In Figure P20.65 (page 722) the rolling axle of length1.50

A flat coil of wire is placed in a uniform magnetic fieldthat is in the y-direction. The magnetic flux throughthe coil is a maximum if the plane of the coil is where?(a) in the xy-plane (b) in the yz-plane (c) in the xzplane(d) in any orientation, because it is a constant

A 10-turn square coil 0.50 m on a side lies in the xyplane.A uniform magnetic field with magnitude 1.0 Tin the negative z-direction changes steadily to 3.0 Tin the positive z-direction, a process that takes 2.0 s.What is the magnitude of the induced emf in the coil?(a) 2.0 V (b) 5.0 V (c) 3.0 V (d) 4.0 V (e) 6.0 V

The current in a 5.00-H inductor decreases uniformlyat a rate of 2.00 A/s. What is the voltage dropacross the inductor? (a) 2.50 V (b) 5.00 V (c) 0.400 V(d) 10.0 V (e) 7.50 V

A generator contains a 100-turn coil that rotates 10.0times per second. If each turn has an area of 0.100 m2and the magnetic field through the coils is 0.050 0 T,what is the maximum emf induced in the coil?(a) 31.4 V (b) 62.8 V (c) 0.314 V (d) 6.28 V (e) 3.14 V

A small airplane with a wingspan of 12 m flies horizontallyand due north at a speed of 60.0 m/s in aregion where the magnetic field of Earth is 60.0 mTdirected 60.0 below the horizontal. What is the magnitudeof the induced emf between the tips of the airplaneswings? (a) 51 mV (b) 31 mV (c) 37 mV (d) 44 mV(e) 22 mV

The bar in Figure MCQ20.6moves on rails to the rightwith a velocity vS, and a uniform,constant magnetic fieldis directed out of the page.Which of the following statementsare correct? (Morethan one statement may becorrect.) (a) The induced currentin the loop is zero. (b) The induced current in theloop is clockwise. (c) The induced current in the loop iscounterclockwise. (d) An external force is required tokeep the bar moving at constant speed. (e) No force isrequired to keep the bar moving at constant speed.

What happens to the amplitude of the induced emfwhen the rate of rotation of a generator coil is doubled?(a) It becomes four times larger. (b) It becomestwo times larger. (c) It is unchanged. (d) It becomesone-half as large. (e) It becomes one-fourth as large.

When a coil is placed in a constant external magneticfield directed perpendicular to the plane of the coil,which of the following actions does not induce an emfin the coil? (a) Collapsing the coil. (b) Changing thestrength of the magnetic field. (c) Moving the coilthrough the field without changing its orientation withrespect to the field. (d) Rotating the coil. (e) Removingthe coil from the field.

A rectangular loop is placed near a long wire carryinga current I as shown in Figure MCQ20.9. If I decreasesin time, what can be said of the current induced in theloop? (a) The direction of the current depends on thesize of the loop. (b) The current is clockwise. (c) Thecurrent is counterclockwise. (d) The current is zero.(e) Nothing can be said about the current in the loopwithout more information.

Two rectangular loops of wire lie in the same plane,as shown in Figure MCQ20.10. If the current I in theouter loop is counterclockwise and increases with time,what is true of the current induced in the inner loop?(a) It is zero. (b) It is clockwise. (c) It is counterclock-wise. (d) Its magnitude depends onthe dimensions of the loops. (e) Itsdirection depends on the dimensionsof the loops.

A bar magnet is held above thecenter of a wire loop lying in thehorizontal plane, as shown in FigureMCQ20.11. The south end ofthe magnet is toward the loop. After the magnet isdropped, what is the direction of the induced currentin the loop as viewed from above? (a) It is clockwise asthe magnet falls toward the loop. (b) It is counterclockwiseas the magnet falls toward the loop. (c) It is clockwiseafter the magnet has moved through the loop andmoves away from it. (d) It is always clockwise. (e) It isfirst counterclockwise as the magnet approaches theloop and then clockwise after it has passed throughthe loop.

