Consider an electron near the Earth’s equator. In which direction does it tend to deflect if its velocity is (a) directed downward?

(b) Directed northward?

(c) Directed westward?

(d) Directed southeastward?

Zachary W.

Numerade Educator

(a) Find the direction of the force on a proton (a positively charged particle) moving through the magnetic fields in Figure P19.2, as shown

(b) Repeat part (a), assuming the moving particle is an electron.

Zachary W.

Numerade Educator

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

Zachary W.

Numerade Educator

Determine the initial direction of the deflection of charged particles as they enter the magnetic fields, as shown in Figure $\mathrm{P} 19.4$

Zachary W.

Numerade Educator

A laboratory electromagnet produces a magnetic field of magnitude 1.50 T. A proton moves through this field with a speed of $6.00 \times 10^{6} \mathrm{m} / \mathrm{s} .$ (a) Find the magnitude of the maximum magnetic force that could be exerted on the proton. (b) What is the magnitude of the maximum acceleration of the proton? (c) Would the field exert the same magnetic force on an electron moving through the field with the same speed? (d) Would the electron undergo the same acceleration? Explain.

Zachary W.

Numerade Educator

A proton moves perpendicular to a uniform magnetic field $\overrightarrow{\mathbf{B}}$ at a speed of $1.00 \times 10^{7} \mathrm{m} / \mathrm{s}$ and undergoes an acceleration of $2.00 \times 10^{13} \mathrm{m} / \mathrm{s}^{2}$ in the positive $x$ -direction when its velocity is in the positive z - direction. Determine the magnitude and direction of the field.

Zachary W.

Numerade Educator

Electrons and protons travel from the Sun to the Earth at a typical velocity of $4.00 \times 10^{5} \mathrm{m} / \mathrm{s}$ in the positive $x$ -direction. Thousands of miles from Earth, they interact with Earth's magnetic field of magnitude $3.00 \times 10^{-8} \mathrm{T}$ in the positive z - direction. Find the (a) magnitude and (b) direction of the magnetic force on a proton. Find the (c) magnitude and (d) direction of the magnetic force on an electron.

Zachary W.

Numerade Educator

An oxygen ion $\left(\mathrm{O}^{+}\right)$ moves in the $x y$ -plane with a speed of $2.50 \times 10^{3} \mathrm{m} / \mathrm{s}$ . If a constant magnetic field is directed along the $z$ -axis with a magnitude of $2.00 \times 10^{-5} \mathrm{T},$ find $(\mathrm{a})$ the magnitude of the magnetic force acting on the ion and (b) the magnitude of the ion’s acceleration.

Manish H.

Numerade Educator

In A proton moving at $4.00 \times 10^{6} \mathrm{m} / \mathrm{s}$ through a magnetic field of magnitude 1.70 $\mathrm{T}$ experiences a magnetic force of magnitude $8.20 \times 10^{-13} \mathrm{N}$ . What is the angle between the proton's velocity and the field?

Zachary W.

Numerade Educator

Sodium ions ( Na' ) move at 0.851 $\mathrm{m} / \mathrm{s}$ through a bloodstream in the arm of a person standing near a large magnet. The magnetic field has a strength of 0.254 T and makes an angle of $51.0^{\circ}$ with the motion of the sodium ions. The arm contains 100 $\mathrm{cm}^{3}$ of blood with a concentration of $3.00 \times 10^{20}$ $\mathrm{Na}^{+}$ ions per cubic centimeter. If no other ions were present in the arm, what would be the magnetic force on the arm?

Zachary W.

Numerade Educator

At the equator, near the surface of Earth, the magnetic field is approximately 50.0$\mu \mathrm{T}$ northward, and the electric field is about 100. N/C downward in fair weather. Find the gravitational, electric, and magnetic forces on an electron with an instantaneous velocity of $6.00 \times 10^{6} \mathrm{m} / \mathrm{s}$ directed to the east in this environment.

Zachary W.

Numerade Educator

A proton travels with a speed of $5.02 \times 10^{6} \mathrm{m} / \mathrm{s}$ at an angle of $60^{\circ}$ with the direction of a magnetic field of magnitude 0.180 T in the positive x - direction. What are (a) the magnitude of the magnetic force on the proton and (b) the proton’s acceleration?

Zachary W.

