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All Textbook Solutions for University Physics Volume 2

The Hall effect is to be used to find the density of charge carriers in an unknown material. A Hall voltage 40 V for 3-A current is observed in a 3-T magnetic field far a rectangular sample with length 2 cm, width 1.5 cm, and height 0.4 cm, Determine the density of the charge carriers.Show tliat the Hall voltage across wires made of the same material, carrying identical currents, and subjected to the same magnetic field is inversely proportional to their diameters. (Hint: Consider haw drift velocity depends on wire diameter.)A velocity selector in a mass spectrometer uses a 0.100-T magnetic field, (a) What electric field strength is needed to select a speed of 4.0106m/s fb) What is the voltage between the plates if they are separated by 1.00 cm?Find the radius of curvature of the path of a 25.0-MeV proton moving peipendicularly to the 1,20-T field of a cyclotron.Unreasonable results To construct a non-mechanical water meter, a 0.500-T magnetic field is placed across the supply water pipe to a home and the Hall voltage is recorded, (a) Find the flow rate through a 3.00-cm-diameter pipe if the Hall voltage is 60.0 mV. (b) What would the Hail voltage be for the same flow rate through a 10,0-cm- diameter pipe with tire same field applied?Unreasonable results A charged particle having mass 6.641027kg (that of a helium atom) moving at 8.70105m/s perpendicular to a 1.50-T magnetic field travels in a circular path of radius 16.0 mm. (a) What is the charge of the particle? (b) What is unreasonable about this result? (c) Which assumptions are responsible?Unreasonable results An inventor wants to generate 120-V power by moving a 1.00-m-long wire perpendicular to Earth’s 5.00105T field. (a) Find the speed with which the wire must move. (b) What Is unreasonable about this result? (c) Which assumption is responsible?Unreasonable results Frustrated by the small Hall voltage obtained in blood flow measurements, a medical physicist decides to increase the applied magnetic field strength to get a 0.500-V output for blood moving at 30.0 cm/s in a 1.50-cm-diameter vessel. (a) What magnetic field strength is needed? (b) What is unreasonable about this result? (C) Which premise is responsible?A particle of charge +q and mass m moves with velocity v0 pointed In the +y-direction as It crosses the x-axis at x= R at a particular time. There is a negative charge -Q fixed at the origin, and there exists a uniform magnetic field B0 pointed in the +z-direction. It is found that the particle describes a circle of radius R about -Q. Find B0 in terms of the given quantities.A proton of speed v=6105m/s enters a region of uniform magnetic field of B = 0.5 T at an angle of q = 30° to the magnetic field. In the region of magnetic field proton describes a helical path with radius R and pitch p (distance between loops.) Find R and p.A particle’s path is bent when it passes through a region of non-zero magnetic field although its speed remains unchanged. This is very useful for “beam steering’’ in particle accelerators. Consider a proton of speed 4106m/s entering a region of uniform magnetic field 0.2 T over a 5-cm-wide region. Magnetic field is perpendicular to the velocity of the particle. By how much angle will the path of the proton be bent? (Hint: the particle comes out tangent to a circle.In a region a non-uniform magnetic field exists such that Bx=0,By=0,andBz=ax where a is a Constant. At some time t, a wire of length L is carrying a current I is located along the x-axis from origin to x = L. Find the magnetic force on the wine at this instant in time.A copper rod of mass in and length L is hung from the ceiling using two springs of spring constant ft. A uniform magnetic field of magnitude pointing perpendicular to the rod and spring [coming out of tire page in tire figure) exists in a region of space covering a length w of the copper rod. The ends of the rod are then connected by flexible copper wire across the terminals of a battery of voltage V. Determine the change in the length of the springs when a current I runs through the copper rod in the direction shown in figure, (Ignore any force by the flexible wire.)The accompanied figure shows an arrangement for measuring mass of ions by an instrument called the mass spectrometer. An ion of mass m and charge +q is produced essentially at rest in source S, a chamber in which a gas discharge is taking place. The ion is accelerated by a potential difference and allowed to enter a region of constant magnetic field B q. In the uniform magnetic field region, the ion moves in a semicircular path striking a photographic plate at a distance x from the entry point. Derive a formula for mass m in terms of B0, q, Vacc, and x.A wire ismade into a circular shape of radius R and pivoted along a central support.The two ends of the sire are touching a banish that is connected to a &power source. The stricture is between the poles of a magnet such that we can assume there is a uniform magnetic field on the wire. In terms of a coordinate system with origin at the center ofthe ring, magneticfieldisBx=B0,By=Bz= 0. and the ring rotates about the z-axis. Find the torque on the ring siren it is not in the xz-plane.A long-rigid wire lies along the x-axis and cairns a current of 2.5 A in the positive x-direction. Around the wire is the magnetic field B=2.0i+5.0x2j, With x in meters and B in millitesla. Calculate the magnetic force on the segment of wire between x=2.0m and x=4.0m .A circular loop of wire of area 10 cm2 carries a current of 25 A. At a particular instant, the loop lies in the xy-plane and is subjected to a magnetic field B=(2.0i+6.0j+8.0k)103T . As viewed from above the xy-plane, the current is circulating clockwise. (a) What is the magnetic dipole moment of the current loop? (b) At this instant, what is the magnetic torque on the loop?Check Your Understanding Using Example 12.1, at what distance would P have to be to measure a magnetic field half of tire given answer?Check Your Understanding The wire loop forms a full circle of radius R and current I. What is the magnitude of the magnetic field at the center?12.3 Check Your Understanding Using Example 12.3, keeping the currents the same in wires 1 and 3, what should the current be in wire 2 to counteract the magnetic fields horn wires 1 and 3 so that there is no net magnetic field at point P?12.4 Check Your Understanding Two wires, both carrying current out of the page, have a current of magnitude 2.0 mA and 3.0 mA, respectively. The first wire is located at (0.0 cm, 5.0 cm) while the other wire is located at (12.0 cm, 0.0 cm). What is the magnitude of the magnetic force per unit length of the first wire on the second and the second wire on the first?Check Your Understanding Using Example 12.5, at what distance would you have to move the first coil to have zero measurable magnetic field at point P?Check Your Understanding Consider using Ampere’s law to calculate the magnetic fields of a finite straight wire and of a circular loop of wire. Why is it not useful for these calculation?12.7 Check Your Understanding What is the ratio of the magnetic field produced from using a finite formula over the infinite approximation for an angle of (a) 85°? (b) 89°? The solenoid has 1000 mins in 50 cm with a current of 1.0 A flowing through the coils Check your Understanding Repeat the calculations from the previous example for I0=0.040A .For calculating magnetic fields, what are the advantages and disadvantages of the Biot-Savart law?Describe the magnetic field due to the current in two wires connected to tire two terminals of a source of emf and twisted tightly around each other.How can you decide if a wire is infinite?Identical currents are carried in two circular loops; however, one loop has twice the diameter as the other loop. Compare the magnetic fields created by the loops at the center of each loop,How would you orient two long, straight, current carrying wires so that there is no net magnetic force between them? (Hint: What orientation would lead to one wire not experiencing a magnetic field from the other?)Compare and contrast the electric field of an infinite line of charge and the magnetic field of an infinite line of current.Is B constant in magnitude for points that lie on a magnetic field line?Is the magnetic field of a current loop uniform?What happens to the length of a suspended spring when a current passes through it?Two concentric circular wines with different diameters carry currents in tire same direction. Describe the force cm the inner wire.Is Ampere’s law valid for all closed paths? Why isn’t it normally useful for calculating a magnetic field?Is the magnetic field inside a toroid completely uniform? Almost uniform?Explain why B=0 inside a long, hollow copper pipe that is carrying an