Solved: To monitor the breathing of a hospital patient, a

What is the magnetic fl ux through a rectangle lying in the xy-plane with length 2.0 m and width 1.0 m if a uniform magnetic fi eld has magnitude 3.0 T and makes an angle of 30.0 with respect to the xy-plane? (a) 1.4 Wb (b) 2.2 Wb (c) 3.0 Wb (d) 4.1 Wb (e) 5.2 Wb

A 10-turn square coil 0.5 m on a side lies in the xy-plane. A uniform magnetic fi eld with magnitude 1.0 T in the negative z-direction changes steadily to 3.0 T in 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

A small airplane with a wingspan of 12 m fl ies horizontally and due north at a speed of 60.0 m/s in a region where the magnetic fi eld of Earth is 60.0 mT directed 60.0 below the horizontal. What is the magnitude of the induced emf between the tips of the airplanes wings? (a) 51 mV (b) 31 mV (c) 37 mV (d) 44 mV (e) 22 mV

A generator contains a 100-turn coil that rotates 10.0 times per second. If each turn has an area of 0.100 m2 and the magnetic fi eld 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

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

A 5.0-H inductor is connected in series with a 10.0-V battery, a switch, and a 2.5- resistor. After closing the switch and completing the circuit, what is the estimated length of time required for the current to reach half its maximum value? (Select the closest answer.) (a) 200 s (b) 50 s (c) 20 s (d) 2 s (e) 0.2 s

The current is doubled in an inductor. By what factor does the stored energy change? (a) 12 (b) 1 (c) 2 (d) 4 (e) 14

When a coil is placed in a constant external magnetic fi eld directed perpendicular to the plane of the coil, which of the following actions does not induce an emf in the coil? (a) Collapsing the coil. (b) Changing the strength of the magnetic fi eld. (c) Moving the coil through the fi eld without changing its orientation with respect to the fi eld. (d) Rotating the coil. (e) Removing the coil from the fi eld.

A rectangular loop is placed near a long wire carrying a current I, as shown in Figure MCQ20.9. If I decreases in time, what can be said of the current induced in the loop? (a) The current has a direction that depends on the size of the loop. (b) The current is clockwise. (c) The current is counterclockwise. (d) The current is zero. (e) The magnitude of the current depends on the size of the loop.

Two rectangular loops of wire lie in the same plane, as shown in Figure MCQ20.10. If the current I in the outer 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 counterclockwise. (d) Its magnitude depends on the dimensions of the loops. (e) Its direction depends on the dimensions of the loops.

A bar magnet is held above the center of a wire loop lying in the horizontal plane, as shown in Figure MCQ20.11. The south end of the magnet is toward the loop. After the magnet is dropped, what is true of the current in the resistor as viewed from above? (a) It is clockwise as the as the magnet falls toward the loop. (c) It is clockwise after the magnet has moved through the loop and moves away from it. (d) It is always clockwise. (e) It is fi rst counterclockwise as the magnet approaches the loop and then clockwise after it has passed through the loop.

A circuit consists of a lightbulb connected to two conducting rails and a conducting movable bar, as shown in Figure MCQ20.12. An external magnetic fi eld is directed perpendicular to the plane of the circuit. During which of the following actions will the bulb light up? (More than one statement may apply.) (a) The bar is moved to the left. (b) The bar is moved to the right. (c) The magnitude of the magnetic fi eld increases with time. (d) The magnitude of the magnetic fi eld decreases with time. (e) The magnetic fi eld remains constant in time.

A circular loop is located in a uniform and constant magnetic fi eld. Describe how an emf can be induced in the loop in this situation.

Does dropping a magnet down a copper tube produce a current in the tube? Explain.

A spacecraft orbiting Earth has a coil of wire in it. An astronaut measures a small current in the coil, although there is no battery connected to it and there are no magnets in the spacecraft. What is causing the current?

A loop of wire is placed in a uniform magnetic fi eld. For what orientation of the loop is the magnetic fl ux a maximum? For what orientation is the fl ux zero?

As the conducting bar in Figure CQ20.5 moves to the right, an electric fi eld directed downward is set up. If the bar were moving to the left, explain why the electric fi eld would be upward.

As the bar in Figure CQ20.5 moves perpendicular to the fi eld, is an external force required to keep it moving with constant speed?

Wearing a metal bracelet in a region of strong magnetic fi eld could be hazardous. Discuss this statement.

How is electrical energy produced in dams? (That is, how is the energy of motion of the water converted to AC electricity?)

