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**Hugh D. Young, College Physics, 9th edition, Addison Wesley 2011**

**Chapter 21**

Multiple-Choice Problems

1. A square loop of wire is pulled upward out of the space between the poles of a magnet, as shown in Figure 21.35. As this is done, the current induced in this loop, as viewed from the N pole of the magnet, will be directed

A. clockwise. B. counterclockwise. C. zero.

2. The two solenoids in Figure 21.36 are coaxial and fairly close to each other. While the resistance of the variable resistor in the left-hand solenoid is increased at a constant rate, the induced current through the resistor R will

A. flow from a to b. B. flow from b to a. C. be zero because the rate is constant.

3. A metal ring is oriented with the plane of its area perpendicular to a spatially uniform magnetic field that increases at a steady rate. After the radius of the ring is doubled, while the rate of increase of the field is cut in half, the emf induced in the ring

A. remains the same.

B. increases by a factor of 2.

C. increases by a factor of 4.

D. decreases by a factor of 2.

4. The slide wire of the variable resistor in Figure 21.37 is moved steadily to the right, increasing the resistance in the circuit. While this is being done, the current induced in the small circuit A is directed

A. clockwise. B. counterclockwise. C. zero.

5. The slide wire on the variable resistor in Figure 21.38 is moved steadily to the left. While this is being done, the current induced in the small circuit A is directed

A. clockwise. B. counterclockwise. C. zero.

6. A metal loop moves at constant velocity toward a long wire carrying a steady current I, as shown in Figure 21.39. The current induced in the loop is directed

A. clockwise. B. counterclockwise. C. zero.

7. The primary coil of an ideal transformer carries a current of 2.5 A, while the secondary coil carries a current of 5.0 A. The ratio of number of turns of wire in the primary to that in the secondary is

A. 1:1 B. 1:2 C. 2:1

8. A metal loop is held above the S pole of a bar magnet, as shown in Figure 21.40, when the magnet is
suddenly dropped from rest. Just after the magnet is dropped, the induced current in the loop, as viewed from above it, is directed

A. clockwise. B. counterclockwise. C. zero.

9. A steady current of 1.5 A flows through the solenoid shown in Figure 21.41. The current induced in the loop, as viewed from the right, is directed

A. clockwise. B. counterclockwise. C. zero.

10. A vertical bar moves horizontally at constant velocity through a uniform magnetic field, as shown in Figure 21.42. We observe that point b is at a higher potential than point a. We can therefore conclude that the magnetic field must have a component that is directed

A. vertically downward. C. perpendicular to the plane of the paper, outward.

B. vertically upward. D. perpendicular to the plane of the paper, inward.

11. The vertical loops A and C in Figure 21.43 are parallel to each other and are centered on the same horizontal line that is perpendicular to both of them. Just after the switch S is closed, loop A will

A. not be affected by loop C. B. be attracted by loop C.

C. be repelled by loop C. D. move upward.

E. move downward.

12. After the switch S in Figure 21.43 has been closed for a very long time, loop A will

A. not be affected by loop C. B. be attracted by loop C.

C. be repelled by loop C. D. move upward.

E. move downward.

13. After the switch S in the circuit in Figure 21.44 is closed,

A. The current is zero 1.5 ms (one time constant) later.

B. The current is zero for a very long time afterward.

C. The largest current is 5.0 A and it occurs just after S is closed.

D. The largest current is 5.0 A and it occurs a very long time after S has been closed.

14. A square metal loop is pulled to the right at a constant velocity perpendicular to a uniform

magnetic field, as shown in Figure 21.45. The current induced in this loop is directed

A. clockwise. B. counterclockwise. C. zero.

15. A metal loop is being pushed at a constant velocity into a uniform magnetic field, as shown in Figure 21.46, but is only partway into the field. As a result of this motion,

A. End a of the resistor R is at a higher potential than end b.

B. End b of the resistor R is at a higher potential than end a.

C. Ends a and b are at the same potential.

**Problems**

1. A circular area with a radius of 6.50 cm lies in the x-y plane. What is the magnitude of the magnetic flux through this circle due to a uniform magnetic field B = 0.230 T that points (a) in the +z direction? (b) at an angle of 53.1° from the +z direction? (c) in the +y direction?