electric current parallel to the axis. Is B=0 outside the pipe?A diamagnetic material is brought dose to a permanent magnet. What happens to the material?If you cut a bar magnet into two pieces, will you end up with one magnet with an isolated north pole and another magnet with an isolated south pole? Explain your answer.A 10-A current flows through the wire shown. What is the magnitude of the magnetic field due to a 0.5-mm segment of wire as measured at (a) point A and (b) point B?Ten amps flow through a square loop where each side is 20 cm in length. At each comer of the loop is a 0.01-cm segment that connects the longer wires as shown. Calculate the magnitude of the magnetic field at the center of the loop.What is the magnetic field at P due to the current I in the wire shown?The accompanying figure shows a current loop consisting of two concentric circular arcs and two perpendicular radial lines. Determine the magnetic field at point P.Find the magnetic field at the center C of the rectangular loop of wire shown in the accompanying figure.Two long wires, one of which has a semicircular tend of radius R, are positioned as shown in the accompanying figure. If both wires carry a current I, how far apart must then parallel sections be so that the net magnetic field at P is zero? Does the current in the straight wire flow up or down?A typical currant in a lightning bolt is 104 A. Estimate the magnetic field 1 m from the bolt.The magnitude of the magnetic field 50 cm from a long, thin, straight wire is 8.0T . What is the current through the long wire?A transmission line strung 7.0 m above the ground carries a current of 500 A. What is the magnetic field on the ground directly below the wire? Compare your' answer with the magnetic field of Earth.A long, straight, horizontal wire carries a left-to-right current of 20 A. If the wire is placed in a uniform magnetic field of magnitude 4.0105 T that is directed vertically downward, what is tire resultant magnitude of the magnetic field 20 cm above the wire? 20 cm below the wire?The two long, parallel wires shown in the accompanying figure carry currents in the same direction. If I1= 10 A and I2= 20 A, what is tire magnetic field at point P?The accompanying figure shows two long, straight, horizontal wires that are parallel and a distance 2a apart. If both wires carry current I in the same direction, (a) what is the magnetic field at P1? (b) P2?Repeat the calculations of the preceding problem with the direction of the current in the lower wire reversed.Consider the area between the wires of the preceding problem. At what distance from tire top wire is the net magnetic field a minimum? Assume that the currents are equal and flow in opposite directions.Two long, straight wires are parallel and 25 cm apart. (a) If each wire carries a current of 50 A in the same direction, what is the magnetic force per meter exerted on each wire? (b) Does tire force pull the wires together or push them apart? (c) What happens if the currents flow in opposite directions?Two long, straight wires are parallel and 10 cm apart. One cans a current of 2.0 A, the other a current of 5.0 A. (a) If the two currents flow in opposite directions, what is the magnitude and direction of the force pet unit length of one wire on the other? (b) What is the magnitude and direction of the force per unit length if the currents flow in the same direction?Two long, parallel wires are hung by cords of length 5.0 cm, as shown in the accompanying figure. Each wire has a mass per unit length of 30 g/m, and they carry the same current in opposite directions. What is the current if the cords hang at 6.0° with respect to the vertical?A circuit with current I has two long parallel wire sections that carry current in opposite directions. Find magnetic field at a point P near these wires that is a distance a from one wire and b from the other wire as shown in the figure.The infinite, straight wire shown in the accompanying figure cans a current I1. The rectangular loop, whose long sides are parallel to the wire, carries a current I2. What are the magnitude and direction of the force on the rectangular loop due to the magnetic field of the wire?When the current through a circular loop is 6.0 A, the magnetic field at its center is 2.0104 T. What is the radius of the loop?How many turns must be wound on a flat, circular coil of radius 20 cm in order to produce a magnetic field of magnitude 4.0105 T at the center of the coil when the current through it is 0.85 A?A flat, circular loop has 20 turns. The radius of the loop is 10.0 cm and the current through the wire is 0.50 A. Determine the magnitude of the magnetic field at the center of the loop.A circular loop of radius R carries a current I. At what distance along the axis of the loop is the magnetic field one- half its value at the center of the loop?Two flat, circular coils, each with a radius R and wound with JV turns, ace mounted along the same axis so that they are parallel a distance d apart. What is the magnetic field at the midpoint of the common axis if a current I flows in the same direction through each coil?For the coils in the preceding problem, what is the magnetic field at the center of either coil?A current 1 flows around the rectangular loop shown in the accompanying figure. Evaluate BdI for the paths A, B, C, and D.Evaluate BdI for each of the cases shown in the accompanying figure.The coil whose lengthwise cross section is shown in the accompanying figure carries a currents I and has N evenly spaced turns distributed along the length I. Evaluate BdI for the paths indicated.A superconducting wire of diameter 0.25 cm carries a current of 1000 A. What is the magnetic field just outside the wire?A long, straight wire of radius R caries a current I that is distributed uniformly over the cross-section of the wire. At what distance from the axis of the wire is the magnitude of the magnetic field a maximum?The accompanying figure shows a cross-section of a long, hollow, cylindrical conductor of inner radius r1= 3.0 cm and outer radius r2= 5.0 cm. A 50-A current distributed uniformly over the cross-section flows into the page. Calculate the magnetic field at r = 2.0 cm. r = 4.0 cm. and r = 6.0 cm.A long, solid, cylindrical conductor of radius 3.0 cm carries a current of 50 A distributed uniformly over its cross-section. Plot the magnetic field as a function of the radial distance r from the center of the conductor.A portion of a long, cylindrical coaxial cable is shown in the accompanying figure. A current I flows down the center conductor, and this current is returned in the outer conductor. Determine the magnetic field in the regions (a)A solenoid is wound with 2000 turns pet meter. When the cuiient is 5.2 A, what is the magnetic field within the solenoid?A solenoid has 12 turns per centimeter. What current will produce a magnetic field of 2.0102 T within the solenoid?If a current is 2.0 A, bow many turns per centimeter must be wound on a solenoid in order to produce a magnetic field of 2.0103 within it?A solenoid is 40 cm long, has a diameter of 3.0 cm, and is wound with 500 turns. If the current through the windings is 4.0 A, what is the magnetic field at a point on the axis of the solenoid that is (a) at the center of the solenoid, (b) 10.0 cm from one end of the solenoid, and (c) 5.0 cm from one end of the solenoid? (d) Compare these answers with the infinite-solenoid case.Determine the magnetic field on the central axis at the opening of a semi-infinite solenoid. (That is, take the opening to be at x = 0 and the other end to be at x= .)By how much is the approximation B=0nI in error at the center of a solenoid that is 15.0 cm long, has a diameter of 4.0 cm, is wrapped with n turns per meter, and carries a current I?A solenoid with 25 turns per centimeter carries a current I. An electron moves within the solenoid in a circle that has a radius of 2.0 cm and is perpendicular to the axis of the solenoid. If the speed of the electron is 2.0105 , what is I?A toroid has 250 trims of wire and carries a current of 20 A. Its inner and outer radii are 8.0 and 9.0 cm. What are the values of its magnetic field at r = 8.1, 8.5, and 8.9 cm?A toroid with a square cross section 3.0cm3.0cm has an inner radius of 25.0 cm. It is wound with 500 turns of wire, and it carries a current of 2.0 A. What is the strength of the magnetic field at the center of the square cross section?The magnetic field in the core of an air-filled solenoid is 1.50 T, By how much will this magnetic field decrease if the ail is pumped out of the core while the current is held constant?A solenoid has a ferromagnetic core, n = 1000 turns per meter, and I = 5.0 A. If B inside the solenoid is 2.0 T, what is for the core material?A 20-A current flows through a solenoid with 2000 turns per meter. What is the magnetic field inside the solenoid if its core is (a) a