Eddy currents are induced currents set up in a piece of metal when it moves through a nonuniform magnetic fi eld. For example, consider the fl at metal plate swinging at the end of a bar as a pendulum, as shown in Figure CQ20.9. At position 1, the pendulum is moving from a region where there is no magnetic fi eld into a region where the fi eld B S in is directed into the paper. Show that at position 1 the direction of the eddy current is counterclockwise. Also, at position 2 the pendulum is moving out of the fi eld into a region of zero fi eld. Show that the direction of the eddy current is clockwise in this case. Use right-hand rule number 2 to show that these eddy currents lead to a magnetic force on the plate directed as shown in the fi gure. Because the induced eddy current always produces a retarding force when the plate enters or leaves the fi eld, the swinging plate quickly comes to rest.

A bar magnet is dropped toward a conducting ring lying on the fl oor. As the magnet falls toward the ring, does it move as a freely falling object?

A piece of aluminum is dropped vertically downward between the poles of an electromagnet. Does the magnetic fi eld affect the velocity of the aluminum? Hint: See Conceptual Question 9.

Is it possible to induce a constant emf for an infi nite amount of time?

Why is the induced emf that appears in an inductor called a back (counter) emf?

A magneto is used to cause the spark in a spark plug in many lawn mowers today. A magneto consists of a permanent magnet mounted on a fl ywheel so that it spins past a fi xed coil. Explain how this arrangement generates a large enough potential difference to cause the spark.

A uniform magnetic fi eld of magnitude 0.50 T is directed perpendicular to the plane of a rectangular loop having dimensions 8.0 cm by 12 cm. Find the magnetic fl ux through the loop.

Find the fl ux of Earths magnetic fi eld of magnitude 5.00 _ 10_5 T through a square loop of area 20.0 cm2 (a) when the fi eld is perpendicular to the plane of the loop, (b) when the fi eld makes a 30.0 angle with the normal to the plane of the loop, and (c) when the fi eld makes a 90.0 angle with the normal to the plane.

A circular loop of radius 12.0 cm is placed in a uniform magnetic fi eld. (a) If the fi eld is directed perpendicular to the plane of the loop and the magnetic fl ux through the loop is 8.00 _ 10_3 T _ m2, what is the strength of the magnetic fi eld? (b) If the magnetic fi eld is directed parallel to the plane of the loop, what is the magnetic fl ux through the loop?

A long, straight wire carrying a current of 2.00 A is placed along the axis of a cylinder of radius 0.500 m and a length of 3.00 m. Determine the total magnetic fl ux through the cylinder.

A long, straight wire lies in the plane of a circular coil with a radius of 0.010 m. The wire carries a current of 2.0 A and is placed along a diameter of the coil. (a) What is the net fl ux through the coil? (b) If the wire passes through the center of the coil and is perpendicular to the plane of the coil, what is the net fl ux through the coil?

A 400-turn solenoid of length 36.0 cm and radius 3.00 cm carries a current of 5.00 A. Find (a) the magnetic fi eld strength inside the coil at its midpoint and (b) the magnetic fl ux through a circular cross-sectional area of the solenoid at its midpoint.

A cube of edge length _ _ 2.5 cm is positioned as shown in Figure P20.7. There is a uniform magnetic fi eld throughout the region with components Bx _ _5.0 T, By _ _4.0 T, and Bz _ _3.0 T. (a) Calculate the fl ux through the shaded face of the cube. (b) What is the total fl ux emerging from the volume enclosed by the cube (i.e., the total fl ux through all six faces)?

Transcranial magnetic stimulation (TMS) is a noninvasive technique used to stimulate regions of the human brain. A small coil is placed on the scalp, and a brief burst of current in the coil produces a rapidly changing magnetic fi eld inside the brain. The induced emf can be suffi cient to stimulate neuronal activity. One such device generates a magnetic fi eld within the brain that rises from zero to 1.5 T in 120 ms. Determine the induced emf within a circle of tissue of radius 1.6 mm and that is perpendicular to the direction of the fi eld.

A rectangular coil is located in a uniform magnetic fi eld of magnitude 0.30 T directed perpendicular to the plane of the coil. If the area of the coil increases at the rate of 5.0 _ 10_3 m2/s, what is the magnitude of the emf induced in the coil?

The fl exible loop in Figure P20.10 has a radius of 12 cm and is in a magnetic fi eld of strength 0.15 T. The loop is grasped at points A and B and stretched until its area is nearly zero. If it takes 0.20 s to close the loop, what is the magnitude of the average induced emf in it during this time?