2. The magnetic field B in a certain region is 0.128 T, and its direction is that of the +z axis in Figure 21.47. (a) What is the magnetic flux across the surface abcd in the figure? (b) What is the magnetic flux across the surface befc? (c) What is the magnetic flux across the surface aefd? (d) What is the net flux through all five surfaces that enclose the shaded volume?

3. An open plastic soda bottle with an opening diameter of 2.5 cm is placed on a table. A uniform 1.75 T magnetic field directed upward and oriented 25° from vertical encompasses the bottle. What is the total magnetic flux through the plastic of the soda bottle?

4. A single loop of wire with an area of 0.0900 m2 is in a uniform magnetic field that has an initial value of 3.80 T, is perpendicular to the plane of the loop, and is decreasing at a constant rate of 0.190 T/s (a) What emf is induced in this loop? (b) If the loop has a resistance of 0.600 Ω find the current induced in the loop.

5. A coil of wire with 200 circular turns of radius 3.00 cm is in a uniform magnetic field along the axis of the coil. The coil has R = 40.0 Ω. At what rate, in teslas per second, must the magnetic field be changing to induce a current of 0.150A in the coil?

6. In a physics laboratory experiment, a coil with 200 turns enclosing an area of 12 cm2 is rotated from a position where its plane is perpendicular to the earth’s magnetic field to one where its plane is parallel to the field. The rotation takes 0.040 s. The earth’s magnetic field at the location of the laboratory is 6.0 × 10-5 T (a) What is the total magnetic flux through the coil before it is rotated? After it is rotated? (b) What is the average emf induced in the coil?

7. A closely wound rectangular coil of 80 turns has dimensions of 25.0 cm by 40.0 cm. The plane of the coil is rotated from a position where it makes an angle of 37.0° with a magnetic field of 1.10 T to a position perpendicular to the field. The rotation takes 0.0600 s. What is the average emf induced in the coil?

8. A very long, straight solenoid with a cross-sectional area of 6.00 m2 is wound with 40 turns of wire per centimeter, and the windings carry a current of 0.250 A. A secondary winding of 2 turns encircles the solenoid at its center. When the primary circuit is opened, the magnetic field of the solenoid becomes zero in 0.0500 s. What is the average induced emf in the secondary coil?

9. A 30.0 cm × 60.0 cm rectangular circuit containing a 15 Ω resistor is perpendicular to a uniform magnetic field that starts out at 2.65 T and steadily decreases at 0.25 T/s (See Figure 21.48.) While this field is changing, what does the ammeter read?

10. A circular loop of wire with a radius of 12.0 cm is lying flat on a tabletop. A magnetic field of 1.5 T is directed vertically upward through the loop (Figure 21.49). (a) If the loop is removed from the field region in a time interval of 2.0 ms, find the average emf that will be induced in the wire loop during the extraction process. (b) If the loop is viewed looking down on it from above, is the induced current in the loop clockwise or counterclockwise?

11. A flat, square coil with 15 turns has sides of length 0.120 m. The coil rotates in a magnetic field of 0.0250 T. (a) What is the angular velocity of the coil if the maximum emf produced is 20.0 mV? (b) What is the average emf at this angular velocity?

12. A cardboard tube is wrapped with two windings of insulated wire, as shown in Figure 21.50. Is the induced current in the resistor R directed from left to right or from right to left in the following circumstances? The current in winding A is directed (a) from a to b and is increasing, (b) from b to a and is decreasing, (c) from b to a and is increasing, and (d) from b to a and is constant.

13. A circular loop of wire is in a spatially uniform magnetic field, as shown in Figure 21.51. The magnetic field is directed into the plane of the figure. Determine the direction (clockwise or counterclockwise) of the induced current in the loop when (a) B is increasing; (b) B is decreasing; (c) B is constant with a value of B0. Explain your reasoning.

