In 1832, Faraday proposed that the apparatus shown in

A magnetic eld of strength 0.30 T is directed perpendicular to a plane circular loop of wire of radius 25 cm. Find the magnetic ux through the area enclosed by this loop.

Find the ux of the Earths magnetic eld of magnitude 5.00  105 T through a square loop of area 20.0 cm2 (a) when the eld is perpendicular to the plane of the loop, (b) when the eld makes a 30.0 angle with the normal to the plane of the loop, and (c) when the eld makes a 90.0 angle with the normal to the plane.

A square loop 2.00 m on a side is placed in a magnetic eld of magnitude 0.300 T. If the eld makes an angle of 50.0 with the normal to the plane of the loop, nd the magnetic 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 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 ux through the coil? (b) If the wire passes through the center of the coil and is perpendicular to the plane of the coil, nd the net ux through the coil.

A solenoid 4.00 cm in diameter and 20.0 cm long has 250 turns and carries a current of 15.0 A. Calculate the magnetic ux through the circular cross-sectional area of the solenoid.

A cube of edge length   2.5 cm is positioned as shown in Figure P20.7. There is a uniform magnetic eld throughout the region with components Bx  5.0T, By  4.0 T, and Bz  3.0 T. (a) Calculate the ux through the shaded face of the cube. (b) What is the total ux emerging from the volume enclosed by the cube (i.e., the total 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 eld inside the brain. The induced emf can be sufcient to stimulate neuronal activity. One such device generates a magnetic 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 eld.

A square, single-turn coil 0.20 m on a side is placed with its plane perpendicular to a constant magnetic eld. An emf of 18 mV is induced in the coil winding when the area of the coil decreases at the rate of 0.10 m2/s. What is the magnitude of the magnetic eld?

The exible loop in Figure P20.10 has a radius of 12 cm and is in a magnetic 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, nd 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 eld of magnitude 0.30 T is perpendicular to the loop. The 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 500-turn circular-loop coil 15.0 cm in diameter is initially aligned so that its axis is parallel to the Earths magnetic eld. In 2.77 ms, the coil is ipped so that its axis is perpendicular to the Earths magnetic eld. If an average voltage of 0.166 V is thereby induced in the coil, what is the value of the Earths magnetic eld at that location?

The plane of a rectangular coil, 5.0 cm by 8.0 cm, is perpendicular to the direction of a magnetic eld . If the coil has 75 turns and a total resistance of 8.0 , at what rate must the magnitude of 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, nd the ux through the loop. (b) If the current in the solenoid is reduced to zero in 3.00 s, nd the magnitude of the average induced emf in the loop.

A 300-turn solenoid with a length of 20 cm and a radius of 1.5 cm carries a current of 2.0 A. A second coil of four turns is wrapped tightly about this solenoid so that it can be considered to have the same radius as the solenoid. Find (a) the change in the magnetic ux through the coil and (b) the magnitude of the average induced emf in the coil when the current in the solenoid increases to 5.0 A in a period of 0.90 s.

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 eld points perpendicularly upward through the plane of the coil. The direction of the eld then reverses so that the nal magnetic eld has a magnitude of 1.1 T and points downward through the coil. If the time required for the eld to reverse directions is 0.10 s, what average current 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 the Earths magnetic eld is 50.0  T and makes an angle of 28.0 with the plane of the coil. Assuming a patient takes 1.80 s to inhale, nd the magnitude of the average induced emf in the coil during that time.

Consider the arrangement shown in Figure P20.18. Assume that R  6.00  and   1.20 m, and that a uniform 2.50-T magnetic 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 Boeing 747 jet with a wingspan of 60.0 m is ying horizontally at a speed of 300 m/s over Phoenix, Arizona, at a location where the Earths magnetic eld is 50.0  T at 58.0 below the horizontal. What voltage is generated between the wingtips?

A 12.0-m-long steel beam is accidentally dropped by a construction crane from a height of 9.00 m. The horizontal component of the Earths magnetic eld over the region is 18.0  T. What is the induced emf in the beam just before impact with the Earth? Assume the long dimension of the beam remains in a horizontal plane, oriented perpendicular to the horizontal component of the Earths magnetic eld.

An automobile has a vertical radio antenna 1.20 m long. The automobile travels at 65.0 km/h on a horizontal road where the Earths magnetic eld is 50.0  T, directed toward the north and downwards at an angle of 65.0 below the horizontal. (a) Specify the direction the automobile should move in order 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.

A helicopter has blades of length 3.0 m, rotating at 2.0 rev/s about a central hub. If the vertical component of Earths magnetic eld is 5.0  105 T, what is the emf induced between the blade tip and the central hub?

A bar magnet is positioned near a coil of wire as shown in Figure P20.23. What is the direction of the current in the resistor when the magnet is moved (a) to the left? (b) to the right?