A wire loop of radius 0.30 m lies so that an external magnetic fi eld of magnitude 0.30 T is perpendicular to the loop. The fi eld reverses its direction, and its magnitude changes to 0.20 T in 1.5 s. Find the magnitude of the average induced emf in the loop during this time.

A circular loop of wire of radius 12.0 cm is placed in a magnetic fi eld directed perpendicular to the plane of the loop, as shown in Figure P20.10. If the fi eld decreases at the rate of 0.050 0 T/s in some time interval, what is the magnitude of the emf induced in the loop during this interval?

The plane of a rectangular coil, 5.0 cm by 8.0 cm, is perpendicular to the direction of a magnetic fi eld B S . If the coil has 75 turns and a total resistance of 8.0 , at what rate must the magnitude of B S change to induce a current of 0.10 A in the windings of the coil?

A square, single-turn wire loop 1.00 cm on a side is placed inside a solenoid that has a circular cross section of radius 3.00 cm, as shown in Figure P20.14. The solenoid is 20.0 cm long and wound with 100 turns of wire. (a) If the current in the solenoid is 3.00 A, what is the fl ux through the loop? (b) If the current in the solenoid is reduced to zero in 3.00 s, what is the magnitude of the average induced emf in the loop?

A 300-turn solenoid with a length of 20.0 cm and a radius of 1.50 cm carries a current of 2.00 A. A second coil of four turns is wrapped tightly around this solenoid, so it can be considered to have the same radius as the solenoid. The current in the 300-turn solenoid increases steadily to 5.00 A in 0.900 s. (a) Use Amperes law to calculate the initial magnetic fi eld in the middle of the 300-turn solenoid. (b) Calculate the magnetic fi eld of the 300-turn solenoid after 0.900 s. (c) Calculate the area of the 4-turn coil. (d) Calculate the change in the magnetic fl ux through the 4-turn coil during the same period. (e) Calculate the average induced emf in the 4-turn coil. Is it equal to the instantaneous induced emf? Explain. (f) Why could contributions to the magnetic fi eld by the current in the 4-turn coil be neglected in this calculation?

A circular coil enclosing an area of 100 cm2 is made of 200 turns of copper wire. The wire making up the coil has resistance of 5.0 , and the ends of the wire are connected to form a closed circuit. Initially, a 1.1-T uniform magnetic fi eld points perpendicularly upward through the plane of the coil. The direction of the fi eld then reverses so that the fi nal magnetic fi eld has a magnitude of 1.1 T and points downward through the coil. If the time required for the fi eld to reverse directions is 0.10 s, what average current fl ows through the coil during that time?

To monitor the breathing of a hospital patient, a thin belt is girded around the patients chest as in Figure P20.17. The belt is a 200-turn coil. When the patient inhales, the area encircled by the coil increases by 39.0 cm2. The magnitude of Earths magnetic fi eld is 50.0 mT and makes an angle of 28.0 with the plane of the coil. Assuming a patient takes 1.80 s to inhale, fi nd the magnitude of the average induced emf in the coil during that time.

An N-turn circular wire coil of radius r lies in the xy-plane, as shown in Figure P20.10. A uniform magnetic fi eld is turned on, increasing steadily from 0 to B0 in the positive z-direction in t 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 positive z-axis, is the direction of the induced current clockwise or counterclockwise? (c) If each loop has resistance R, fi nd an expression for the magnitude of the induced current, I.)

A pickup truck has a width of 79.8 in. If it is traveling north at 37 m/s through a magnetic fi eld with vertical component of 35 mT, what magnitude emf is induced between the driver and passenger sides of the truck?

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

An automobile has a vertical radio antenna 1.20 m long. The automobile travels at 65.0 km/h on a horizontal road where Earths magnetic fi eld is 50.0 mT, directed toward the north and downward at an angle of 65.0 below the horizontal. (a) Specify the direction the automobile should move so as to generate the maximum motional emf in the antenna, with the top of the antenna positive relative to the bottom. (b) Calculate the magnitude of this induced emf.

An astronaut is connected to her spacecraft by a 25-m-long tether cord as she and the spacecraft orbit Earth in a circular path at a speed of 3.0 _ 103 m/s. At one instant, the voltage measured between the ends of a wire embedded in the cord is measured to be 0.45 V. Assume the long dimension of the cord is perpendicular to the vertical component of Earths magnetic fi eld at that instant. (a) What is the magnitude of the vertical component of Earths fi eld at this location? (b) Does the measured voltage change as the system moves from one location to another? Explain.