14. Using Lenz’s law, determine the direction of the current in resistor ab of Figure 21.52 when (a) switch S is opened after having been closed for several minutes; (b) coil B is brought closer to coil A with the switch closed; (c) the resistance of R is decreased while the switch remains closed.

15. A solenoid carrying a current I is moving toward a metal ring, as shown in Figure 21.53. In what direction, clockwise or counterclockwise (as seen from the solenoid) is a current induced in the ring? In what direction will the induced current be if the solenoid now stops moving toward the ring, but the current in it begins to decrease?

16. A metal bar is pulled to the right perpendicular to a uniform magnetic field. The bar rides on parallel metal rails connected through a resistor, as shown in Figure 21.54, so the apparatus makes a complete circuit. Find the direction of the current induced in the circuit in two ways: (a) by looking at the magnetic force on the charges in the moving bar and (b) using Lenz’s law.

17. Two closed loops A and C are close to a long wire carrying a current I. (See Figure 21.55.) Find the direction (clockwise or counterclockwise) of the current induced in each of these loops if I is steadily increasing.

18. A bar magnet is held above a circular loop of wire as shown in Figure 21.56. Find the direction (clockwise or counterclockwise, as viewed from below the loop) of the current induced in this loop in each of the following cases. (a) The loop is dropped. (b) The magnet is dropped. (c) Both the loop and magnet are dropped at the same instant.

19. The current in Figure. 21.57 obeys the equation Ie = I0e-2bt where b > 0. Find the direction (clockwise or counterclockwise) of the current induced in the round coil for t > 0.

20. A bar magnet is close to a metal loop. When this magnet is suddenly moved to the left away from the loop, as shown in Figure 21.58, a counterclockwise current is induced in the coil, as viewed by an observer looking through the coil toward the magnet. Identify the north and south poles of the magnet.

21. A very thin 15.0 cm copper bar is aligned horizontally along the east–west direction. If it moves horizontally from south to north at 11.5 m/s in a vertically upward magnetic field of 1.22 T, (a) what potential difference is induced across its ends, and (b) which end (east or west) is at a higher potential? (c) What would be the potential difference if the bar moved from east to west instead?

22. When a thin 12.0 cm iron rod moves with a constant velocity of 4.50 m/s perpendicular to the rod in the direction shown in Figure 21.59, the induced emf across its ends is measured to be 0.450V. (a) What is the magnitude of the magnetic field? (b) Which point is at a higher potential, a or b? (c) If the bar is rotated clockwise by 90° in the plane of the paper, but keeps the same velocity, what is the potential difference induced across its ends?

23. You’re driving at 95 km/h in a direction 35° east of north, in a region where the earth’s magnetic field of 5.5 × 10-5 T is horizontal and points due north. If your car measures 1.5 m from its underbody to its roof, calculate the induced emf between roof and underbody. (You can assume the sides of the car are straight and vertical.) Is the roof of the car at a higher or lower potential than the underbody?

24. A 1.41 m bar moves through a uniform, 1.20 T magnetic field with a speed of 2.50 m/s (Figure 21.60). In each case, find the emf induced between the ends of this bar and identify which, if any, end (a or b) is at the higher potential. The bar moves in the direction of (a) the +x-axis; (b) the –y-axis;

(c) the +z-axis. (d) How should this bar move so that the emf across its ends has the greatest possible value with b at a higher potential than a, and what is this maximum emf?

25. The conducting rod ab shown in Figure 21.61 makes frictionless contact with metal rails ca and db. The apparatus is in a uniform magnetic field of 0.800 T, perpendicular to the plane of the figure. (a) Find the magnitude of the emf induced in the rod when it is moving toward the right with a speed 7.50 m/s. (b) In what direction does the current flow in the rod? (c) If the resistance of the circuit abdc is a constant 1.50 Ω, find the magnitude and direction of the force required to keep the rod moving to the right with a constant speed of 7.50 m/s.