A conducting rectangular loop of mass M, resistance R, and dimensions w by  falls from rest into a magnetic eld as shown in Figure P20.24. During the time interval before the top edge of the loop reaches the eld, the loop approaches a terminal speed vT. (a) Show that (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 P20.25. The coil moves into a uniform magnetic eld with constant velocity . What are the magnitude and direction of the total magnetic force on the coil (a) as it enters the magnetic eld, (b) as it moves within the eld, and (c) as it leaves the eld? v

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 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 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 100-turn square wire coil of area 0.040 m2 rotates about a vertical axis at 1 500 rev/min, as indicated in Figure P20.30. The horizontal component of the Earths magnetic eld at the location of the loop is 2.0  105 T. Calculate the maximum emf induced in the coil by the Earths eld.

Considerable scientic work is currently underway to determine whether weak oscillating magnetic elds such as those found near outdoor electric power lines can effect human health. One study indicated that a magnetic eld of magnitude 1.0  103 T, oscillating at 60 Hz, might stimulate red blood cells to become cancerous. If the diameter of a red blood cell is 8.0  m, determine the maximum emf that can be generated around the perimeter of the cell.

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 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 the Earths magnetic eld is 55  T. 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 the Earths magnetic eld and (b) parallel to the Earths magnetic eld? (Note that the area of an ellipse is given by A   ab, where a is the length of the semimajor axis and b is the length of the semiminor axis.)

A 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 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 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 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  (  /32) s? Assume that 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 coiled telephone cord forms a spiral with 70.0 turns, a diameter of 1.30 cm, and an unstretched length of 60.0 cm. Determine the self-inductance of one conductor in the unstretched cord.

A coil has an inductance of 3.0 mH, and the current in it changes from 0.20 A to 1.5 A in 0.20 s. Find the magnitude of the average induced emf in the coil during this period.

how that the two expressions for inductance given by and 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 ux through each turn of the coil at an instant when the current is 4.00 A?

Show that the SI units for the inductive time constant   L/R are seconds.

An RL circuit with L  3.00 H and an RC circuit with C  3.00  F 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 6.0-V battery is connected in series with a resistor and an inductor. The series circuit has a time constant of 600  s, and the maximum current is 300 mA. What is the value of the inductance?

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, 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 nal value in 3.00 s.

Consider the circuit shown in Figure P20.46. Take  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  s after the switch is closed. (c) What is the value of the nal steady-state current? (d) How long does it take the current to reach 80.0% of its maximum value?

How much energy is stored in a 70.0-mH inductor at an instant when the current is 2.00 A?

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.

What is the time constant for (a) the circuit shown in Figure P20.50a and (b) the circuit shown in Figure P20.50b?

In Figure P20.51, the bar magnet is being moved toward the loop. Is (Va  Vb) positive, negative, or zero during this motion? Explain.

Your physics teacher asks you to help her set up a demonstration of Faradays law for the class. The apparatus consists of a strong permanent magnet that has a field of 0.10 T, a small 10-turn coil of radius 2.0 cm cemented on a wood frame with a handle, some flexible connecting wires, and an ammeter, as in Figure P20.52. The idea is to pull the coil out of the center of the magnetic field as quickly as possible and read the average current registered on the meter. The combined resistance of the coil, leads, and meter is 2.0 , and you must flip the coil out of the field in about 0.20 s. The ammeter you must use has a full-scale sensitivity of 1 000  A. Will this meter be sensitive enough to show the induced current clearly?

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 P20.53 (page 690). Both coil and solenoid have crosssectional areas of 1.00  104 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 self-inductance of the solenoid during this interval. The magnetic eld produced by the solenoid at the location of the coil is one-half as strong as the eld at the center of the solenoid. (c) Determine the average rate of change in magnetic flux through each turn of the coil during the stated interval. (d) Find the magnitude of the average induced current in the coil.

Figure P20.54 is a graph of induced emf versus time for a coil of N turns rotating with angular speed  in a uniform magnetic eld directed perpendicular to the axis of rotation of the coil. Copy this sketch (increasing the scale), and, on the same set of axes, show the graph of emf versus t when (a) the number of turns in the coil is doubled, (b) the angular speed is doubled, and (c) the angular speed is doubled while the number of turns in the coil is halved.

The plane of a square loop of wire with edge length a  0.200 m is perpendicular to the Earths magnetic eld at a point where B  15.0  T, as in Figure P20.55. The total resistance of the loop and the wires connecting it to the ammeter is 0.500 . If the loop is suddenly collapsed by horizontal forces as shown, what total charge passes through the ammeter?

A novel method of storing electrical energy has been proposed. A huge underground superconducting coil 1.00 km in diameter would be fabricated. It would carry a maximum current of 50.0kA through each winding of a 150-turn Nb3Sn solenoid. (a) If the inductance of this huge coil were 50.0 H, what is the total stored energy? (b) What is the compressive force per meter acting between two adjacent windings 0.250 m apart? [Hint: Because the radius of the coil is so large, the magnetic eld created by one winding and acting on an adjacent turn can be considered to be that of a long, straight wire.]