vacuum and (b) filled with liquid oxygen at 90 K?The magnetic dipole moment of the iron atom is about 2.11023Am2 . (a) Calculate the maximum magnetic dipole moment of a domain consisting of 1019 iron atoms, (b) What current would have to flow through a single circular loop of wire of diameter 1.0 cm to produce this magnetic dipole moment?Suppose you wish to produce 1.2-T magnetic field in a toroid with an iron core for which =4.0103 . The toroid has a mean radius of 15 cm and is wound with 500 turns. What current is required?A current of 1.5 A flows through the windings of a large, thin toroid with 200 turns per meter. If the toroid is filled with iron for which =3.0103 , what is the magnetic field within it?A solenoid with an iron core is 25 cm long and is wrapped with 100 turns of wire. When the current through the solenoid is 10 A, the magnetic field inside it is 2.0 T. For this current, what is the permeability of the iron? If the current is turned off and then restored to 10 A, will the magnetic field necessarily return to 2.0 T?Three long, straight, parallel wires, all carrying 20 A, are positioned as shown in the accompanying figure. What is the magnitude of the magnetic field at the point P?A current I flows around a wire bent into the shape of a square of side a. What is tire magnetic field at the point P that is a distance z above the center of the square (see the accompanying figure)?The accompanying figure shows a long, straight wire carrying a current of 10 A. What is the magnetic force on an electron at the instant it Is 20 cm from tire wire, traveling parallel to the wire with a speed of 2.0105m/s ? Describe qualitatively the subsequent motion of the election, For this current, what is tire permeability of the iron? If the current is turned off and then restored to 10 A, will the magnetic field necessarily return to 2.0 T?Current flows along a thin, infinite sheet as shown in the accompanying figure. The current per unit length along the sheet is J in amperes per meter, (a) Use the Biot-Savart law to show that B=0J/2 on either side of the sheet. What is the direction of B on each side? (b) Now use Ampere’s law to calculate the field.(a) Use the result of the previous problem to calculate the magnetic field between, above, and below the pair of infinite sheets shown in the accompanying figure, (b) Repeat your calculations if the direction of the current in the lower sheet is reversed.We often assume that the magnetic field is uniform in a legion and zero everywhere else. Show that in reality' it is impossible for a magnetic field to drop abruptly to zero, as illustrated in the accompanying figure. (Hint; Apply Ampere's law over the path shown.)How is the percentage change in the strength of the magnetic field across the face of the toroid related to the percentage change in the radial distance from the axis of the toroid?Show that the expression for the magnetic field of a toroid reduces to tlrat for the field of an infinite solenoid in the limit that the cenmal radius goes to infinity.A toroid with an inner radius of 20 cm and an outer radius of 22 cm is tightly wound with one layer of wire that has a diameter of 0.25 mm. (a) How many turns are there on the toroid? (b) If the current through the toroid windings is 2.0 A, what is the strength of the magnetic field at the center of the toroid?A wire element has dI,IdI=JAdl=Jdv , where A and dv are the cross-sectional area and volume of the element, respectively. Use this, the Biot-Savart law, and J=nev to show that the magnetic field of a moving point charge q is given by: B=04qvrr2 A reasonably uniform magnetic field over a limited region of space can be produced with the Helmholtz coil, which consists of two parallel coils centered on the same axis. The coils are connected so that they carry the same current I. Each coil has N turns and radius R, which is also the distance between the coils, (a) Find tire magnetic field at any point on the z-axis shown in the accompanying figure, (b) Show that dB/dz and d2Bdz2 are both zero at z = 0. (These vanishing derivatives demonstrate that the magnetic field varies only slightly near z = 0.)A charge of 4.0C .s distributed uniformly around a thin ring of insulating material. The ring has a radius of 0.20 m and rotates at 2.0104 rev/min around die axis that passes through its center and is perpendicular to the plane of the ring. What is the magnetic field at the center of the ring?A thin, nonconducting disk of radius R is free to rotate around the axis that passes through its center and is perpendicular to the face of the disk. The disk is charged uniformly with a total charge q. If the disk rotates at a constant angular velocity , what is tire magnetic field at its center?Consider the disk in the previous problem. Calculate tlie magnetic field at a point on its central axis that is a distance y above tlie disk,Consider the axial magnetic field Bv=0IR2/2(y2+R2)3/2 of the circular current loop shown below. (a) Evaluate aaBydy Also show that limaaaBydy=0I (b) Can you deduce this limit without evaluating the integral? (Hint: See the accompanying figure.)The current density in the long, cylindrical wire shown in the accompanying figure varies with distance r from the center of the wire according to J = cr. where c is a constant (a) What is the current through the wire? (b) What is the magnetic field produced by this current for r < R? For rR ?A long, straight, cylindrical conductor contains a cylindrical cavity whose axis is displaced by n from the axis of the conductor, as shown in the accompanying figure. The current density in the conductor is given by J=J0k, where J0 is a constant and k is along the axis of the conductor. Calculate the magnetic field at an arbitrary point P in the cavity by superimposing the field of a solid cylindrical conductor with radius R1and current density Jonto the field of a solid cylindrical conductor with radius R2and current density J . Then use the fact that the appropriate azimuthal unit vectors can be expressed as 1=kr1and 2=kr2 to show that everywhere inside the cavity the magnetic field is given by the constant B=120J0ka , where a=r1r2 and r1=r1r1 is the position of P relative to the center of the conductor and r2=r2r2 is the position of P relative to the center of the cavity.Between the two ends of a horseshoe magnet the field is uniform as shown in the diagram. As you move out to outside edges, the field bends. Show by Ampere’s law that the field must bend and thereby the field weakens due to these bends. B-field Show that the magnetic field of a thin wire and that of a current loop are zero if you are infinitelv far away.An Ampere loop is chosen as shown by dashed lines for a parallel constant magnetic field as shown by solid arrows. Calculate BdI for each side of the loop then find the entire BdI . Can you think of an Ampere loop that would make the problem easier? Do those results match these?, A ray long, thick, cylindrical wire of radius R carries a current density J that varies across its cross-section, The magnitude of the current density7 at a point a distance r from the center of the wine is given by J=J0rR where J0 is a constant. Find the magnetic field (a) at a point outside the wire and (b) at a point inside the wire. Write your answer in terms of the net current I through the wire.A very long, cylindrical wire of radius a has a circular hole of radius b in it at a distance d from the center. The wire carries a uniform current of magnitude I through it. The direction of the current in the figure is out of the paper. Find the magnetic field (a) at a point at the edge of the hole closest to the center of the thick wire, (b) at an arbitrary point inside the hole, and (c) at an arbitrary point outside the wire. (Hint: Think of the hole as a sum of two wires carrying current in the opposite directions.)Magnetic field inside a torus. Consider a torus of rectangular cross-section with inner radius a and outer radius b. N turns of an insulated thin wire are wound evenly on the toms tightly all around the torus arid connected to a battery producing a steady current f in the wire. Assume that the current on the top and bottom surfaces in the figure is radial, and the current on the inner and outer radii surfaces is vertical. Find the magnetic field inside the toms as a function of radial distance r from the axis.Two long coaxial copper tubes, each of length L, are connected to a battery of voltage V. The inner tube has inner radius o and outer radius b, and the outer tube has inner radius c and outer radius d. The tubes are then disconnected from the battery and rotated in the same direction at angular speed of radians per second about their common axis. Find the magnetic field (a) at a point inside the space enclosed by the inner tube r < a, and fb) at a point between the tubes b < r < c, and (c) at a point outside the tubes r > d. (Hint: Hunk of copper tubes as a capacitor and find the charge density based on the voltage applied, Q=VC, C=20LIn(c/b) .)The accompanying figure shows a flat, infinitely long sheet of width a that carries a current I uniformly distributed across it. Find the magnetic field at the point P, which is in the plane of the sheet and at a distance x from one edge. Test your result for the limit a ? 