Numerade Educator

An electron moves in a circular path perpendicular to a magnetic field of magnitude 0.235 T. If the kinetic energy of the electron is $3.30 \times 10^{-19} \mathrm{J},$ find (a) the speed of the electron and (b) the radius of the circular path.

Zachary W.

Numerade Educator

Figure P19.14a is a diagram of a device called a velocity selector, in which particles of a specific velocity pass through undeflected while those with greater or lesser velocities are deflected either upwards or downwards. An electric field is directed perpendicular to a magnetic field, producing an electric force and a magnetic force on the charged particle that can be equal in magnitude and opposite in direction (Fig. P19.14b) and hence cancel. Show that particles with a speed of $v=E / B$ will pass through the velocity selector undeflected.

Zachary W.

Numerade Educator

Consider the mass spectrometer shown schematically in Figure P19.15. The electric field between the plates of the velocity selector is $9.50 \times 10^{2} \mathrm{V} / \mathrm{m},$ and the magnetic fields in both the velocity selector and the deflection chamber have magnitudes of 0.930 T. Calculate the radius of the path in the system for a singly charged ion with mass $m=2.18 \times 10^{-26} \mathrm{kg}$ .

Hint: See Problem 14.

Zachary W.

Numerade Educator

A mass spectrometer is used to examine the isotopes of uranium. Ions in the beam emerge from the velocity selector at a speed of $3.00 \times 10^{5} \mathrm{m} / \mathrm{s}$ and enter a uniform magnetic field 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 $^{235} \mathrm{U}$ and $^{238} \mathrm{U?}$

Zachary W.

Numerade Educator

Jupiter’s magnetic field occupies a volume of space larger than the Sun and contains ionized particles ejected from sources including volcanoes on Io, one of Jupiter’s moons. A sulfur ion $\left(\mathrm{S}^{+}\right)$ in Jupiter's magnetic field has mass $5.32 \times 10^{-26} \mathrm{kg}$

and kinetic energy 75.0 $\mathrm{eV}$ . (a) Find the maximum magnetic force on the ion from Jupiter’s magnetic field of magnitud $4.28 \times 10^{-4} \mathrm{T} .$ (b) Find the radius of the sulfur ion's circular path, assuming its velocity is perpendicular to Jupiter’s magnetic field.

Zachary W.

Numerade Educator

Electrons in Earth’s upper atmosphere have typical speeds near $6.00 \times 10^{5} \mathrm{m} / \mathrm{s} .$ (a) Calculate the magnitude of Earth's magnetic field if an electron’s velocity is perpendicular to the magnetic field and its circular path has a radius of $7.00 \times 10^{-2} \mathrm{m}$ (b) Calculate the number of times per second that an electron circles around a magnetic field line.

Zachary W.

Numerade Educator

A proton is at rest at the plane vertical boundary of a region containing a uniform vertical magnetic field B (Fig. P19.19). An alpha particle moving horizontally makes a head- on elastic collision with the proton.

Immediately after the collision, both particles enter the magnetic field, moving perpendicular to the direction of the field. The radius of the proton’s trajectory is R. The mass of the alpha particle is four times that of the proton, and its charge is twice that of the proton. Find the radius of the alpha particle’s trajectory.

Zachary W.

Numerade Educator

EA A proton (charge $+e,$ mass $m_{p} ),$ a deuteron (charge $+e,$ mass $2 m_{p} ),$ and an alpha particle (charge $+2 e,$ mass 4$m_{p} )$ are accelerated from rest through a common potential difference $\Delta V$ . Each of the particles enters a uniform magnetic field $\overrightarrow{\mathbf{B}}$ , with its velocity in a direction perpendicular to $\overrightarrow{\mathbf{B}}$ . The proton moves in a circular path of radius $r_{p} .$ In terms of $r_{p},$ determine

(a) the radius $r_{d}$ of the circular orbit for the deuteron and

(b) the radius $r \alpha$ for the alpha particle.

Zachary W.

Numerade Educator

A particle passes through a mass spectrometer as illustrated in Figure P19.15. The electric field between the plates of the velocity selector has a magnitude of 8 250 V/m, and the magnetic fields in both the velocity selector and the deflection chamber have magnitudes of 0.093 1 T. In the deflection chamber the particle strikes a photographic plate 39.6 cm removed from its exit point after traveling in a semicircle.