A Boeing 747 jet with a wingspan of 60.0 m is fl ying horizontally at a speed of 300 m/s over Phoenix, Arizona, at a location where Earths magnetic fi eld is 50.0 mT at 58.0 below the horizontal. What voltage is generated between the wingtips?

Consider the arrangement shown in Figure P20.24. Assume R _ 6.00 , _ _ 1.20 m, and a uniform 2.50-T magnetic fi eld is directed into the page. At what speed should the bar be moved to produce a current of 0.500 A in the resistor?

A bar magnet is positioned near a coil of wire, as shown in Figure P20.25. What is the direction of the current in the resistor when the magnet is moved (a) to the left and (b) to the right?

In Figure P20.26 what is the direction of the current induced in the resistor at the instant the switch is closed?

A copper bar is moved to the right while its axis is maintained in a direction perpendicular to a magnetic fi eld, as shown in Figure P20.27. If the top of the bar becomes positive relative to the bottom, what is the direction of the magnetic fi eld?

Find the direction of the current in the resistor shown in Figure P20.28 (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.

Find the direction of the current in the resistor R shown in Figure P20.29 after each of the following steps (taken in the order given): (a) The switch is closed. (b) The variable resistance in series with the battery is decreased. (c) The circuit containing resistor R is moved to the left. (d) The switch is opened.

A conducting rectangular loop of mass M, resistance R, and dimensions w by _ falls from rest into a magnetic fi eld B S , as shown in Figure P20.30. During the time interval before the top edge of the loop reaches the fi eld, the loop approaches a terminal speed vT . (a) Show that vT 5 MgR B2w2 (b) Why is vT proportional to R? (c) Why is it inversely proportional to B2?

A rectangular coil with resistance R has N turns, each of length _ and width w, as shown in Figure P20.31. The coil moves into a uniform magnetic fi eld B S with constant velocity vS . What are the magnitude and direction of the total magnetic force on the coil (a) as it enters the magnetic fi eld, (b) as it moves within the fi eld, and (c) as it leaves the fi eld?

A conducting bar of length _ moves to the right on two frictionless rails, as shown in Figure P20.24. A uniform magnetic fi eld directed into the page has a magnitude of 0.30 T. Assume _ _ 35 cm and R _ 9.0 . (a) At what constant speed should the bar move to produce an 8.5-mA current in the resistor? What is the direction of this induced current? (b) At what rate is energy delivered to the resistor? (c) Explain the origin of the energy being delivered to the resistor.

A generator produces 24.0 V when turning at 900 rev/min. What potential difference does it produce when turning at 500 rev/min?

A 100-turn square wire coil of area 0.040 m2 rotates about a vertical axis at 1 500 rev/min, as indicated in Figure P20.34. The horizontal component of Earths magnetic fi eld at the location of the loop is 2.0 _ 10_5 T. Calculate the maximum emf induced in the coil by Earths fi eld.

Considerable scientifi c work is currently under way to determine whether weak oscillating magnetic fi elds such as those found near outdoor electric power lines can affect human health. One study indicated that a magnetic fi eld of magnitude 1.0 _ 10_3 T, oscillating at 60 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 generated around the perimeter of the cell.

A fl at coil enclosing an area of 0.10 m2 is rotating at 60 rev/s, with its axis of rotation perpendicular to a 0.20-T magnetic fi eld. (a) If there are 1 000 turns on the coil, what is the maximum voltage induced in the coil? (b) When the maximum induced voltage occurs, what is the orientation of the coil with respect to the magnetic fi eld?

In a model AC generator, a 500-turn rectangular coil 8.0 cm by 20 cm rotates at 120 rev/min in a uniform magnetic fi eld of 0.60 T. (a) What is the maximum emf induced in the coil? (b) What is the instantaneous value of the emf in the coil at t _ (p/32) s? Assume the emf is zero at t _ 0. (c) What is the smallest value of t for which the emf will have its maximum value?

A motor has coils with a resistance of 30 and operates from a voltage of 240 V. When the motor is operating at its maximum speed, the back emf is 145 V. Find the current in the coils (a) when the motor is fi rst turned on and (b) when the motor has reached maximum speed. (c) If the current in the motor is 6.0 A at some instant, what is the back emf at that time?