0.A hypothetical current flowing in the z-direction creates the field B=C[(x/y2)i+(1/y)j] in the rectangular region of the xy-plane shown in the accompanying figure. Use Ampere’s law to find the current through the rectangle.A nonconducting hard rubber circular disk of radius R is painted with a uniform surface charge density tr. It is rotated about its axis with angular speed . (a) Find the magnetic field produced at a point on the axis a distance h meters from the center of the disk, (b) Find the numerical value of magnitude of the magnetic field when =1C/m2 , R = 20 cm, h = 2 cm, and magnetic field of Earth, which is about 1/2 Gauss. =400rad/sec , and compare it with the magnitude of Chek sour Understanding A closely und coil has a radius of 4.0 cm. 50 turns, and a total ieslstancc of 40 . At what rae must a magnetic field peipendicular lathe face of the coil change En euler to prodice Joule hearing in the cull a a tale of 2.0 mW Check ‘sour Und.rtanding Find the dhectlon of the hiduced cunent n the wire loop shoii below as the magnet enteis, passes though, and leaves the Loop.Check Your UnderstAnding Verify the directions of the induced cuffents in Figure Figure 133 (a)Qosig the switch of ciicuit 1 produces ashort4ived cuicenz surge in circuit2 (b) If the switch remains closed, no current is observed in cituit 1 (c) Opening the switch again produces a short-lived cuirent in circuit 2 but in the opposite direclion fiom before.Check Your Understanding Shown below is a rod of length l that is rotated counterclockwise around the axis through O by the torque due to mg . Assuming that the rod is in a uniform magnetic field B , what is the emf induced between the ends of the rod when its angular velocity is ? Which end of the rod is at a higher potential?Check Your Understanding A rod of length 10cm moves at aspeed of 10 m/sperpendicularly though a 1.5-T magneticfield. What is the potential difference betweenthe ends of the rod?Check Your understanding Suppose that the coil of Example 13.2 is a square rather than circular. Can Equation 13.12 be used to calculate (a) the induced emf and (b) the induced electric field? =EdI=dmdt13.12 Check Your Understanding What Is the magnitude of the induced electric field in Example 13.8 at t=0 if r= 6.0cm, R= 2.0cm,n = 2000 turnspermeter, I0= 2.0A, and a = 200s-1?Check your Understanding Themagneticfield shown below Is confined to the cylindrical region shown and is changing withtime. Identity those paths for which =EdI0 .Check Your Understanding A long solenoid of cross-section area 5.0 cm2is wound with 25 turns of wire pet centimeter. It is placed in the middle of a closely wrapped coil of 10 turns and radius 25 cm, as shown below. (a) What is the emfInduced In the coil when the current through the solenoid Is decreasing at a rate dl/dt= -0.20A/s? (b) What Is the electric field Induced In the coil?A stationary coil is in a magnetic field that is changing with time. Does the emf induced in the coil depend on the actual values of the magnetic field?In Faraday’s experiments, what would be the advantage of using coils with many turns?A copper ring and a wooden ring of the same dimensions are placed in magnetic fields so that there is the same change in magnetic flux through them. Compare the induced electric fields and coffins in the rings.Discuss the factors determining the induced emf in a :losed loop of wire.a. Does the induced emf in a circuit depend on the resistance of the circuit? (b) Does the induced current depend on the resistance of the circuit?How would changing the radius of loop D shown below affect its emf, assuming C and D are much closer together compared to their radii?Can there be an induced emf in a circuit at an instant when the magnetic flux through the circuit is zero?Does the induced emf always act to decrease the magnetic flux through a circuit?How would you position a flat loop of wire in a changing magnetic field so that there is no Induced emf in the loop?The normal to tt plane of a single-turn conducting loop is directed at an angle to a spatially uniform magnetic field B. It has a fixed area and orientation relative to the magnetic fleck Show that the emf induced In the loop is given by i = (dB/dt)(Acos),where A is the area of the loop.The circular conducting loops shown in the accompanying figure are parallel, perpendicular to the plane of the page, and coaxial. (a) When the switch S is dosed, what is the direction of the cun’ent induced in D? (b) When the switch is opened, what is the direction of the current induced in loop D?The north pole of a mag’iet is moved toward a copper loop, as shown below. If you are looking at the loop from above the magnet, will you say the induced convent is circulating c1ockise or counterclockwise?The accompanying figure shows a conducting ring at various positions as It moves througli a magnetic field. What is the sense of the induced em! for each of those positions?Show that and dm/dt have the same units.State the direction of the induced current for each case shown be1o obseMng from the sick of the magnet A bar magnet falls under the influence of gravity along the axis of a long copper tube. If air resistance is negligible, will there be a force to oppose the descent of the magnet? If so, will the magnet reach a terminal velocity?Around the geographic North Pole (or magnetic South Pole), Earth’s magnetic field is almost vertical. If an airplane is flying northward in this region, which side of the wing is positively charged and which is negatively charged?A wire loop moves translationally (no rotation) in a uniform magnetic field. Is there an emf induced in the loop?Is the work required to accelerate a rod from rest to a speed v in a magnetic field greater than the final kinetic energy of the rod? Why?The copper sheet shown below is partially in a magnetic field. When it is pulled to the right, a resisting force pulls it to the left. Explain. What happen if the sheet is pushed to the left?A conducting sheet lies in a plane perpendicular to a magnetic field B that is below the sheet. If B oscillates at a high frequency and the conductor is made of a material of low resistivity, the region above the sheet is effectively shielded from B . Explain why. Will the conductor shield this region from static magnetic fields?Electromagnetic braking can be achieved by applying a strong magnetic field to a pinning metal disk attached to a shaft. (a) How can a magnetic field slow the spinning of a disk? (b) Would the brakes work if the disk was made of plastic instead of metal?A coil is moved through a magnetic field as shown below. The field is uniform inside the rectangle and zero outside. What is the direction of the induced current and what is the direction of the magnetic force on the coil at each position shown?A 50-turn coil has a diameter of 15 cm. The coil is placed in a spatially uniform magnetic field of magnitude 0.50 T so that the face of the coil and the magnetic field are perpendicular. Find the magnitude of the emf induced in the coil if the magnetic field is reduced to zero uniformly in (a) 0.10 s, (b) 1.0 s, and (c) 60 s.Repeat your calculations of the preceding problem’s time of 0.1 s with the plane of the coil making an angle of (a) 30°, (b) 60°, and (c) 90° with the magnetic field.A square loop whose sides are 6.0-cm long is made with copper wire of radius 1.0 mm. If a magnetic field perpendicular to the loop is changing at a rate of 5.0 mT/s, what is the current in the loop?The magnetic field through a circular loop of radius 10.0 cm varies with time as shown below. The field is perpendicular to the loop. Plot the magnitude of the induced emf in the loop as a inaction of time.The accompanying figure shows a single-turn rectangular coil that has a resistance of 2. . The magnetic field at all points inside the coil varies according to B=B0eat,B0=0.25T and a = 200 Hz. What is the current induced in the coil at (a) t = 0.001 s, (b) 0.002 s, (c) 2.0 s?How would the answers to the preceding problem change if the coil consisted of 20 closely spaced turns&A long solenoid with n= 10 turns per centimeter has a cross-sectional area of 5.0 cm2and carries a current of 0.25 A. A coil with five turns encircles the solenoid. When the current through the solenoid is turned off, it decreases to zero in 0.050 s. What is the average emf induced in the coil?A rectangular wire loop with length a and width b lies in the xy-plane, as shownbelow. Within the loop there is a