(a) What is the mass- to- charge ratio of the particle? (b) What is the mass of the particle if it is doubly ionized? (c) What is its identity, assuming it’s an element?

Zachary W.

Numerade Educator

In Figure P19.2, assume 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, find the direction of the magnetic force acting on the wire.

Zachary W.

Numerade Educator

A current I 5 15 A is directed along the positive x - axis and perpendicular to a magnetic field. 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 field in the region through which the current passes.

Zachary W.

Numerade Educator

A straight wire carrying a 3.0- A current is placed in a uniform magnetic field of magnitude 0.28 T directed perpendicular to the wire. (a) Find the magnitude of the magnetic force on a section of the wire having a length of 14 cm. (b) Explain why you can’t determine the direction of the magnetic force from the information given in the problem.

Zachary W.

Numerade Educator

In Figure P19.3, assume 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, find the direction of the magnetic field that will produce the magnetic force shown.

Zachary W.

Numerade Educator

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 field needed to lift this wire vertically upward?

Zachary W.

Numerade Educator

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 field of strength 0.300 T. Find the magnetic force on a 5.00- m length of the wire.

Zachary W.

Numerade Educator

At a certain location, Earth has a magnetic field of $0.60 \times$ $10^{-4} \mathrm{T},$ pointing $75^{\circ}$ below the horizontal in a north-south 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?

Zachary W.

Numerade Educator

A wire with a mass of 1.00 g/cm is placed on a horizontal surface with a coefficient 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 field that enables the wire to move in this fashion

Zachary W.

Numerade Educator

Mass m 5 1.00 kg is suspended vertically at rest by an insulating string connected to a circuit partially

immersed in a magnetic field as in Figure P19.30. The magnetic field has magnitude $B_{\text { in }}=2.00 \mathrm{T}$ and the length $\ell=0.500 \mathrm{m}$ .

(a) Find the current $I .$

(b) If $\boldsymbol{E}=115 \mathrm{V},$ find the required resistance $R$ .

Zachary W.

Numerade Educator

Consider the system pictured in Figure P19.31. A 15- cm length of conductor of mass 15 g, free to move vertically, is placed between two thin, vertical conductors, and a uniform magnetic field acts perpendicular to the page. When a 5.0- A current is directed as shown in the figure, the horizontal wire moves upward at constant velocity in the presence of gravity. (a) What forces act on the horizontal wire, and under what condition is the wire able to move upward at constant velocity? (b) Find the magnitude and direction of the minimum magnetic field required to move the wire at constant speed. (c) What happens if the magnetic field exceeds this minimum value? (The wire slides without friction on the two vertical conductors.)

Nicole S.

Numerade Educator

A metal rod of mass $m$ carrying a current I glides on two horizontal rails a distance $d$ apart. If the coefficient of kinetic friction between the rod and rails is $\mu_{k},$ what vertical magnetic field is required to keep the rod moving at a constant speed?

Zachary W.

Numerade Educator

In Figure P19.33, the cube is 40.0 cm on each edge. Four straight segments of wire $-a b, b c,$

$c d,$ and $d a-$ form a closed loop that carries a current $I=5.00 \mathrm{A}$ in the direction shown. A uniform magnetic field of magnitude B 5 0.020 0 T is in the positive y- direction. Determine the magnitude and direction of the magnetic force on each segment.

Zachary W.

Numerade Educator

A horizontal power line of length 58 m carries a current of 2.2 kA as shown in Figure P19.34. Earth’s magnetic field at this location has a magnitude equal to $5.0 \times 10^{-5} \mathrm{T}$ and makes

an angle of $65^{\circ}$ with the power line. Find the magnitude and direction of the magnetic force on the power line.

Zachary W.

Numerade Educator

A wire is formed into a circle having a diameter of 10.0 cm and is placed in a uniform magnetic field of 3.00 mT. The wire carries a current of 5.00 A. Find the maximum torque on the wire.

Zachary W.

Numerade Educator

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

Zachary W.

Numerade Educator

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. P19.37). The coil lies in the plane of the page and carries a clockwise current of 6.00 A. If the coil is in a uniform magnetic field of $2.00 \times$ $10^{-4}$ 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 $A=\pi a b,$ where $a$ and $b$ are, respectively, the semimajor and semiminor axes of the ellipse.