A coil of 10.0 turns is in the shape of an ellipse having a 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 the magnitude of Earths magnetic fi eld is 55 mT. What is the maximum voltage induced in the coil if the axis of rotation of the coil is along its major axis and is aligned (a) perpendicular to Earths magnetic fi eld and (b) parallel to Earths magnetic fi eld? (Note: The area of an ellipse is given by A _ pab, where a is the length of the semimajor axis and b is the length of the semiminor axis.)

A technician wraps wire around a tube of length 36 cm having a diameter of 8.0 cm. When the windings are evenly spread over the full length of the tube, the result is a solenoid containing 580 turns of wire. (a) Find the self-inductance of this solenoid. (b) If the current in this solenoid increases at the rate of 4.0 A/s, what is the selfinduced emf in the solenoid?

The current in a coil changes from 3.5 A to 2.0 A in 0.50 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 inductance given by L 5 NFB I and L 5 2e DI/Dt have the same units.

A solenoid of radius 2.5 cm has 400 turns and a length of 20 cm. Find (a) its inductance and (b) the rate at which current must change through it to produce an emf of 75 mV.

An emf of 24.0 mV is induced in a 500-turn coil when the current is changing at a rate of 10.0 A/s. What is the magnetic fl ux through each turn of the coil at an instant when the current is 4.00 A? RL

A battery is connected in series with a 3.0- resistor and a 12-mH inductor. The maximum current in the circuit is 150 mA. Find (a) the time constant of this circuit and (b) the emf of the battery.

An RL circuit with L _ 3.00 H and an RC circuit with C _ 3.00 mF have the same time constant. If the two circuits have the same resistance R, (a) what is the value of R and (b) what is this common time constant?

A battery is connected in series with a 0.30- resistor and an inductor, as shown in Figure 20.27. The switch is closed at t _ 0. The time constant of the circuit is 0.25 s, and the maximum current in the circuit is 8.0 A. Find (a) the emf of the battery, (b) the inductance of the circuit, (c) the current in the circuit after one time constant has elapsed, and (d) the voltage across the resistor and the voltage across the inductor after one time constant has elapsed.

A 25-mH inductor, an 8.0- resistor, and a 6.0-V battery are connected in series. The switch is closed at t _ 0. Find the voltage drop across the resistor (a) at t _ 0 and (b) after one time constant has passed. Also, fi nd the voltage drop across the inductor (c) at t _ 0 and (d) after one time constant has elapsed.

Calculate the resistance in an RL circuit in which L _ 2.50 H and the current increases to 90.0% of its fi nal value in 3.00 s.

Consider the circuit shown in Figure P20.50. Take e _ 6.00 V, L _ 8.00 mH, and R _ 4.00 . (a) What is the inductive time constant of the circuit? (b) Calculate the current in the circuit 250 ms after the switch is closed. (c) What is the value of the fi nal steady-state current? (d) How long does it take the current to reach 80.0% of its maximum value?

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

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

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

A 60.0-m length of insulated copper wire is wound to form a solenoid of radius 2.0 cm. The copper wire has a radius of 0.50 mm. (a) What is the resistance of the wire? (b) Treating each turn of the solenoid as a circle, how many turns can be made with the wire? (c) How long is the resulting solenoid? (d) What is the self-inductance of the solenoid? (e) If the solenoid is attached to a battery with an emf of 6.0 V and internal resistance of 350 m, compute the time constant of the circuit. (f) What is the maximum current attained? (g) How long would it take to reach 99.9% of its maximum current? (h) What maximum energy is stored in the inductor? ADDITIONAL PROBLEMS

In Figure P20.55 the bar magnet is being moved toward the loop. Is (Va _ Vb) positive, negative, or zero during this motion? Explain.

A 500-turn circular-loop coil 15.0 cm in diameter is initially aligned so that its axis is parallel to Earths magnetic fi eld. In 2.77 ms the coil is fl ipped so that its axis is perpendicular to Earths magnetic fi eld. If an average voltage of 0.166 V is thereby induced in the coil, what is the value of the Earths magnetic fi eld at that location?

An 820-turn wire coil of resistance 24.0 is placed on top of a 12 500-turn, 7.00-cm-long solenoid, as in Figure P20.57. Both coil and solenoid have cross-sectional areas of 1.00 _ 10_4 m2. (a) How long does it take the solenoid current to reach 0.632 times its maximum value? (b) Determine the average back emf caused by the selfinductance of the solenoid during this interval. The magnetic fi eld produced by the solenoid at the location of the coil is one-half as strong as the fi eld at the center of the solenoid. (c) Determine the average rate of change in magnetic fl ux through each turn of the coil during the stated interval. (d) Find the magnitude of the average induced current in the coil.