26. Measuring blood flow. Blood contains positive and negative ions and therefore is a conductor. A blood vessel, therefore, can be viewed as an electrical wire. We can even picture the flowing blood as a series of parallel conducting slabs whose thickness is the diameter d of the vessel moving with speed v (a) If the blood vessel is placed in a magnetic field B perpendicular to the vessel, as in the figure, show that the motional potential difference induced across it is E = vBd. (b) If you expect that the blood will be flowing at for a vessel 5.0 mm in diameter, what strength of magnetic field will you need to produce a potential difference of 1.0 mV? (c) Show that the volume rate of flow (R) of the blood is equal to R = πEd/(4B).

27. A toroidal solenoid has a mean radius of 10.0 cm and a cross-sectional area of 4.00 cm2 and is wound uniformly with 100 turns. A second coil with 500 turns is wound uniformly on top of the first. What is the mutual inductance of these coils?

28. A 10.0-cm-long solenoid of diameter 0.400 cm is wound uniformly with 800 turns. A second coil with 50 turns is wound around the solenoid at its center. What is the mutual inductance of the combination of the two coils?

29. Two coils are wound around the same cylindrical form, like the coils in Example 21.8. When the current in the first coil is decreasing at a rate of 0.242 A/s, the induced emf in the second coil has magnitude 1.65 mV. (a) What is the mutual inductance of the pair of coils? (b) If the second coil has 25 turns, what is the average magnetic flux through each turn when the current in the first coil equals 1.20 A? (c) If the current in the second coil increases at a rate of 0.360 A/s, what is the magnitude of the induced emf in the first coil?

30. One solenoid is centered inside another. The outer one has a length of 50.0 cm and contains 6750 coils, while the coaxial inner solenoid is 3.0 cm long and 0.120 cm in diameter and contains 15 coils. The current in the outer solenoid is changing at 37.5 A/s. (a) What is the mutual inductance of these solenoids? (b) Find the emf induced in the inner solenoid.

31. Two toroidal solenoids are wound around the same form so that the magnetic field of one passes through the turns of the other. Solenoid 1 has 700 turns, and solenoid 2 has 400 turns. When the current in solenoid 1 is 6.52 A, the average flux through each turn of solenoid 2 is 0.0320 Wb. (a) What is the mutual inductance of the pair of solenoids? (b) When the current in solenoid 2 is 2.54 A, what is the average flux through each turn of solenoid 1?

32. A 4.5 mH toroidal inductor has 125 identical equally spaced coils. (a) If it carries an 11.5 A current, how much magnetic flux passes through each of its coils? (b) If the potential difference across its ends is 1.16 V, at what rate is he current in it changing?

33. At the instant when the current in an inductor is increasing at a rate of 0.0640 A/s, the magnitude of the self-induced emf is 0.0160 V. What is the inductance of the inductor?

34. An inductor has inductance of 0.260 H and carries a current that is decreasing at a uniform rate of 18.0 mA/s. Find the self-induced emf in this inductor.

35. A 2.50 mH toroidal solenoid has an average radius of 6.00 cm and a cross-sectional area of 2.00 cm2 (a) How many coils does it have? (Make the same assumption as in Example 21.10.) (b) At what rate must the current through it change so that a potential difference of 2.00 V is developed across its ends?

36. Self-inductance of a solenoid. A long, straight solenoid has N turns, a uniform cross-sectional area A, and length l. Use the definition of self-inductance expressed by Equation 21.13 to show that the inductance of this solenoid is given approximately by the equation L = μ0AN2/l. Assume that the magnetic field is uniform inside the solenoid and zero outside.

37. When the current in a toroidal solenoid is changing at a rate of 0.0260 A/s, the magnitude of the induced emf is 12.6 mV. When the current equals 1.40 A, the average flux through each turn of the solenoid is 0.00285 Wb. How many turns does the solenoid have?

38. A transformer consists of 275 primary windings and 834 secondary windings. If the potential difference across the primary coil is 25.0 V, (a) what is the voltage across the secondary coil, and (b) what is the effective load resistance of the secondary coil if it is connected across a 125 Ω resistor?

39. Off to Europe! You plan to take your hair blower to Europe, where the electrical outlets put out 240 V instead of the 120V seen in the United States. The blower puts out 1600 W at 120 V. (a) What could you do to operate your blower via the 240 V line in Europe? (b) What current will your blower draw from a European outlet? (c) What resistance will your blower appear to have when operated at 240 V?