Zachary W.

Numerade Educator

A current-carrying rectangular wire loop with width $a=0.120 \mathrm{m}$ and length $b=0.200 \mathrm{m}$ is in the $x y$ -plane, supported by a non- conducting, frictionless axle of negligible weight. A current of $I=3.00 \mathrm{A}$ travels counterclockwise in the circuit (Fig. $\mathrm{P} 19.38 )$ Calculate the magnitude and direction of the force exerted on the (a) left and (b) right segments of wire by a uniform magnetic field of 0.250 T that points in the positive x - direction. Find the magnetic force exerted on the (c) top and (d) bottom segments. (e) Find the magnitude of the net torque on the loop about the axle.

Zachary W.

Numerade Educator

A 6.00 -turn circular coil of wire is centered on the origin in the $x y$ -plane. The coil has radius $r=0.200 \mathrm{m}$ and carries a counterclockwise current $I=1.60 \mathrm{A}$ (Fig. $\mathrm{P} 19.39 ) .$ (a) Calculate the magnitude of the coil’s magnetic moment. (b) Find the magnitude of the magnetic torque on the coil due to a 0.200 - T magnetic field that is directed at an angle $\theta=60.0^{\circ}$ from the positive $z$ -direction and has components only in the $x z$ -plane.

Zachary W.

Numerade Educator

The orientation of small satellites is often controlled using torque from current - carrying coils in Earth’s magnetic field. Suppose a multiturn coil has a cross-sectional area of $6.36 \times$

$10^{-4} \mathrm{m}^{2},$ dissipates 0.200 $\mathrm{W}$ of electrical power from a $5.00-\mathrm{V}$ power supply, and provides a magnetic moment of magnitude 0.0200 $\mathrm{A} \cdot \mathrm{m}^{2} .(\mathrm{a})$ Find the coil current $I$ (b) Calculate the number of turns in the coil. (c) Calculate the maximum magnitude of torque if Earth’s magnetic field has magnitude $3.75 \times 10^{-5} \mathrm{T}$ at the satellite's location.

Zachary W.

Numerade Educator

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 field 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.

Zachary W.

Numerade Educator

A rectangular loop has dimensions 0.500 m by 0.300 m. The loop is hinged along the x - axis and lies in the xy - plane (Fig. P19.42). A uniform magnetic field of 1.50 T is directed at an angle of 40.0° with respect to the positive y - axis and lies parallel everywhere to the yz - plane. The loop carries a current of 0.900 A in the direction shown. (Ignore gravitation.) (a) In what direction is magnetic force exerted on wire segment ab? What is the direction of the magnetic torque associated with this force, as computed with respect to the x - axis? (b) What is the direction of the magnetic force exerted on segment cd ? What is the direction of the magnetic torque associated with this force, again computed with respect to the x- axis? (c) Can the forces examined in parts (a) and (b) combine to cause the loop to rotate around the x- axis? Can they affect the motion of the loop in any way? Explain. (d) What is the direction (in the yz - plane) of the magnetic force exerted on segment bc ? Measuring torques with respect to the x - axis, what is the direction of the torque exerted by the force on segment bc ? (e) Looking toward the origin along the positive x- axis, will the loop rotate clockwise or counterclockwise? (f ) Compute the magnitude of the magnetic moment of the loop. (g) What is the angle between the magnetic moment vector and the magnetic field? (h) Compute the torque on the loop using the values found for the magnetic moment and magnetic field.

Zachary W.

Numerade Educator

A lightning bolt may carry a current of $1.00 \times 10^{4}$ A for a short time. What is the resulting magnetic field $1.00 \times 10^{2} \mathrm{m}$ from the bolt? Suppose the bolt extends far above and below the point of observation.

Zachary W.

Numerade Educator

A long, straight wire going through the origin is carrying a current of 3.00 A in the positive z - direction (Fig. P19.44). At a point a distance r 5 1.20 m from the origin on the positive x - axis, find the (a) magnitude and (b) direction of the magnetic field. At a point the same distance from the origin on the negative y - axis, find the (c) magnitude and (d) direction of the magnetic field.

Zachary W.