A circuit consists of a conducting movable bar and alightbulb connected to two conducting rails, as shownin Figure MCQ20.12. An external magnetic field isdirected perpendicular to the plane of the circuit.Which of the following actions will make the bulblight up? More than one statement may be correct.(a) The bar is moved to the left. (b) The bar is movedto the right. (c) The magnitude of the magnetic field isincreased. (d) The magnitude of the magnetic field isdecreased. (e) The bar is lifted off the rails.

Assume the battery in Figure CQ20.12a is replacedby an AC source and the switch is held closed. If helddown, the metal ring on top of the solenoid becomeshot. Why?

A magneto is used to cause the spark in a spark plugin many lawn mowers today. A magneto consists of apermanent magnet mounted on a flywheel so that itspins past a fixed coil. Explain how this arrangementgenerates a large enough potential difference to causethe spark.

A bar magnet is positionednear a coil of wire, as shownin Figure P20.15. What is thedirection of the current in theresistor when the magnet ismoved (a) to the left and (b) tothe right?

Find the direction of the current in the resistor shownin Figure P20.16 (a) at the instant the switch is closed,(b) after the switch has been closed for several minutes,and (c) at the instant the switch is opened.

A circular loop of wire lies below a long wire carryinga current that is increasing as in Figure P20.17a.(a) What is the direction of the induced current inthe loop, if any? (b) Now suppose the loop is next tothe same wire as in P20.17b. What is the direction ofthe induced current in the loop, if any? Explain youranswers.

A square, single-turn wireloop , 5 1.00 cm on a sideis placed inside a solenoidthat has a circular cross sectionof radius r 5 3.00 cm,as shown in the end view ofFigure P20.18. The solenoidis 20.0 cm long and woundwith 100 turns of wire. (a) Ifthe current in the solenoid is3.00 A, what is the flux through the square loop? (b) Ifthe current in the solenoid is reduced to zero in 3.00 s,what is the magnitude of the average induced emf inthe square loop?

A 300-turn solenoid with a length of 20.0 cm anda radius of 1.50 cm carries a current of 2.00 A. A secondcoil of four turns is wrapped tightly around thissolenoid, so it can be considered to have the sameradius as the solenoid. The current in the 300-turnsolenoid increases steadily to 5.00 A in 0.900 s. (a) UseAmperes law to calculate the initial magnetic field inthe middle of the 300-turn solenoid. (b) Calculate themagnetic field of the 300-turn solenoid after 0.900 s.(c) Calculate the area of the 4-turn coil. (d) Calculatethe change in the magnetic flux through the 4-turncoil during the same period. (e) Calculate the averageinduced emf in the 4-turn coil. Is it equal to theinstantaneous induced emf? Explain. (f) Why couldcontributions to the magnetic field by the current inthe 4-turn coil be neglected in this calculation?

A circular coil enclosing an area of 100 cm2 ismade of 200 turns of copper wire. The wire makingup the coil has resistance of 5.0 V, and the ends of thewire are connected to form a closed circuit. Initially,a 1.1-T uniform magnetic field points perpendicularlyupward through the plane of the coil. The directionof the field then reverses so that the final magneticfield has a magnitude of 1.1 T and points downwardthrough the coil. If the time required for the field toreverse directions is 0.10 s, what is the average currentin the coil during that time?

To monitor the breathing of a hospital patient,a thin belt is girded around the patients chest as inFigure P20.21. The belt is a 200-turn coil. When thepatient inhales, the area encircled by the coil increasesby 39.0 cm2. The magnitude of Earths magnetic fieldis 50.0 mT and makes an angle of 28.0 with the planeof the coil. Assuming a patient takes 1.80 s to inhale,find the magnitude of the average induced emf in thecoil during that time.

An N-turn circular wire coil of radius r liesin the xy-plane (the plane of the page), as in FigureP20.10. A uniform magnetic field is turned on, increasingsteadily from 0 to B0 in the positive z-direction int seconds. (a) Find a symbolic expression for the emf,e, induced in the coil in terms of the variables given.(b) Looking down on at the xy-plane from the positivez-axis, is the direction of the induced current clockwiseor counterclockwise? (c) If each loop has resistance R,find an expression for the magnitude of the inducedcurrent, I.