A spacecraft is in a circular orbit of radius 3.0 _ 104 km around a 2.0 _ 1030 kg pulsar. The magnetic fi eld of the pulsar at that radial distance is 1.0 _ 102 T directed perpendicular to the velocity of the spacecraft. The spacecraft is 0.20 km long with a radius of 0.040 km and moves counterclockwise in the xy-plane around the pulsar. (a) What is the speed of the spacecraft? (b) If the magnetic fi eld points in the positive z-direction, is the emf induced from the back to the front of the spacecraft or from side to side? (c) Compute the induced emf. (d) Describe the hazards for astronauts inside any spacecraft moving in the vicinity of a pulsar.

A conducting rod of length _ moves on two horizontal frictionless rails, as in Figure P20.24. A constant force of magnitude 1.00 N moves the bar at a uniform speed of 2.00 m/s through a magnetic fi eld B S that is directed into the page. (a) What is the current in an 8.00- resistor R? (b) What is the rate of energy dissipation in the resistor? (c) What is the mechanical power delivered by the constant force?

When the coil of a motor is rotating at maximum speed, the current in the windings is 4.0 A. When the motor is fi rst turned on, the current in the windings is 11 A. If the motor is operated at 120 V, fi nd (a) the resistance of the windings and (b) the back emf in the coil at maximum speed.

The bolt of lightning depicted in Figure P20.61 passes 200 m from a 100-turn coil oriented as shown. If the current in the lightning bolt falls from 6.02 _ 106 A to zero in 10.5 ms, what is the average voltage induced in the coil? Assume the distance to the center of the coil determines the average magnetic fi eld at the coils position. Treat the lightning bolt as a long, vertical wire.

The square loop in Figure P20.62 is made of wires with a total series resistance of 10.0 . It is placed in a uniform 0.100-T magnetic fi eld directed perpendicular into the plane of the paper. The loop, which is hinged at each corner, is pulled as shown until the separation between points A and B is 3.00 m. If this process takes 0.100 s, what is the average current generated in the loop? What is the direction of the current?

The magnetic fi eld shown in Figure P20.63 has a uniform magnitude of 25.0 mT directed into the paper. The initial diameter of the kink is 2.00 cm. (a) The wire is quickly pulled taut, and the kink shrinks to a diameter of zero in 50.0 ms. Determine the average voltage induced between endpoints A and B. Include the polarity. (b) Suppose the kink is undisturbed, but the magnetic fi eld increases to 100 mT in 4.00 _ 10_3 s. Determine the average voltage across terminals A and B, including polarity, during this period.

An aluminum ring of radius 5.00 cm and resistance 3.00 _ 10_4 is placed around the top of a long air-core solenoid with 1 000 turns per meter and a smaller radius of 3.00 cm, as in Figure P20.64. If the current in the sole- noid is increasing at a constant rate of 270 A/s, what is the induced current in the ring? Assume the magnetic fi eld produced by the solenoid over the area at the end of the solenoid is one-half as strong as the fi eld at the center of the solenoid. Assume also the solenoid produces a negligible fi eld outside its cross-sectional area.

In Figure P20.65 the rolling axle, 1.50 m long, is pushed along horizontal rails at a constant speed v _ 3.00 m/s. A resistor R _ 0.400 is connected to the rails at points a and b, directly opposite each other. (The wheels make good electrical contact with the rails, so the axle, rails, and R form a closed-loop circuit. The only signifi cant resistance in the circuit is R.) A uniform magnetic fi eld B _ 0.800 T is directed vertically downward. (a) Find the induced current I in the resistor. (b) What horizontal force F S is required to keep the axle rolling at constant speed? (c) Which end of the resistor, a or b, is at the higher electric potential? (d) After the axle rolls past the resistor, does the current in R reverse direction?

An N-turn square coil with side _ and resistance R is pulled to the right at constant speed v in the positive x-direction in the presence of a uniform magnetic fi eld B acting perpendicular to the coil, as shown in Figure P20.66. At t _ 0, the right side of the coil is at the edge of the fi eld. After a time t has elapsed, the entire coil is in the region where B _ 0. In terms of the quantities N, B, _, v, and R, fi nd symbolic expressions for (a) the magnitude of the induced emf in the loop during the time interval t, (b) the magnitude of the induced current in the coil, (c) the power delivered to the coil, and (d) the force required to remove the coil from the fi eld. (e) What is the direction of the induced current in the loop? What is the direction of the magnetic force on the loop while it is being pulled out of the fi eld?