time-dependent magnetic field given by B(t)=C((xcost)i+(ysint)k) ,with B(t) intesla. Determine the emf induced in the loop as a function of time.The magnetic field perpendicular to a single sire loop of diameter 10.0 cm decreases fron 0.50 T to zero. The re Is made of copper and has a diameter of 2.0 mm and length 1.0 cm. How much charge moves thrnugh the re while tt field is changing?A single-turn circular loop of wire of radius 50 mm lies in a plane perpendicular to a spatially uniform magnetic field. During a 0.10-s time interval, the magnitude of the field Increases uniformly from 200 to 300 mT. (a) Determine the emf Induced in the ioop. (b) If the magnetic field is directed out of the page, what is the direction of the cun-ent induced in the loop?When a magnetic field is first turned on, t1 flux through a 20-turn loop varies with time according to m=5.0t22.0t ,where m is in milliwebers, tis in seconds, and the loop is in the plane of the page with the unit normal pointing outward. (a) What is the emf induced in the loop as a function of time? What is the direction of the induced current at (b) t= 0, (c) 0.10, (d) 1.0, and (e) 2.0 s?The magnetic flux through the loop shown in the accompanying figure varies with time according to m=2.00e3tsin(120t) , where m is in milliwebers. What are the direction and magnitude of the current through the 5.00 resistor at (a) t= 0; (b) t=2.17102 s and (c)t=3.00 s?Use Lenz’s law to determine tl direction of induced current in each case.An automobile with a radio antenna 1.0 m long travels at 100.0 km/h in a location where theEarth’s horizontal magnetic field is 5.5105T . What is the maximum possible emf induced in the antenna due to this motion?38P Suppose the magnetic field of the preceding problem oscillates with time according to B=B0sint . What then isthe emf Induced in the loop when its trailing side Is a distance d from the right edge of the magnetic field region?A coil of 1000 turns encloses an area of 25 cm2. It is rotated in 0.010 s from a position where its plane is perpendicular to Earth’s magnetic field to one where its plane is parallel to the field. If the strength of the field is 6.0 x I0 T. what is the average emf induced in the coil?In the circuit sho in the accompanying figure, the rtd slides along the conducting rails at a constant VelOCity v. The velocity is in the same plane as the rails and directed at an angle 6 to them. A uniform magnetic field Bis directed out of the page. What is the emf induced in the r?The rod shown in the accompanying figure is moving through a uniform magnetic field of strength B=0.50T with a constant velocity of magnitude v=8.0m/s . What is the potential difference between the ends of the rod? Which end of the rod is at a higher potential? B A 25-cm nod moves at 5.0 m/s in a plane perpendicular to a magnetic field of strength 0.25 T. The rod, velocity vector, and magnetic field vector are mutually perpendicular, as indicated in tire accompanying figure. Calculate (a) the magnetic force on an electron in tire rod, (b) the electric field in the rod, and (c) the potential difference between the ends of the rod. (d) What is the speed of the rod if the potential difference is 1.0 V?In the accompanying figure, the rails, connecting end piece, and rod all have a resistance per unit length of . The rod moves to the left at v = 3.0 m/s. If everywhere in the region, what is tire current in the circuit (a) when a = 8.0 cm? (b) when a = 5.0 cm ? Specify also the sense of the current flow.The rod shown below moves to the right on essentially zero-resistance rails at a speed v=3.0m/s . If B = 0.75 T eveywhere in the region, what is the current through the 5.0 resistor? Does the current circulate clockwise or counterclockwise?Shown below is a conducting rod that slides along metal rails. The apparatus is in a uniform magnetic field of strength 0.25 T, which is directly into the page. The rod is pulled to the right at a constant speed of 5.0 m/s by a force . The only significant resistance in the circuit comes from the 2.0resistor shown. (a) What is the emf induced in the circuit? (b) What is the induced current? Does it circulate clockwise or counter clockwise? (c) What is the magnitude of(d) What are the power output of and the power dissipated in the resistor?Calculate the induced electric field in a 50-tuni coil with a diameter of 15 cm that is placed in a spatially uniform magnetic field of magnitude 0.50 T so that the face of the coil and the magnetic field are perpendicular. This magnetic field is reduced to zero in 0.10 seconds. Assume drat the magnetic field is cylindrically symmetric with respect to the central axis of the coil.The magnetic field through a circular loop of radius 10.0 cm varies with time as shown in the accompanying figure. The field is perpendicular to the loop. Assuming cylindrical symmetry with respect to the central axis of the loop, plot tire induced electric field in the loop as a function of time.The current I through a long solenoid with n trims per meter and radius R is changing with time as given by dI/dt. Calculate the induced electric field as a function of distance r from the central axis of the solenoid.Calculate the electric field induced both inside and outside the solenoid of the preceding problem if I=I0sint. .51P The magnetic field at all points within the cylindrical region whose cross-section is indicated in the accompanying figure starts at 1.0 T and decreases uniformly to zero in 20 s. What is the electric field (both magnitude and direction) as a function of r, the distance from the geometric center of the region?The current in a long solenoid of radius 3 cm is varied with time at a rate of 2 A/s. A circular loop of wire of radius 5 cm and resistance 2 surrounds the solenoid. Find the electrical current induced in the loop.The current in a long solenoid of radius 3 cm and 20 turns cm is varied with time at a rate of 2 A/s. Find the electric field at a distance of 4 cm from die center of the solenoid.Design a current loop that, when rotated in a uniform magnetic field of strength 0.10 T, will produce an emf =0 sin t. where 0=110V and 0=110V .A flat, square coil of 20 turns that has sides of length 15.0 cm is rotating in a magnetic field of strength 0.050 T. If tlie maximum emf produced in die coil is 30.0 mV, what is the angular velocity of the coil?A 50-turn rectangular coil with dimensions 0.15m0.40m rotates in a uniform magnetic field of magnitude 0.75 T at 3600 rev/min. [a) Determine the emf induced in the coil as a function of time, (b) If the coil is connected to a 1000 resistor, what is the power as a function of time required to keep the coil turning at 3600 rpm? (c) Answer part (b) if the coil is connected to a 2000 resistor.The square armature coil of an alternating current generator has 200 turns and is 20.0 cm on side. When it rotates at 3600 rpm, its peak output voltage is 120 V. (a) Wliat is the frequency' of the output voltage? (b) What is the strength of the magnetic field in which the coil is turning?A flip coil is a relatively simple device used to measure a magnetic field, It consists of a circular coil of N turns wound with fine conducting wire. The coil is attached to a ballistic galvanometer, a device that measures the total charge that passes through it. The coil is placed in a magnetic field B such that its face is perpendicular to the field. It is then flipped through 180°, and tire total charge Q that flows through the galvanometer is measured. (a) If the total resistance of tire coil and galvanometer Is R, what is the relationship between B and Q? Because the coil is very small, you can assume that Bis uniform over it. (b) How can you determine whether or not tire magnetic field is perpendicular to the face of the coil?The flip coil of the preceding problem has a radius of 3.0 cm and is wound with 40 turns of copper wire. The total resistance of tire coil and ballistic galvanometer is 0.20 When the coil is flipped through 180° in a magnetic fielda change of 0.090 C flows through the ballistic galvanometer. (a) Assuming that and the face of the coil are initially perpendicular, what is tire magnetic field? (b) If the coil is flipped through 90°, what is tire reading of the galvanometer?A 120-V, series-wound motor has a field resistance of 80 and an armature resistance of 10. When it is operating at full speed, a back emf of 75 V is generated, (a) What is the initial current drawn by the motor? When the motor is operating at full speed, where are (b) the current drawn by the motor, (c) the power output of the source, [ d) the power output of the motor, and (e) the power dissipated in the two resistances?A small series-wound dc motor is operated from a 12-V car battery. Under a normal load, the motor draws 4.0 A, and when the armature is clamped so that it cannot turn, the motor draws 24 A. What is the back emf when tire