A truck is carrying a steel beam of length 15.0 m on afreeway. An accident causes the beam to be dumpedoff the truck and slide horizontally along the ground ata speed of 25.0 m/s. The velocity of the center of massof the beam is northward while the length of the beammaintains an eastwest orientation. The vertical componentof the Earths magnetic field at this locationhas a magnitude of 35.0 mT. What is the magnitude ofthe induced emf between the ends of the beam?

A 2.00-m length of wire is held in an eastwest directionand moves horizontally to the north with a speedof 15.0 m/s. The vertical component of Earths magneticfield in this region is 40.0 mT directed downward.Calculate the induced emf between the ends of thewire and determine which end is positive.

A pickup truck has a width of 79.8 in. If it is travelingnorth at 37 m/s through a magnetic field with verticalcomponent of 35 mT, what magnitude emf is inducedbetween the driver and passenger sides of the truck?

A bar magnet is held stationary while a circularloop of wire is moved toward the magnet at constantvelocity at position A as in Figure P20.26. The looppasses over the magnet at position B and moves awayfrom the magnet at position C. Find the directionof the induced current in the loop using Lenzs law(a) at position A and (b) at position C. (c) What is theinduced current in the loop at position B? Explain.

An automobile has a vertical radio antenna 1.20 mlong. The automobile travels at 65.0 km/h on a horizontalroad where Earths magnetic field is 50.0 mT,directed toward the north and downward at an angleof 65.0 below the horizontal. (a) Specify the directionthe automobile should move so as to generate the maximummotional emf in the antenna, with the top of theantenna positive relative to the bottom. (b) Calculatethe magnitude of this induced emf.

An astronaut is connected to her spacecraft by a25-m-long tether cord as she and the spacecraft orbitEarth in a circular path at a speed of 3.0 3 103 m/s. Atone instant, the voltage measured between the ends ofa wire embedded in the cord is measured to be 0.45 V.Assume the long dimension of the cord is perpendicularto the vertical component of Earths magneticfield at that instant. (a) What is the magnitude of thevertical component of Earths field at this location?(b) Does the measured voltage change as the systemmoves from one location to another? Explain.

Figure P20.29 shows a bar of mass m 5 0.200 kg thatcan slide without friction on a pair of rails separated by a distance , 5 1.20 m and located on an inclinedplane that makes an angle u 5 25.0 with respect to theground. The resistance of the resistor is R 5 1.00 V,and a uniform magnetic field of magnitude B 50.500 T is directed downward, perpendicular to theground, over the entire region through which the barmoves. With what constant speed v does the bar slidealong the rails?

Consider the arrangement shown in Figure P20.30.Assume R 5 6.00 V, , 5 1.20 m, and a uniform 2.50-Tmagnetic field is directed into the page. At what speedshould the bar be moved to produce a current of0.500 A in the resistor?

A square coil of wire of side 2.80 cm is placed in auniform magnetic field of magnitude 1.25 T directedinto the page as in Figure P20.31. The coil has 28.0turns and a resistance of 0.780 V. If the coil is rotatedthrough an angle of 90.0 about the horizontal axisshown in 0.335 s, find (a) the magnitude of the averageemf induced in the coil during this rotation and(b) the average current induced in the coil during thisrotation.

A 100-turn square wire coil ofarea 0.040 m2 rotates about avertical axis at 1 500 rev/min, asindicated in Figure P20.32. Thehorizontal component of Earthsmagnetic field at the location ofthe loop is 2.0 3 1025 T. Calculatethe maximum emf inducedin the coil by Earths field.

Considerable scientific workis currently under way to determinewhether weak oscillating magnetic fields suchas those found near outdoor electric power lines can affect human health. One study indicated that a magneticfield of magnitude 1.0 3 1023 T, oscillating at60 Hz, might stimulate red blood cells to become cancerous.If the diameter of a red blood cell is 8.0 mm,determine the maximum emf that can be generatedaround the perimeter of the cell.

A flat coil enclosing an area of 0.10 m2 is rotating at60 rev/s, with its axis of rotation perpendicular to a0.20-T magnetic field. (a) If there are 1 000 turns onthe coil, what is the maximum voltage induced in thecoil? (b) When the maximum induced voltage occurs,what is the orientation of the coil with respect to themagnetic field?