40. You need a transformer that will draw 15 W of power from a 220 V (rms) power line, stepping the voltage down to 6.0 V (rms). (a) What will be the current in the secondary coil? (b) What should be the resistance of the secondary circuit? (c) What will be the equivalent resistance of the input circuit?

41. A step-up transformer. A transformer connected to a 120 V (rms) ac line is to supply 13,000 V (rms) for a neon sign. To reduce the shock hazard, a fuse is to be inserted in the primary circuit and is to blow when the rms current in the secondary circuit exceeds 8.50 mA. (a) What is the ratio of secondary to primary turns of the transformer? (b) What power must be supplied to the transformer when the rms secondary current is 8.50 mA? (c)What current rating should the fuse in the primary circuit have?

42. An air-filled toroidal solenoid has a mean radius of 15.0 cm and a cross-sectional area of 5.00 cm2. When the current is 12.0 A, the energy stored is 0.390 J. How many turns does the winding have?

43. Energy in a typical inductor. (a) How much energy is stored in a 10.2 mH inductor carrying a 1.15 A current? (b) How much current would such an inductor have to carry to store 1.0 J of energy? Is this a reasonable amount of current for ordinary laboratory circuit elements?

44. (a) What would have to be the self-inductance of a solenoid for it to store 10.0 J of energy when a 1.50 A current runs through it? (b) If this solenoid’s cross-sectional diameter is 4.00 cm, and if you could wrap its coils to a density of 10 coils/mm, how long would the solenoid be? (See problem 36.) Is this a realistic length for ordinary laboratory use?

45. A solenoid 25.0 cm long and with a cross-sectional area of 0.500 cm2 contains 400 turns of wire and carries a current of 80.0 A. Calculate: (a) the magnetic field in the solenoid; (b) the energy density in the magnetic field if the solenoid is filled with air; (c) the total energy contained in the coil’s magnetic field (assume the field is uniform); (d) the inductance of the solenoid.

46. Large inductors have been proposed as energy-storage devices. (a) How much electrical energy is converted to light and thermal energy by a 200 W lightbulb in one day? (b) If the amount of energy calculated in part (a) is stored in an inductor in which the current is 80.0 A, what is the inductance?

47. When a certain inductor carries a current I, it stores 3.0 mJ of magnetic energy. How much current (in terms of I ) would it have to carry to store 9.0 mJ of energy?

48. A 12.0 V dc battery having no appreciable internal resistance, a 150.0 Ω resistor, an 11.0 mH inductor, and an open switch are all connected in series. After the switch is closed, what are (a) the time constant for this circuit, (b) the maximum current that flows through it, (c) the current 73.3 μs after the switch is closed, and (d) the maximum energy stored in the inductor?

49. An inductor with an inductance of 2.50 H and a resistor with a resistance of 8.00 Ω are connected to the terminals of a battery with an emf of 6.00 V and negligible internal resistance. Find (a) the initial rate of increase of the current in the circuit, (b) the initial potential difference across the inductor, (c) the current 0.313 s after the circuit is closed, and (d) the maximum current.

50. In Figure 21.63, both switches S1 and S2 are initially open. S1 is then closed and left closed until a constant current is established. Then S2 is closed just as S1, is opened, taking the battery out of the circuit. (a) What is the initial current in the resistor just after S2 is closed and S1 is opened? (b) What is the time constant of the circuit? (c) What is the current in the resistor after a large number of time constants have elapsed?

51. In the circuit shown in Figure 21.64, the battery and the inductor have no appreciable internal resistance and there is no current in the circuit. After the switch is closed, find the readings of the ammeter (A) and voltmeters (V1 and V2) (a) the instant after the switch is closed; (b) after the switch has been closed for a very long time. (c) Which answers in parts (a) and (b) would change if the inductance were 24.0 mH instead?

52. A 35.0 V battery with negligible internal resistance, a 50.0 V resistor, and a 1.25 mH inductor with negligible resistance are all connected in series with an open switch. The switch is suddenly closed. (a) How long after closing the switch will the current through the inductor reach one-half of its maximum value? (b) How long after closing the switch will the energy stored in the inductor reach one-half of its maximum value?