motor is operating normally?Shown in the following figure is a long, straight wire and a single-turn rectangular loop, both of which he in the plane of the page. The wire is parallel to the long sides of the loop and is 0.50 m away from the closer side. At an instant when the emf induced in the loop is 2.0 V, what is the time rate of change of the current in the wire?A metal bar of mass 500 g slides outward at a constant speed of 1.5 cm/s over two parallel rails separated by a distance of 30 cm which are pail of a U-shaped conductor. There is a uniform magnetic field of magnitude 2 T pointing out of the page over the entire area. The railing and metal bar have an equivalent resistance of 150 . (a) Determine the induced current, both magnitude and direction, (b) Find the direction of tire induced current if the magnetic field is pointing into the page, (c) Find the direction of the induced current if the magnetic field is pointed into the page and the bar moves inwards.A current is induced in a circular loop of radius 1.5 cm between two poles of a horseshoe electromagnet when the current in the electromagnet is varied. The magnetic Field in the area of the loop is perpendicular to the area and has a uniform magnitude. If the rate of change of magnetic field is 10 T/s, find the magnitude and direction of the induced current if resistance of the loop is 25 .A metal bar of length 25 cm is placed perpendicular to a uniform magnetic field of strength 3 T. (a) Determine the induced emf between the ends of the rod when it is not moving, (b) Determine the emf when the rod is moving perpendicular to its Length and magnetic field with a speed of 50 cm/s.A coil with 50 turns and area 10cm2 is oriented with its plane perpendicular to a 0.75-T magnetic field. If the coil is flipped over (rotated through 180°) in 0.20 s, what is the average emf induced in it?A 2-turn planer loop of flexible wire is placed inside a long solenoid of n turns per meter that canes a constant current I0 The area A of tire loop is changed by pulling on its sides while ensuring that tire plane of the loop always remains perpendicular to tire axis of the solenoid. If n = 500 turns per meter, I0=20A, and A=20cm2 , what is the emf induced in the loop when dA/dt=100 ?The conducting rod shown in the accompanying figure moves along parallel metal rails that are 25-cm apart. The system is in a uniform magnetic field of strength 0.75 T, which is directed into the page. The resistances of the rod and the rails are negligible, but the section PQ has a resistance of 0.25 . (a) What is the emf (including its sense) induced in the rod when it is moving to tire right with a speed of 5.0 m/s? (b) What force is required to keep the rod moving at this speed? (c) What is the rate at which work is done by this force? (d) What is the power dissipated in the resistor?A circular loop of wire of radius 10 cm is mounted on a vertical shaft and rotated at a frequency of 5 cycles per second in a region of uniform magnetic field of 2 Gauss perpendicular to the axis of rotation, (a) Find an expression for the time-dependent flux through the ring, (b) Determine the time-dependent current through the ring if it has a resistance of 10 The magnetic field between the poles of a horseshoe electromagnet is uniform and has a cylindrical symmetry about an axis from the middle of the South Pole to the middle of the North Pole. The magnitude of the magnetic field changes as a rate of dB/dt due to the changing current through the electromagnet, Determine the electric field at a distance r from the center.A long solenoid of radius a with n turns per unit length is carrying a time-dependent current I(t)=I0sin(t) , where I0and are constants. The solenoid is surrounded by a wire of resistance R that has two circular- loops of radius b with b>a (see the following figure). Find the magnitude and direction of current induced in the outer A 120-V, series-wound dc motor draws 0.50 A from its power source when operating at full speed, and it draws 2.0 A when it starts. The resistance of the armature coils is 10 , (a) What is the resistance of the field coils? (b) What is tire back emf of the motor when it is running at full speed? (c) The motor operates at a different speed and draws 1.0 A from the source. What is the back emf in this case?The armature and field coils of a series-wound motor have a total resistance of 3.0 . When connected to a 120-V source and running at normal speed, the motor draws 4.0 A. (a) How large is tire back emf? (b) What current will the motor draw just after it is turned on? Can you suggest a way to avoid this large initial current?A copper wire of Length I is fashioned into a circular coil with N turns. When the magnetic field through the coil changes with time, for what value of N is the induced emf a maximum?A 0.50-kg copper sheet drops through a uniform horizontal magnetic field of 1.5 T, and it reaches a terminal velocity of 2.0 m's. (a) What is the net map,-, eh: force on the sheet after it reaches terminal velocity? (b) Describe the mechanism responsible for this force, (c) How much power is dissipated as Joule heating while the sheet moves at terminal velocity?A circular copper disk of radius 7.5 on rotates at 2400 rpm around the axis through its center and perpendicular to its face. The disk is in a uniform magnetic field B of strength 1.2 T that is directed along the axis. What is the potential difference between the rim and the axis of the disk?A short rod of length a moves with its velocity vparallel to an infinite wire carrying a current I (see below). If the end of the rod nearer the wire is a distance b from the wire, what is the emf induced in the rod?A rectangular circuit containing a resistance R is pulled at a constant velocity V away from a long, straight wire carrying a current I0 (see below). Derive an equation that gives the current induced in the circuit as a function or the distance x between the near side or the circuit and the wire.Two infinite solenoids cross the plane of the circuit as shown below. The radii of the solenoids are 0.10 and 020 m, respectively, and the current in each solenoid is changing such that dB/dt50.0T/s . What are die currents in die resistors of the circuit?An eight-turn coil is tightly wrapped around the outside of the long solenoid as shown below. The radius of the solenoid is 2.0 cm and it has 10 turns per centimeter. The current through the soleNOID increases according to I=I0(1eat) where I0=4.0A and a=2.0102s1 . What is the emf induced in the coil when (a) t=0, (b) t=1.0102s and (c) t ?Shown below is a long rectangular loop of width w, length l, mass m, and resistance R. The loop starts tram rest at the edge of a uniform magnetic held B and is pushed into the Held by a constant force F function of time.A square bar of mass m and resistance R is sliding without friction down very long, parallel conducting rails of negligible resistance I see below) The two rails are a distance ' apart and are connected to each other at the bottom of the incline by a zero-resistance wire. The rails are inclined at an angle , and there is a uniform vertical magnetic field B throughout the region, la) Show that the bar acquires a terminal velocity given by vmgRsinB2I2cos2 Calculate the work per unit time done by the force of gravity. (c) Compare this with the power dissipated in the Joule heating of the bar (d) What would happen if B were reversed?The accompanying figure shows a metal disk of inner radius r1 and other radius r2 rotating at an angular velocity while in a uniform magnetic field directed parallel to the rotational axis. The brush leads of voltmeter are connected to the dark's inner and outer surfaces as shown. What is the reading of the voltmeter ?A long solenoid with 10 turns per centimeter is placed inside a copper ring such that both objects hove the same central axis. The radius of the ring is 10.0 cm. and the radius of the solenoid is 5.0 cm. (a) What is the emf induced in the ring when the current 2 through the solenoid is 5.0 A and changing at a rate of 100 A/s? (b) What is the emf induced in the ring when 1=2.0A and. dI/dt=100A/s ? (c) What is the electric field inside the ring for these two cases? id: Suppose the ring is moved so that its central axis and the central axis of the solenoid are still parallel but no longer coincide. (You should assume that the solenoid is still inside die ring.) New what is the emf induced in the ring? (el Can you calculate the electric field in die ring as you did in part (c)?The current in the long, straight wire shown in the accompanying figure is given by, where I=I0sint I0=15Aand=120rad/s. What is she current induced in the rectangular loop at (a) t = 0 and (b) t=2.1103s The resistance of loop is 2.0 .A 500-turn coil with a 0.250m2 area is spun in Earth’s 5.00105T magnetic field, producing a 12.0-kV maximum emf. (a) As what angular