Numerade Educator

Neurons in our bodies carry weak currents that produce detectable magnetic fields. A technique called magnetoencephalography, or MEG, is used to study electrical activity

in the brain using this concept. This technique is capable of detecting magnetic fields as weak as $1.0 \times 10^{-15} \mathrm{T}$ . Model the neuron as a long wire carrying a current and find the current

it must carry to produce a field of this magnitude at a distance of 4.0 cm from the neuron.

Zachary W.

Numerade Educator

In 1962 measurements of the magnetic field of a large tornado were made at the Geophysical Observatory in Tulsa, Oklahoma. If the magnitude of the tornado's field was $B=1.50 \times 10^{-8} \mathrm{T}$ 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.

Zachary W.

Numerade Educator

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

Zachary W.

Numerade Educator

The two wires shown in Figure $\mathrm{P} 19.48$ are separated by $d=$ 10.0 $\mathrm{cm}$ and carry currents of $I=5.00 \mathrm{A}$ in opposite directions. Find the magnitude and direction of the net magnetic field (a) at a point midway between the wires; (b) at point $P_{1}$ , 10.0 $\mathrm{cm}$ to the right of the wire on the right; and $(\mathrm{c})$ at point $P_{2}, 2 d=20.0 \mathrm{cm}$ to the left of the wire on the left.

Zachary W.

Numerade Educator

Four long, parallel conductors carry equal currents of $I=5.00 \mathrm{A}$ Figure P19.49 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 field at point P, located at the center of the square with edge of length 0.200 m.

Zachary W.

Numerade Educator

Two long, parallel wires carry currents of $I_{1}=3.00 \mathrm{A}$ and $I_{2}=5.00 \mathrm{A}$ in the direction indicated in Figure $\mathrm{P} 19.50$ . (a) Find the magnitude and direction of the magnetic field at a point midway between the wires $(d=20.0 \mathrm{cm}) .$ (b) Find the magnitude and direction of the magnetic field at point $P$ , located $d=20.0 \mathrm{cm}$ above the wire carrying the 5.00 -A current.

Zachary W.

Numerade Educator

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 P19.51. What is the magnetic field atpoint P, located at x 5 4.00 m, y 5 3.00 m?

Zachary W.

Numerade Educator

The magnetic field 40.0 $\mathrm{cm}$ away from a long, straight wire carrying current 2.00 $\mathrm{A}$ is 1.00$\mu \mathrm{T} .$ (a) At what distance is it 0.100$\mu \mathrm{T} ?$ At one instant, the two conductors in a long household extension cord carry equal $2.00-\mathrm{A}$ currents in opposite directions. The two wires are 3.00 $\mathrm{mm}$ apart. Find the magnetic field 40.0 $\mathrm{cm}$ 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 field does the cable create at points outside?

Zachary W.

Numerade Educator

A long, straight wire lies on a horizontal table in the $x y$ -plane and carries a current of 1.20$\mu \mathrm{A}$ in the positive $x$ -direction along the $x$ -axis. A proton is traveling in the negative $x$ -direction at speed $2.30 \times 10^{4} \mathrm{m} / \mathrm{s}$ a distance $d$ above the wire (i.e., $z=d ) .$ What is the direction of the magnetic field of the wire at the position of the proton? (b) What

is the direction of the magnetic force acting on the proton? (c) Explain why the direction of the proton’s motion doesn’t change. (d) Using Newton’s second law, find a symbolic expression for d in terms of the acceleration of gravity g, the proton mass $m,$ its speed $v,$ charge $q,$ and the current $I$ .

(e) Find the numeric answer for the distance $d$ using the results of part (d).

Zachary W.

Numerade Educator

Two long, parallel wires separated by a distance 2d carry equal currents in the same direction. An end view of the two wires is shown in Figure P19.54, where the currents are out of the page. magnetic field at $P$ on the $x$ -axis set up by the two wires? (b) Find an expression for the magnitude of the(a) What is the direction of the field at P. (c) From your result to part (b), determine the field at a point midway between the two wires. Does your result meet with your expectation? Explain.

Zachary W.

Numerade Educator

Two long, parallel wires separated by 2.50 cm carry currents in opposite directions. The current in one wire is 1.25 A, and the current in the other is 3.50 A. (a) Find the magnitude of the force per unit length that one wire exerts on the other. (b) Is the force attractive or repulsive?

Zachary W.