In a model AC generator, a 500-turn rectangular coil8.0 cm by 20 cm rotates at 120 rev/min in a uniformmagnetic field of 0.60 T. (a) What is the maximumemf induced in the coil? (b) What is the instantaneousvalue of the emf in the coil at t 5 (p/32) s? Assume theemf is zero at t 5 0. (c) What is the smallest value of tfor which the emf will have its maximum value?

A motor has coils with a resistance of 30 V and operatesfrom a voltage of 240 V. When the motor is operatingat its maximum speed, the back emf is 145 V.Find the current in the coils (a) when the motor is firstturned on and (b) when the motor has reached maximumspeed. (c) If the current in the motor is 6.0 A atsome instant, what is the back emf at that time?

A coil of 10.0 turns is in the shape of an ellipse havinga major axis of 10.0 cm and a minor axis of 4.00 cm.The coil rotates at 100 rpm in a region in which themagnitude of Earths magnetic field is 55 mT. What isthe maximum voltage induced in the coil if the axis ofrotation of the coil is along its major axis and is aligned(a) perpendicular to Earths magnetic field and(b) parallel to Earths magnetic field? Note: The area ofan ellipse is given by A 5 pab, where a is the length ofthe semimajor axis and b is the length of the semiminoraxis.

A technician wraps wire around a tube of length 36 cmhaving a diameter of 8.0 cm. When the windings areevenly spread over the full length of the tube, the resultis a solenoid containing 580 turns of wire. (a) Find theself-inductance of this solenoid. (b) If the current inthis solenoid increases at the rate of 4.0 A/s, what is theself-induced emf in the solenoid?

The current in a coil drops from 3.5 A to 2.0 A in0.50 s. If the average emf induced in the coil is 12 mV,what is the self-inductance of the coil?

Show that the two expressions for inductancegiven byL 5NFBIand L 52eDI/Dt

A solenoid of radius 2.5 cm has 400 turns and alength of 20 cm. Find (a) its inductance and (b) therate at which current must change through it to producean emf of 75 mV.

An emf of 24.0 mV is induced in a 500-turn coil whenthe current is changing at a rate of 10.0 A/s. What isthe magnetic flux through each turn of the coil at aninstant when the current is 4.00 A? RL Circuits

An electromagnet can bemodeled as an inductor inseries with a resistor. Considera large electromagnetof inductance L 5 12.0 Hand resistance R 5 4.50 Vconnected to a 24.0 V batteryand switch as in FigureP20.43. After the switch isclosed, find (a) the maximum current carried by theelectromagnet, (b) the time constant of the circuit,and (c) the time it takes the current to reach 95.0% ofits maximum value.

An RL circuit with L 5 3.00 H and an RC circuit withC 5 3.00 mF have the same time constant. If the twocircuits have the same resistance R, (a) what is thevalue of R and (b) what is this common time constant?

A battery is connected in series with a 0.30-V resistorand an inductor, as in Figure P20.43. The switchis closed at t 5 0. The time constant of the circuit is0.25 s, and the maximum current in the circuit is 8.0 A.Find (a) the emf of the battery, (b) the inductance ofthe circuit, (c) the current in the circuit after one timeconstant has elapsed, (d) the voltage across the resistorafter one time constant has elapsed, and (e) thevoltage across the inductor after one time constant haselapsed.

A 25-mH inductor, an 8.0-V resistor, and a 6.0-V batteryare connected in series as in Figure P20.43. Theswitch is closed at t 5 0. Find the voltage drop acrossthe resistor (a) at t 5 0 and (b) after one time constanthas passed. Also, find the voltage drop across theinductor (c) at t 5 0 and (d) after one time constanthas elapsed.

Calculate the resistance in an RL circuit in which L 52.50 H and the current increases to 90.0% of its finalvalue in 3.00 s.

Consider the circuit shown in Figure P20.43. Take e 56.00 V, L 5 8.00 mH, and R 5 4.00 V. (a) What is theinductive time constant of the circuit? (b) Calculatethe current in the circuit 250 ms after the switch isclosed. (c) What is the value of the final steady-statecurrent? (d) How long does it take the current to reach80.0% of its maximum value?