velocity must the coil be spun? (b) What is unreasonable about this result? (c) Which assumption or premise is responsible?A circular loop of wire of radius 10 cm. is mounted on a vertical shaft and rotated at a frequency of 5 cycles per second in a region of uniform magnetic field of 2104T perpendicular to the arcs of rotation, (a) Find an Expression for the time-dependent flux through the ring (b) Determine die time-dependent current through the ring if it has a resistance of 10.A long solenoid of radius a with n turns per unit length is carrying a tune-dependent current I(t)=I0sintwhere I0 and are constants. The solenoid is surrounded by a wire of resistance R that has two circular loops of radius b with b>a. Find the magnitude and direction of current induced m the outer loops at tune t=0.A rectangular copper loop of mass 100 g and resistance 0.2 is in a region of uniform magnetic field that is perpendicular to the area enclosed by the ring and horizontal to Earth's surface (see below). The loop is let go from rest what it is at the edge of the nonzero magnetic field region. (a) Find an expression for the speed when the loop just exits the region of uniform magnetic field. (b) If it was let go at t = 0, what is the tune when it exits the region of magnetic field for the following values, a=25cm,b=50cm,B=3T,g=9.8m/s2 ?Assumethat the magnetic field of the induced current is negligible compared to 3 T.A metal bar of mass m slides without friction over two rails a distance D apart in the region that has a uniform magnetic held of magnitude B0 and direction perpendicular to the rails (see below). The two rails are connected at one end to a resistor whose resistance is much larger than the resistance of the rails and the bar The bar is given an initial speed of vg. It is found to slow down. How far does the bar go before coming to rest? Assume that the magnetic held of the induced current is negligible compared to B0 A time-dependent uniform magnetic field of magnitude B(t) is confined in a cylindrical region of radius R. A conducting rod of length 2D is placed in the region, as shown below. Show that the emf between the ends of the rod is given by dBdtDR2D2 . ( Hint: To find the between the ends, we need to integrate the electric field from one end to the other. To find the electric field, use Faraday’s law as “Ampere’s law for E”.)Check Your Understanding. A current I(t)=(5.0A)sin((200rad/s)t) flows through the solenoid of part (b) of Example 14.1. What is the maximum emf induced in the surrounding coil?Check Your Understanding. Current flows through the inductor in Figure 14.8 from B to A instead of from A to B as shown. Is the current increasing or decreasing in order to produce the emf given in diagram (a)? In diagram(b)?Check Your Understanding. A changing current induces an emf of 10 V across a 0.25-H inductor. What is the rate at which the current is changing?Check Your Understanding (a) Calculate the self-inductance of a solenoid that is tightly wound with wire of diameter 0.10 cm, has a cross-sectional area of 0.90 cm2, and is 40 cm long. (b) If the current through the solenoid decreases uniformly from 10 to 0 A in 0.10 s, what is the emf induced between the ends of the solenoid?Check Your Understanding (a) What is the magnetic flux through one turn of a solenoid of self- inductance 8.0 × 10-5 H when a current of 3.0 A flows through it? Assume that the solenoid has 1000 turns and is wound from wire of diameter 1.0 mm. (b) What is the cross-sectional area of the solenoid?Check Your Understanding How much energy is stored in the inductor of Example 14.2 after the current reaches its maximum value?Check Your Understanding Verify that RC and L/R have the dimensions of time.Check Your Understanding (a) If the current in the circuit of in Figure 14.12(b) increases to 909 of its final value after 5.0 s, what is the inductive time constant? (b) If R=20 , what is the value of the self-inductance? (c) If the 20 resistor is replaced with a 100 resister, what is the time taken for the current to reach 90% of its final value?Check Your Understanding For the circuit of in Figure 14.12(b), show that when steady state is reached, the difference in the total energies produced by the battery and dissipated in the resistor is equal to the energy stored in the magnetic field of the coil.Check Your Understanding The angular frequency of the oscillations in an LC circuit is 2.0103rad/s . (a) If L = 0.10H, what is C? (b) Suppose that at t = 0, all the is stored in the inductor. What is the value of ? (c) A second identical capacitor is connected in parallel with the original capacitor. What is the angular frequency of this circuit?Check Your Understanding In an RLC circuit, L = 5.0 mH, C=60F , and R=200 . (a) Is the circuit under damped, critically damped, or overdamped? (b) If the circuit starts oscillating with a charge of 3.0103C on the capacitor, how much energy has been dissipated in the resistor by the time the oscillations cease?Show that N m /l and el(dl/dt), which are both expressions for se1f-inductance, have the same units.A 10-H inductor carries a current of 20 A. Describe bow a 50-V emf can be induced across it.The ignition circuit of an automobile is powered by a 12-V battery. How are we able generate large voltages with this power source?When the current through a large inductor is interrupted with a switch, an are appears across the open terminals of the switch. Explain.Does self-inductance depend on the value of the magnetic flux? Does depend on the current through the wire? Correlate your answers with the equation N=LI Would the self-inductance of a 1.0 m long, tightly wound solenoid differ from the self-inductance per meter of in infinite, but otherwise identical, solenoid?Discuss how you might determine the-inductance per unit length of a long, straight wire.The self-inductance of a coil is zero if there is no current passing through the windings. True or false?How does the self- inductance per unit length near the center of a solenoid (away from the ends) compare with its value near the end of the solenoid?Solve that I I 2 /2 has units of energy.Use Lenz’s law to explain why the initial current in the RL circuit of Figure 14.12(b) is aero.When the current in the RL circuit of Figure 14.12(b) reaches its final value /R, what is the voltage across the inductor? Across the resistor?Does the time required for the current in an RL circuit to reach any fraction of its steady-state value depend on the emf of the battery?An inductor is connected across the terminals of a battery. Does the current stance of the battery? Does the time required for the current to reach its final value depend on this resistance?At what time is the voltage across the inductor of the RL circuit of Figure 14.12(b) a maximum?In the simple RL circuit of Figure 14.12(b), can the emf induced across the inductor ever be greater than the emf of the battery used to produce the current?If emf of the battery of Figure 14.12(b) is reduced by a factor of 2, by how much does the steady-state energy stored in the magnetic field of the inductor change?A steady current flows through a circuit with a large induct3ve time constant. When a switch in the circuit is opened, a large spark across the terminals of the switch. Explain.Describe how the currents through R1and R2, shown below vary with time after switch S is closed.Discuss possible practical applications of RL circuits.Do Kirchhoff’s rules apply to circuits that contain inductors and Capacitors?Can a circuit e1eent have both capacitance and inductance?In an LC circuit, what determines the frequency and the amplitude of the energy oscillations in either the inductor or capacitor?When a wire is connected between the two ends of a solenoid, the resulting circuit can oscillate like an RLC circuit. Describe what causes the capacitance in this circuit.Describe what effect the resistance of the connecting wires has on an oscillating LC circuit.Suppose you wanted to design an LC circuit with a frequency of 0.01 Hz. What problems might you encounter?A radio receiver uses an RLC circuit to pick out particular frequencies to listen to in your house or car without hearing other unwanted frequencies. How would someone design such a circuit?When the current in one coi1 changes at a rate of 5.6 A/s an emf of 6.3 × 10-3 is induced in a second, nearby coil. What is the mutual inductance of the two coils?An emf of 9.7 × 10-3 V is induced in a coil while the current in a nearby coil is decreasing at a rate of 2.7 A/s. What is the mutual inductance of the two coils?Two coils close to each other have a mutual inductance of 32 mH. If the current in one coil decays according to I=I0eat , where I0= 5.0 A and =2.0103s1 , what is the emf induced in the second coil immediately after the current starts to decay? At t=10.0103s? ?A coil of 40 turns is wrapped around a long