Numerade Educator

Two parallel wires separated by 4.0 cm repel each other with a force per unit length of $2.0 \times 10^{-4} \mathrm{N} / \mathrm{m}$ . The current in one wire is 5.0 $\mathrm{A}$ . ( a ) Find the current in the other wire. (b) Are the currents in the same direction or in opposite directions? (c) What would happen if the direction of one current were reversed and doubled?

Zachary W.

Numerade Educator

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.

Zachary W.

Numerade Educator

In Figure $P 19.58$ the current in the long, straight wire is $I_{1}=5.00 \mathrm{A}$ , and the wire lies in the plane of the rectangular loop, which carries 10.0 A. The dimensions shown are $c=0.100 \mathrm{m}, a=0.150 \mathrm{m},$ and $\ell=0.450 \mathrm{m} .$ Find the magnitude and direction of the net force exerted by the magnetic field due to the straight wire on the loop.

Zachary W.

Numerade Educator

A long solenoid that has $1.00 \times 10^{3}$ turns uniformly distributed over a length of 0.400 $\mathrm{m}$ produces a magnetic field of magnitude $1.00 \times 10^{-4} \mathrm{T}$ at its center. What current is required in the windings for that to occur?

Zachary W.

Numerade Educator

A certain superconducting magnet in the form of a solenoid of length 0.50 m can generate a magnetic field of 9.0 T in its core when its coils carry a current of 75 A. The windings, made of a niobium–titanium alloy, must be cooled to 4.2 K. Find the number of turns in the solenoid.

Zachary W.

Numerade Educator

It is desired to construct a solenoid that will have a resistance of 5.00$\Omega\left(\text { at } 20^{\circ} \mathrm{C}\right)$ and produce a magnetic field of $4.00 \times$ $10^{-2} \mathrm{T}$ at its center when it carries a current of 4.00 $\mathrm{A}$ . 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.

Zachary W.

Numerade Educator

Certain experiments must be performed in the absence of any magnetic fields. Suppose such an experiment is located at the center of a large solenoid oriented so that a current of $I=1.00 \mathrm{A}$ produces a magnetic field that exactly cancels Earth's $3.50 \times 10^{-5}$ T magnetic field. Find the solenoid's number of turns per meter.

Zachary W.

Numerade Educator

An electron is moving at a speed of $1.0 \times 10^{4} \mathrm{m} / \mathrm{s}$ in a circular path of radius 2.0 cm inside a solenoid. The magnetic field of the solenoid is perpendicular to the plane of the electron’s path. Find (a) the strength of the magnetic field inside the solenoid and (b) the current in the solenoid if it has 25 turns per centimeter.

Zachary W.

Numerade Educator

Figure P19.64 is a setup that can be used to measure magnetic fields. A rectangular coil of wire contains N turns and has a width w. The coil is attached to one arm of a balance and is suspended between the poles of a magnet. The field is uniform and perpendicular to the plane of the coil. The system is first balanced when the current in the coil is zero. When the switch is closed and the coil carries a current I, a mass m must be added to the right side to balance the system. (a) Find an expression for the magnitude of the magnetic field and determine its direction. (b) Why is the result independent of the vertical dimension of the coil? (c) Suppose the coil has 50 turns and width of 5.0 cm. When the switch is closed, the coil carries a current of 0.30 A, and a mass of 20.0 g must be added to the right side to balance the system. What is the magnitude of the magnetic field?

Zachary W.

Numerade Educator

Two coplanar and concentric circular loops of wire carry currents of $I_{1}=5.00 \mathrm{A}$ and $I_{2}=3.00 \mathrm{A}$ in opposite directions as in Figure $\mathrm{P} 19.65 .$ (a) If $r_{1}=12.0 \mathrm{cm}$ and $r_{2}=9.00 \mathrm{cm},$ what are $(\mathrm{a})$ the magnitude and (b) the direction of the net magnetic field at the center of the two loops? $(\mathrm{c})$ Let $r_{1}$ remain fixed at 12.0 $\mathrm{cm}$ and let $r_{2}$ be a variable. Determine the value of $r_{2}$ such that the net field at the center of the loop is zero.

Zachary W.