(a) If an inductor carrying a 1.70-A current stores anenergy of 0.300 mJ, what is its inductance? (b) Howmuch energy does the same inductor store if it carriesa 3.0-A current?

A 300-turn solenoid has a radius of 5.00 cm and alength of 20.0 cm. Find (a) the inductance of the solenoidand (b) the energy stored in the solenoid whenthe current in its windings is 0.500 A.

A 24-V battery is connected in series with a resistorand an inductor, with R 5 8.0 V and L 5 4.0 H, respectively.Find the energy stored in the inductor (a) whenthe current reaches its maximum value and (b) onetime constant after the switch is closed.

A 60.0-m length of insulated copper wire is woundto form a solenoid of radius 2.0 cm. The copper wirehas a radius of 0.50 mm. (a) What is the resistance ofthe wire? (b) Treating each turn of the solenoid as acircle, how many turns can be made with the wire?(c) How long is the resulting solenoid? (d) What is theself-inductance of the solenoid? (e) If the solenoid isattached to a battery with an emf of 6.0 V and internalresistance of 350 mV, compute the time constant ofthe circuit. (f) What is the maximum current attained?(g) How long would it take to reach 99.9% of its maximumcurrent? (h) What maximum energy is stored inthe inductor?

Two circular loops of wire surround an insulating rodas in Figure P20.53. Loop 1 carries a current I in theclockwise direction when viewed from the left end. Ifloop 1 moves toward loop 2, which remains stationary,what is the direction of the induced current in loop 2when viewed from the left end?

A circular loop of wire of resistance R 5 0.500 V andradius r 5 8.00 cm is in a uniform magnetic fielddirected out of the page as in Figure P20.54. If a clockwisecurrent of I 5 2.50 mA is induced in the loop,(a) is the magnetic field increasing or decreasing intime? (b) Find the rate at which the field is changingwith time.

A rectangular coil with resistance R has N turns, eachof length , and width w, as shown in Figure P20.55.The coil moves into a uniform magnetic field BSin withconstant velocity vS. What are the magnitude and directionof the total magnetic force on the coil (a) as itenters the magnetic field, (b) as it moves within thefield, and (c) as it leaves the field?

A conducting bar of length , moves to the righton two frictionless rails, as shown in Figure P20.30.A uniform magnetic field directed into the page hasa magnitude of 0.30 T. Assume , 5 35 cm and R 59.0 V. (a) At what constant speed should the bar moveto produce an 8.5-mA current in the resistor? What isthe direction of this induced current? (b) At what rateis energy delivered to the resistor? (c) Explain the originof the energy being delivered to the resistor.

An 820-turn wire coil of resistance 24.0 V is placed ontop of a 12 500-turn, 7.00-cm-long solenoid, as in FigureP20.57. Both coil and solenoid have cross-sectionalareas of 1.00 3 1024 m2. (a) How long does it take thesolenoid current to reach 0.632 times its maximumvalue? (b) Determine the average back emf caused bythe self-inductance of the solenoid during this interval.The magnetic field produced by the solenoid at thelocation of the coil is one-half as strong as the field atthe center of the solenoid. (c) Determine the averagerate of change in magnetic flux through each turn of the coil during the stated interval. (d) Find the magnitudeof the average induced current in the coil.

A spacecraft is in a circular orbit of radius equalto 3.0 3 104 km around a 2.0 3 1030 kg pulsar. Themagnetic field of the pulsar at that radial distance is1.0 3 102 T directed perpendicular to the velocity ofthe spacecraft. The spacecraft is 0.20 km long with aradius of 0.040 km and moves counterclockwise inthe xy-plane around the pulsar. (a) What is the speedof the spacecraft? (b) If the magnetic field points inthe positive z-direction, is the emf induced from theback to the front of the spacecraft or from side to side?(c) Compute the induced emf. (d) Describe the hazardsfor astronauts inside any spacecraft moving in thevicinity of a pulsar.