solenoid of cross-sectional area 7.5 × 10-3 m2. The solenoid is 0.50 m long and has 500 turns. (a) What is the mutual inductance of this system? (b) The outer coil is replaced by a coil of 40 turns whose radius is three times that of the solenoid. What is the mutual inductance of this configuration?A 600-turn solenoid is 0.55 m long and 4.2 cm in diameter. Inside the solenoid, a small (1.1 cm × 1.4 cm), single-turn rectangular coil is fixed in place with its face perpendicular to the long axis of the solenoid. What is the mutual inductance of this system?A toroidal coil has a mean radius of 16 cm and a cross-sectional area of 0.25 cm2; it is wound uniformly with 1000 turns. A second toroidal coil of 750 turns is wound uniformly over the first coil. Ignoring the variation of the magnetic field within a toroid, determine the mutual inductance of the two coils.A solenoid of N1turns has length l1and radius R1, and a second smaller solenoid of N2turns has length l2and radius R2. The smaller solenoid is placed completely inside the larger solenoid so that their long axes coincide. What is the mutual inductance of the two solenoids?An emf of 0.40 V is induced across a coil when the current through it changes uniformly from 0.10 to 0.60 A in 0.30 s. What is the self-inductance of the coil?The current shown in part (a) below is increasing, whereas that shown in part (b) is decreasing. In each case, determine which end of the inductor is at the higher potential.What is the rate at which the current though a 0.30-H coil is changing if an emf of 0.12 V is induced across the coil?When a camera uses a flash, a fully charged capacitor discharges through an inductor. In what time must the 0.100-A current through a 2.00-mH inductor be switched on or off to induce a 500-V emf?A coil with a self-inductance of 2.0 H carries a current that varies with time according to I(t) = (2.0 A)sin 120t . Find an expression for the emf induced in the coil.A solenoid 50 cm long is wound with 500 turns of wire. The cross-sectional area of the coil is 2.0 cm2. What is the self-inductance of the solenoid?A coil with a self-inductance of 3.0 H carries a current that decreases at a uniform rate dl/dt = -0.050 A/s. What is the emf induced in the coil? Describe the polarity of the induced emf.The current I(t) through a 5.0-mH inductor varies with time, as shown below. The resistance of the inductor is 5.0 . Calculate the voltage across the inductor at t = 2.0 ms, r = 4.0 ms, and t = 8.0 ms.A long, cylindrical solenoid with 100 turns per centimeter has a radius of 1.5 cm. (a) Neglecting end effects, that is the self-inductance per unit length of the solenoid? (b) If the current through the solenoid changes at the rate 5.0 AJs, what is the emf induced per unit length?Suppose that a rectangular toroid has 2000 windings and a self-inductance of 0.040 H. If h = 0.10 m, what is the ratio of its outer radius to its inner radius?What is the self-inductance per meter of a coaxial cable whose inner radius is 0.50 mm and whose outer radius is 4.00 mm?At the instant a current of 0.20 A is flowing through a coil of wire, the energy stored in its magnetic field is 6.0 × 10-3 J. What is the self-inductance of the coil?Suppose that a rectangular toroid has 2000 windings and a self-inductance of 0.040 H. If h = 0.10 m, what is the current flowing through a rectangular toroid when the energy in its magnetic field is 2.0 × 10-6 J?Solenoid A is tightly wound while solenoid B has windings that are evenly spaced with a gap equal to the diameter of the wire. The solenoids are otherwise identical. Determine the ratio of the energies stored per unit length of these solenoids when the same current flows through each.A 10-H inductor carries a current of 20 A. How much ice at 0ºC could be melted by the energy stored in the magnetic field of the inductor? (Hint: Use the value Lf= 334 J/g for ice.)A coil with a self-inductance of 3.0 H and a resistance of 100 2 carries a steady current of 2.0 A. (a) What is the energy stored in the magnetic field of the coil? (b) What is the energy per second dissipated in the resistance of the coil?A current of 1.2 A is flowing in a coaxial cable whose outer radius is five times its inner radius. What is the magnetic field energy stored in a 3.0-m length of the cable?In Figure 14.12, =12V , L = 20 mH, and R=5.0. Determine (a) the time constant of the circuit, (b) the initial current through the resistor, (C) the final current through the resistor, (d) the current through the resistor when t=2L , and (e) the voltages across the inductor and the resistor when t=2L .For the circuit shown below, =20V , L = 4.0 mH, and R = 5.0 . After steady state is reached with S1 closed and S2 open, S2 is closed and immediately thereafter (at t = 0) S1 is opened. Determine (a) the current through L at t = 0, (b) the current through L at r = 4.0 x 10-4 s, and (c) the voltages across L and R at t = 4.0 x 10-4 s.The current in the RL circuit shown here increases to 40% of its steady-state value in 2.0 s. What is the time constant of the circuit?How long after switch S1 is thrown does it take the current in the circuit shown to reach half its maximum value? Express your answer in terms of the time constant of the circuit.Examine the circuit shown below in part (a). Determine dl/dt at the instant after the switch is thrown in the circuit of (a), thereby producing the circuit of (b). Show that if I were to continue to increase at this initial rate, It would reach its maximum /R in one time constant.The current in the RL circuit shown below reaches half maximum value in l.75 ms after the switch S1is town. Determine (a) the tine constant of circuit and (b) the resistance of the circuit if L = 250 mH.Consider the circuit shown below. Find l1, l2and l3when (a) the switch S is first closed, (b) after the currents have reached steady-state values, and (c) at the instant the switch is reopened (after being closed for a long time).For the circuit shown below, =50V , R1= 10 , and L= 2.0 mH. Find the values of l1and l2(a) immediately after switch S is closed, (b) a long time after S is closed, (C) immediately after S is reopened, and (d) a long time after S is reopened.For the circuit shown below, find the current through the inductor 2.0105 s after the switch is reopened.Show that for the circuit shown below, the initial energy stored in the inductor, LI2(0)/2 , is equal to the total energy eventually dissipated in the resistor 0I2(t)Rdt.A 5000-pF capacitor is charged to 100 V and then quickly connected to an 80-mH inductor. Determine (a) the maximum energy stored in the magnetic field of the inductor, (b) the peak value of the current, and (c) the frequency of oscillation of the circuit.The self-inductance and capacitance of an LC circuit e 0.20 mH and 5.0 pF. What is the angular frequency at which the circuit oscillates?What is the self-inductance of an LC circuit that oscillates at 60 Hz 1ien the capacitance is 10 F ?In an oscillating LC circuit the maximum charge on the capacitor is 2.0 × 10-6 C and the maximum current through the inductor is 8.0 mA. (a) What is the period of the oscillations? (b) How much time elapses between an instant when the capacitor is uncharged and the next instant when it is fully charged?The self-inductance and capacitance of an oscillating LC circuit are L = 20 mH and C = 1.0 F , respectively. (a) What is the frequency of the oscillation? (b) If the maximum potential difference the plates of the capacitor is 50 V, what is the maximum current in the circuit?In an oscillating LC circuit, the maximum charge on the capacitor is qm. Determine the charge on the capacitor and the current through the inductor den energy is shared equally between the electric and magnetic fields. Express your answer in terms of qm, L, and C.In the circuit shown below, S1is opened and S2is dosed simultaneously. Determine (a) the frequency of the resulting oscillations, (b) the maximum charge on the capacitor, (c) the maximum current through the inductor, and (d) the electromagnetic energy of the oscillating circuit.An LC circuit in an AM tuner (in a car stereo) uses a coil with an inductance of 2.5 mH and a variable capacitor. If the natural frequency of the circuit is to be adjustable over the range 540 to 1600 kHz (the AM broadcast band), that range of capacitance is required?In an oscillating RLC circuit, R=5.0 ,. L=5.0mH , and C = 500 F . What is the angular frequency of the oscillations?In an oscillating RLC circuit with L = 10 mH, C = 1.5 F , and R = 2.0 , how much time elapses before the amplitude of the oscillations drops to half its initial value?What resistance R must be connected in series with a 200-mH inductor of the resulting RLC oscillating circuit is to decay to 50% of its initial value of charge in 50 cycles? To 0.10% of its initial value in 50 cycles?

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