A conducting rod of length  moves on two horizontal frictionless rails, as in Figure P20.18. A constant force of magnitude 1.00 N moves the bar at a uniform speed of 2.00 m/s through a magnetic eld 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?

The square loop in Figure P20.58 is made of wires with a total series resistance of 10.0 . It is placed in a uniform 0.100-T magnetic 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 bolt of lightning depicted in Figure P20.59 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  s, what is the average voltage induced in the coil? Assume that the distance to the center of the coil determines the average magnetic eld at the coils position. Treat the lightning bolt as a long, vertical wire.

The wire shown in Figure P20.60 is bent in the shape of a tent with   60 and L  1.5 m, and is placed in a uniform magnetic eld of 0.30 T directed perpendicular to the tabletop. The wire is hinged at points a and b. If the tent is flattened out on the table in 0.10 s, what is the average induced emf in the wire during this time?

The magnetic eld shown in Figure P20.61 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 eld increases to 100 mT in 4.00  103 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  104  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 P20.62. If the current in the solenoid is increasing at a constant rate of 270 A/s, what is the induced current in the ring? Assume that the magnetic eld produced by the solenoid over the area at the end of the solenoid is one-half as strong as the eld at the center of the solenoid. Assume also that the solenoid produces a negligible eld outside its cross-sectional area.

In Figure P20.63, 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 signicant resistance in the circuit is R.) A uniform magnetic eld B  0.800 T is directed vertically downwards. (a) Find the induced current I in the resistor. (b) What horizontal force 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?

In 1832, Faraday proposed that the apparatus shown in Figure P20.64 could be used to generate electric current from the water owing in the Thames River.2 Two conducting plates of length a and width b are placed facing one another on opposite sides of the river, a distance w apart and immersed entirely. The ow velocity of the river is , and the vertical component of Earths magnetic eld is B. Show that the current in the load resistor R is where  is the resistivity of the water. (b) Calculate the short-circuit current (R  0) if a  100 m, b  5.00 m, v  3.00 m/s, B  50.0  T, and   100 m.

A horizontal wire is free to slide on the vertical rails of a conducting frame, as in Figure P20.65 (page 692). The wire has mass m and length , and the resistance of the circuit is R. If a uniform magnetic eld is directed perpendicular to the frame, what is the terminal speed of the wire as it falls under the force of gravity? (Neglect friction.)

A one-turn coil of wire of area 0.20 m2 and resistance 0.25  is in a magnetic eld that varies with time as shown in Figure P20.66a. The magnetic ux through the coil at t  0 is as shown in Figure P20.66b. (a) When is the induced current the largest? (b) When is it zero? (c) Is the induced current always in the same direction? (d) Find the direction (clockwise or counterclockwise) and magnitude of the current between times t  0 and t  2.0 s, between t  2.0 s and t  4.0 s, and between t  4.0 s and t  6.0 s.

Experimenting with induced currents is not easy. For small magnets and small coils of wire, the resulting induced currents are so small that they are difcult to detect. Thus, you may have to try this exercise several times before you are satised with your results. Wind a coil of wire on a cardboard mailing tube. Use insulated wire with as small a diameter as possible, because you need as many turns as possible on the coil. Connect the coil to a ashlight bulb, and see if you can get it to light by moving a bar magnet into and out of the coil in rapid succession. Why does the speed of movement make a difference? If you are unsuccessful, place two magnets side by side and repeat the experiment. After you have nished experimenting with the bulb, ask your instructor to let you use a galvanometer as a current detector. These devices are capable of measuring very small currents, and they have the added advantage of detecting the direction of the current in a circuit. Use your equipment to observe or test the following: (a) Does the magnitude of the induced current depend on the speed of movement of the magnet? (b) Can you induce a current by holding the magnet still and moving the coil over it? (c) Does the direction of the current depend on whether the magnet is pushed in or pulled out of the coil? (d) Does the direction of the current depend on whether the inserted pole of the magnet is the north pole or the south pole? (e) Can you predict the direction of the current by using Lenzs law? (f) Replace your bar magnet with the electromagnet you constructed in the last chapter, and repeat the preceding observations.

As explained in the text, a cassette tape is made up of tiny particles of metal oxide attached to a long plastic strip. Pull a tape out of a cassette that you do not mind destroying, and see if it is repelled or attracted by a refrigerator magnet. Also, try this with an expendable oppy computer disk.

This experiment takes steady hands, a dime, and a strong magnet. After verifying that a dime is not attracted to the magnet, carefully balance the coin on its edge. (This will not work with other coins, because they require too much force to topple them.) Hold one pole of the magnet within a millimeter of the face of the dime, but do not make contact with it. Now very rapidly pull the magnet straight back away from the coin. Which way does the dime tip? Does the coin fall the same way most of the time? Explain what is going on in terms of Lenzs law.