A conducting rod of length , moves on two horizontalfrictionless rails, as in Figure P20.30. A constantforce of magnitude 1.00 N moves the bar at a uniformspeed of 2.00 m/s through a magnetic field BSthat isdirected into the page. (a) What is the current in an8.00-V resistor R? (b) What is the rate of energy dissipationin the resistor? (c) What is the mechanicalpower delivered by the constant force?

A long solenoid of radius r 5 2.00 cm is wound with3.50 3 103 turns/m and carries a current that changesat the rate of 28.5 A/s as in Figure P20.60. What is themagnitude of the emf induced in the square conductingloop surrounding the center of the solenoid?

The bolt of lightning depicted in Figure P20.61 passes200 m from a 100-turn coil oriented as shown. If thecurrent in the lightning bolt falls from 6.02 3 106 Ato zero in 10.5 ms, what is the average voltage inducedin the coil? Assume the distance to the center of thecoil determines the average magnetic field at the coilsposition. Treat the lightning bolt as a long, verticalwire.

The square loop in FigureP20.62 is made of wireswith a total series resistanceof 10.0 V. It is placed in auniform 0.100-T magneticfield directed perpendicularinto the plane of the paper.The loop, which is hingedat each corner, is pulled asshown until the separation between points A and B is3.00 m. If this process takes 0.100 s, what is the averagecurrent generated in the loop? What is the direction ofthe current?

The magnetic field shown in Figure P20.63 has a uniformmagnitude of 25.0 mT directed into the paper.The initial diameter of the kink is 2.00 cm. (a) Thewire is quickly pulled taut, and the kink shrinks to adiameter of zero in 50.0 ms. Determine the averagevoltage induced between endpoints A and B. Includethe polarity. (b) Suppose the kink is undisturbed, butthe magnetic field increases to 100 mT in 4.00 3 1023 s.Determine the average voltage across terminals A andB, including polarity, during this period.

An aluminum ring of radius 5.00 cm and resistance3.00 3 1024 V is placed around the top of a long aircoresolenoid with 1 000 turns per meter and a smallerradius of 3.00 cm, as in Figure P20.64. If the current inthe solenoid is increasing at a constant rate of 270 A/s,what is the induced current in the ring? Assume themagnetic field produced by the solenoid over the areaat the end of the solenoid is one-half as strong as thefield at the center of the solenoid. Assume also thesolenoid produces a negligible field outside its crosssectionalarea.

In Figure P20.65 (page 722) the rolling axle of length1.50 m is pushed along horizontal rails at a constantspeed v 5 3.00 m/s. A resistor R 5 0.400 V is connected to the rails at points a and b, directly opposite eachother. (The wheels make good electrical contact withthe rails, so the axle, rails, and R form a closed-loopcircuit. The only significant resistance in the circuit isR.) A uniform magnetic field B 5 0.800 T is directedvertically downward. (a) Find the induced current I inthe resistor. (b) What horizontal force FSis required tokeep the axle rolling at constant speed? (c) Which endof the resistor, a or b, is at the higher electric potential?(d) After the axle rolls past the resistor, does the currentin R reverse direction? Explain your answer.

An N-turn square coil with side , and resistanceR is pulled to the right at constant speed v in thepositive x-direction in the presence of a uniform magneticfield B acting perpendicular to the coil, as shownin Figure P20.66. At t 5 0, the right side of the coil isat the edge of the field. After a time t has elapsed, theentire coil is in the region where B 5 0. In terms of thequantities N, B, ,, v, and R, find symbolic expressionsfor (a) the magnitude of the induced emf in the loop during the time interval t, (b) the magnitude of theinduced current in the coil, (c) the power delivered tothe coil, and (d) the force required to remove the coilfrom the field. (e) What is the direction of the inducedcurrent in the loop? (f) What is the direction of themagnetic force on the loop while it is being pulled outof the field?

A conducting rectangular loop of mass M, resistance R, anddimensions w by , falls from restinto a magnetic field BS, as shownin Figure P20.67. During the timeinterval before the top edge of theloop reaches the field, the loopapproaches a terminal speed vT.(a) Show thatvT 5MgRB2w2(b) Why is vT proportional to R? (c) Why is it inverselyproportional to B2?