53. A 1.50 mH inductor is connected in series with a dc battery of negligible internal resistance, a 0.750 kΩ resistor, and an open switch. How long after the switch is closed will it take for (a) the current in the circuit to reach half of its maximum value, (b) the energy stored in the inductor to reach half of its maximum value?

54. A 12.0 μF capacitor and a 5.25 mH inductor are connected in series with an open switch. The capacitor is initially charged to 6.20 μC. What is the angular frequency of the charge oscillations in the capacitor after the switch is closed?

55. A 5.00 μF capacitor is initially charged to a potential of 16.0 V. It is then connected in series with a 3.75 mH inductor. (a) What is the total energy stored in this circuit? (b) What is the maximum current in the inductor? What is the charge on the capacitor plates at the instant the current in the inductor is maximal?

56. A 15.0 μF capacitor is charged to 175 μC and then connected across the ends of a 5.00 mH inductor. (a) Find the maximum current in the inductor. At the instant the current in the inductor is maximal, how much charge is on the capacitor plates? (b) Find the maximum potential across the capacitor. At this instant, what is the current in the inductor? (c) Find the maximum energy stored in the inductor. At this instant, what is the current in the circuit?

57. An inductor is connected to the terminals of a battery that has an emf of 12.0 V and negligible internal resistance. The current is 4.86 mA at 0.725 ms after the connection is completed. After a long time the current is 6.45 mA. What are (a) the resistance R of the inductor and (b) the inductance L of the inductor?

58. A rectangular circuit is moved at a constant velocity of 3.0 m/s into, through, and then out of a uniform 1.25 T magnetic field, as shown in Figure 21.65. The magnetic field region is considerably wider than 50.0 cm. Find the magnitude and direction (clockwise or counterclockwise) of the current induced in the circuit as it is (a) going into the magnetic field, (b) totally within the magnetic field, but still moving, and (c) moving out of the field. (d) Sketch a graph of the current in this circuit as a function of time, including the preceding three cases.

59. The rectangular loop in Figure 21.66, with area A and resistance R, rotates at uniform angular velocity ω about the y axis. The loop lies in a uniform magnetic field B in the direction of the x axis. Sketch graphs of the following quantities, as functions of time, letting t = 0 when the loop is in the position shown in the figure: (a) the magnetic flux through the loop, (b) the rate of change of flux with respect to time, (c) the induced emf in the loop, (d) the induced emf if the angular velocity is doubled.

60. A flexible circular loop 6.50 cm in diameter lies in a magnetic field with magnitude 0.950 T, directed into the plane of the page as shown in Figure 21.67. The loop is pulled at the points indicated by the arrows, forming a loop of zero area in 0.250 s. (a) Find the average induced emf in the circuit. (b) What is the direction of the current in R: from a to b or from b to a? Explain your reasoning.

61. An electromagnetic car alarm. Your latest invention is a car alarm that produces sound at a particularly annoying frequency of 3500 Hz. To do this, the car-alarm circuitry must produce an alternating electric current of the same frequency. That’s why your design includes an inductor and a capacitor in series. The maximum voltage across the capacitor is to be 12.0 V (the same voltage as the car battery). To produce a sufficiently loud sound, the capacitor must store 0.0160 J of energy. What values of capacitance and inductance should you choose for your car-alarm circuit?

62. In the circuit shown in Figure 21.68, S1 has been closed for a long enough time so that the current reads a steady 3.50 A. Suddenly, S2 is closed and S1 is opened at the same instant. (a) What is the maximum charge that the capacitor will receive? (b) What is the current in the inductor at this time?

63. Consider the circuit in Figure 21.69. (a) Just after the switch is closed, what is the current through each of the resistors? (b) After the switch has been closed a long time, what is the current through each resistor? (c) After S has been closed a long time, it is opened again. Just after it is opened, what is the current through the 20.0 Ω resistor?

