direction of the magnetic eld at points A, B, and C.

Two long parallel wires, each with a mass per unit length of 40g/m, are supported in a horizontal plane by 6.0-cm-long strings, as shown in Figure P19.64. Each wire carries the same current I, causing the wires to repel each other so that the angle  between the supporting strings is 16. (a) Are the currents in the same or opposite directions? (b) Determine the magnitude of each current.

Protons having a kinetic energy of 5.00 MeV are moving in the positive x-direction and enter a magnetic eld of 0.050 0 T in the z-direction, out of the plane of the page, and extending from x  0 to x  1.00 m as in Figure P19.65 (page 658). (a) Calculate the y-component of the protons momentum as they leave the magnetic eld. (b) Find the angle  between the initial velocity vector of the proton beam and the velocity vector after the beam emerges from the eld. [Hint: Neglect relativistic effects and note that 1 eV  1.60
1019 J.]

A straight wire of mass 10.0 g and length 5.0 cm is suspended from two identical springs that, in turn, form a closed circuit (Fig. P19.66). The springs stretch a distance of 0.50 cm under the weight of the wire. The circuit has a total resistance of 12 . When a magnetic eld directed out of the page (indicated by the dots in the gure is turned on, the springs are observed to stretch an additional 0.30 cm. What is the strength of the magnetic eld? (The upper portion of the circuit is xed.)

A solenoid 10.0 cm in diameter and 75.0 cm long is made from copper wire of diameter 0.100 cm with very thin insulation. The wire is wound onto a cardboard tube in a single layer, with adjacent turns touching each other. To produce a eld of magnitude 20.0 mT at the center of the solenoid, what power must be delivered to the solenoid?

Assume that the region to the right of a certain vertical plane contains a vertical magnetic eld of magnitude 1.00mT and that the eld is zero in the region to the left of the plane. An electron, originally traveling perpendicular to the boundary plane, passes into the region of the eld. (a) Noting that the path of the electron is a semicircle, determine the time interval required for the electron to leave the eld-lled region. (b) Find the kinetic energy of the electron if the maximum depth of penetration into the eld is 2.00 cm.

Three long wires (wire 1, wire 2, and wire 3) are coplanar and hang vertically. The distance between wire 1 and wire 2 is 20.0 cm. On the left, wire 1 carries an upward current of 1.50 A. To the right, wire 2 carries a downward current of 4.00 A. Wire 3 is located such that when it carries a certain current, no net force acts upon any of the wires. Find (a) the position of wire 3 and (b) the magnitude and direction of the current in wire 3.

Two long parallel conductors separated by 10.0 cm carry currents in the same direction. The rst wire carries a current I1  5.00 A and the second carries I2  8.00 A. (a) What is the magnitude of the magnetic eld created by I1 at the location of I2? (b) What is the force per unit length exerted by I1 on I2? (c) What is the magnitude of the magnetic eld created by I2 at the location of I1? (d) What is the force per length exerted by I2 on I1?

For this activity, you will need a small bar magnet, a small plastic container, and a bowl of water. Tape the magnet to the bottom of the container, and oat the container and magnet on the surface of the bowl as in Figure A19.1. The magnet and the container should rotate and come to equilibrium, with the magnet pointing along a northsouth line. The compass you have constructed is similar to the type used by early sailing vessels. How can you determine which direction is north and which is south?

In the Northern Hemisphere, the direction of Earths magnetic eld becomes more and more nearly vertical the farther north one goes. To nd the variation from the horizontal of the magnetic eld in your locale, try the following: press an unmagnetized needle through a PingPong ball, and balance the structure between two drinking glasses that are lined up along an eastwest line. Next, press a magnetized needle through the ball at right angles to the unmagnetized needle so that the needle points north. The magnetized needle can now rotate in the vertical direction and will point in the direction of Earths magnetic eld, which is at some angle below the horizontal. Take several measurements of this dip angle and obtain an average value.

Construct an electromagnet by wrapping about 1 meter of small-diameter insulated wire around a steel nail. Tape the ends of the wires to a D-cell battery as in Figure A19.3. How many staples or paper clips can you pick up with your electromagnet? How would you increase the magnetic eld set up by the nail? Disconnect the wires from the battery and test how much magnetism is retained by the nail by seeing how many staples it can pick up. A convenient way to test the strength of a magnet is to attach a paper clip to a rubber band. Note how far the rubber band is stretched before the clip comes free of the magnet. Test your electromagnet in this way. Where is the magnetic eld of the electromagnet strongest, at the ends of the nail or near its center? When you have your nail magnetized, bang it against a table or the oor and then check its magnetism. Why does the nail lose its magnetism by this procedure?

You can trace out the eld pattern of a magnet with iron lings. Any machine shop will supply the lings, which should be soaked in a soap solution to remove grit and oil and then dried. Scatter them lightly over the surface of a paper covering the magnet, and then tap the paper gently to jar the lings into alignment. Explain why the lings form their pattern. Examine the eld pattern set up in the following situations: (a) Arrange two bar magnets about 4 cm apart, aligned with opposite poles facing each other. (b) Use two bar magnets about 4 cm apart, aligned with like poles facing each other. (c) Use a horseshoe magnet.