Numerade Educator

An electron moves in a circular path perpendicular to a constant magnetic field of magnitude 1.00 mT. The angular momentum of the electron about the center of the circle is $4.00 \times 10^{-25} \mathrm{kg} \cdot \mathrm{m}^{2} / \mathrm{s} .$ Determine (a) the radius of the circular path and $(\mathrm{b})$ the speed of the electron.

Zachary W.

Numerade Educator

Two long, straight wires cross each other at right angles, as shown in Figure P19.67. (a) Find the direction and magnitude of the magnetic field at point P, which is in the same plane as the two wires. (b) Find the magnetic field at a point 30.0 cm above the point of intersection (30.0 cm out of the page, toward you).

Zachary W.

Numerade Educator

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 field is required to keep the rod moving at a constant speed if the coefficient of kinetic friction between the rod and rails is 0.100?

Zachary W.

Numerade Educator

Using an electromagnetic flowmeter (Fig. P19.69), a heart surgeon monitors the flow 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 field magnitude of 0.040 0 T, a potential difference of 160$\mu \mathrm{V}$ 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.

Zachary W.

Numerade Educator

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 field with a strength of $4.0 \times 10^{-3}$ T. The field 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 0.50 m? Neglect the magnetic field of Earth.

Zachary W.

Numerade Educator

Three long, parallel conductors carry currents of $I=2.0 \mathrm{A}$ . Figure $\mathrm{P} 19.71 \mathrm{is}$ an end view of the conductors, with each current coming out of the page. Given that $a=1.0 \mathrm{cm},$ determine the magnitude and direction of the magnetic field at

points $A, B,$ and $C .$

Zachary W.

Numerade Educator

Two long, parallel wires, each with a mass per unit length of 0.040 kg/m, are supported in a horizontal plane by 6.0- cm- long strings, as shown in Figure P19.72. Each wire carries the same current $I$ , causing the wires to repel each other so that the angle $\theta$ between the supporting strings is $16^{\circ} .$ (a) Are the currents in the same or opposite directions? (b) Determine the magnitude of each current.

Zachary W.

Numerade Educator

Protons having a kinetic energy of 5.00 MeV are moving in the positive $x$ -direction and enter a magnetic field of 0.0500 $\mathrm{T}$ in the $z$ -direction, out of the plane of the page, and extending from $x=0$ to $x=1.00 \mathrm{m}$ as in Figure $\mathrm{P} 19.73$ . (a) Calculate the $y$ -component of the protons' momentum as they leave the magnetic field. (b) Find the angle $\alpha$ between the initial velocity vector of the proton beam and the velocity vector after the beam emerges from the field. Hint: Neglect relativistic effects and note that $1 \mathrm{eV}=1.60 \times 10^{-19} \mathrm{J}$ .

Zachary W.

Numerade Educator

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. P19.74). The springs stretch a distance of 0.50 cm

under the weight of the wire. The circuit has a total resistance of 12$\Omega .$ When a magnetic field directed out of the page (indicated by the dots in the figure) is turned on, the springs are observed to stretch an additional 0.30 cm. What is the strength of the magnetic field? (The upper portion of the circuit is fixed.)

Zachary W.

Numerade Educator

A 1.00 -kg ball having net charge $Q=5.00 \mu \mathrm{C}$ is thrown out of a window horizontally at a speed $v=20.0 \mathrm{m} / \mathrm{s}$ . The window is at a height $h=20.0 \mathrm{m}$ above the ground. A uniform horizontal magnetic field of magnitude $B=0.0100 \mathrm{T}$ is perpendicular to the plane of the ball's trajectory. Find the magnitude of the magnetic force acting on the ball just before it hits the ground. Hint: Ignore magnetic forces in finding the ball's final velocity.

Zachary W.

Numerade Educator

Two long, parallel conductors separated by 10.0 $\mathrm{cm}$ carry currents in the same direction. The first wire carries a current $I_{1}=5.00 \mathrm{A}$ , and the second carries $I_{2}=8.00 \mathrm{A}$ . (a) What is the magnitude of the magnetic field created by $I_{1}$ at the location of $I_{2} ?$ (b) What is the force per unit length exerted by $I_{1}$ on $I_{2}$ ? (c) What is the magnitude of the magnetic field created by $I_{2}$ at the location of $I_{1} ?$ (d) What is the force per length exerted by $I_{2}$ on $I_{1} ?$

Zachary W.

Numerade Educator