**Chapter 22**

**Multiple-Choice Problems**

1. A piece of electrical equipment in an ac circuit draws a root-mean-square current of 5.00 A. The average current over each cycle is

A. 5√2 = 7.07 A B. 5.00 A. C. 5/√2 = 3.54 A D. 0.

2. A sinusoidal current is described by i = Icosωt, where ω = 1.57 rad/s. At some time t', where 2s < t' < 4s, the current is +3.0 A. Which phasor can represent the current at time t'?

3. A lightbulb is the resistance in a series R–L–C circuit having an ac voltage source v = Vcosωt. As the frequency of the source is adjusted closer and closer to the value 1/√(LC) what happens to the brightness of the bulb?

A. It increases. B. It decreases. C. It does not change.

4. A series R–L–C ac circuit with a sinusoidal voltage source of angular frequency ω has a total reactance X. If this frequency is doubled, the reactance becomes

A. 4X. B. 2X. C. X/2 D. X/4 E. none of the above.

5. In a series R–L–C circuit powered by an ac sinusoidal voltage source, which phasor diagram best illustrates the relationship between the current i and the potential drop vR across the resistor?

6. In a series R–L–C circuit powered by an ac sinusoidal voltage source, which phase diagram best illustrates the relationship between the current i and the potential drop vc across the capacitor?

7. In a series R–L–C circuit powered by an ac sinusoidal voltage source, which phase diagram best illustrates the relationship between the current i and the potential drop vL across the inductor?

8. A series circuit contains an inductor, a resistor, a capacitor, and a sinusoidal voltage source of angular frequency ω. If we double this frequency (there may be more than one correct choice),

A. the inductive reactance is doubled.

B. the capacitive reactance is doubled.

C. the total reactance is doubled.

D. the impedance is doubled.

9. In order to double the resonance frequency of a series R–L–C ac circuit, you could

A. double both the inductance and capacitance.

B. double the resistance.

C. cut the resistance in half.

D. cut both the inductance and capacitance in half.

**Problems**

1. You have a special lightbulb with a very delicate wire filament. The wire will break if the current in it ever exceeds 1.50 A, even for an instant. What is the largest root-mean-square current you can run through this bulb?

2. The plate on the back of a certain computer scanner says that the unit draws 0.34 A of current from a 120 V, 60 Hz line. Find (a) the root-mean-square current, (b) the current amplitude, (c) the average current, and (d) the average square of the current.

3. A capacitance C and an inductance L are operated at the same angular frequency. (a) At what angular frequency will they have the same reactance? (b) If L = 5.00 mH and C = 3.50 μF, what is the numerical value of the angular frequency in part (a), and what is the reactance of each element?

4. (a) Compute the reactance of a 0.450 H inductor at frequencies of 60.0 Hz and 600 Hz. (b) Compute the reactance of a 2.50 μF capacitor at the same frequencies. (c) At what frequency is the reactance of a 0.450 H inductor equal to that of a 2.50 μF capacitor?

5. A radio inductor. You want the current amplitude through a 0.450-mH inductor (part of the circuitry for a radio receiver) to be 2.60 mA when a sinusoidal voltage with amplitude 12.0 V is applied across the inductor. What frequency is required?

6. A 2.20 μF capacitor is connected across an ac source whose voltage amplitude is kept constant at 60.0 V, but whose frequency can be varied. Find the current amplitude when the angular frequency is (a) 100 rad/s (b) 1000 rad/s (c) 10000 rad/s.

7. The voltage amplitude of an ac source is 25.0 V, and its angular frequency is 1000 rad/s. Find the current amplitude if the capacitance of a capacitor connected across the source is (a) 0.0100 μF (b) 1.00 μF (c) 100 μF.

8. Find the current amplitude if the self-inductance of a resistanceless inductor that is connected across the source of the previous problem is (a) 0.0100 H, (b) 1.00 H, (c) 100 H.

9. A sinusoidal ac voltage source in a circuit produces a maximum voltage of 12.0 V and an rms current of 7.50 mA. Find (a) the voltage and current amplitudes and (b) the rms voltage of this source.

10. A 65 Ω resistor, an 8.0 μF capacitor, and a 35 mH inductor are connected in series in an ac circuit. Calculate the impedance for a source frequency of (a) 300 Hz and (b) 30.0 Hz.

11. In an R–L–C series circuit, the rms voltage across the resistor is 30.0 V, across the capacitor it is 90.0 V, and across the inductor it is 50.0 V. What is the rms voltage of the source?

12. A 1500 Ω resistor is connected in series with a 350 mH inductor and an ac power supply. At what frequency will this combination have twice the impedance that it has at 120 Hz?

13. (a) Compute the impedance of a series R–L–C circuit at angular frequencies of 1000, 750, and 500 rad/s. Take R = 200 Ω, L = 0.900 H, and C = 2.00 μF. (b) Describe how the current amplitude varies as the angular frequency of the source is slowly reduced from 1000 rad/s to 500 rad/s (c) What is the phase angle of the source voltage with respect to the current when ω = 1000 rad/s? (d) Construct a phasor diagram when ω = 1000 rad/s.

14. A 200 Ω resistor is in series with a 0.100 H inductor and a 0.500 μF capacitor. Compute the impedance of the circuit and draw the phasor diagram (a) at a frequency of 500 Hz, (b) at a frequency of 1000 Hz. In each case, compute the phase angle of the source voltage with respect to the current and state whether the source voltage lags or leads the current.

15. The power of a certain CD player operating at 120 Vrms is 20.0 W. Assuming that the CD player behaves like a pure resistance, find (a) the maximum instantaneous power, (b) the rms current, and (c) the resistance of this player.

16. A series R–L–C circuit is connected to a 120 Hz ac source that has Vrms = 80.0 V. The circuit has a resistance of 75.0 Ω and an impedance of 105 Ω at this frequency. What average power is delivered to the circuit by the source?

17. The circuit in Problem 13 carries an rms current of 0.250 A with a frequency of 100 Hz. (a) What is the average rate at which electrical energy is converted to heat in the resistor? (b) What average power is delivered by the source? (c) What is the average rate at which electrical energy is dissipated (converted to other forms) in the capacitor? in the inductor?

18. A series ac circuit contains a 250 Ω resistor, a 15 mH inductor, a 3.5 μH capacitor, and an ac power source of voltage amplitude 45 V operating at an angular frequency of 360 rad/s (a) What is the power factor of this circuit? (b) Find the average power delivered to the entire circuit. (c) What is the average power delivered to the resistor, to the capacitor, and to the inductor?

19. An ac series R–L–C circuit contains a 120 Ω resistor, a 2.0 μF capacitor, and a 5.0 mH inductor. Find (a) the resonance angular frequency and (b) the length of time that each cycle lasts at the resonance angular frequency.

20. (a) At what angular frequency will a 5.00 μF capacitor have the same reactance as a 10.0 mH inductor? (b) If the capacitor and inductor in part (a) are connected in an L–C circuit, what will be the resonance angular frequency of that circuit?

21. In an R-L-C series circuit, R = 150 Ω, L = 0.750 H, and C = 0.0180 μF. The source has voltage amplitude and a frequency equal to the resonance frequency of the circuit. (a) What is the power factor? (b) What is the average power delivered by the source? (c) The capacitor is replaced by one with C = 0.0360 μF and the source frequency is adjusted to the new resonance value. Then what is the average power delivered by the source?

22. You need to make a series ac circuit having a resonance angular frequency of using a resistor, a capacitor, and an inductor. (a) What should be the inductance of the inductor, and (b) what is the impedance of this circuit when you use it with an ac voltage source having an angular frequency of 1525 Hz?

23. A series circuit consists of an ac source of variable frequency, a 115 Ω resistor, a 1.25 μF capacitor, and a 4.50 mH inductor. Find the impedance of this circuit when the angular frequency of the ac source is adjusted to (a) the resonance angular frequency, (b) twice the resonance angular frequency, and (c) half the resonance angular frequency.

24. In a series R–L–C circuit, R = 400 Ω, L = 0.350 H, and C = 0.0120 μF (a) What is the resonance angular frequency of the circuit? (b) The capacitor can withstand a peak voltage of 550 V. If the voltage source operates at the resonance frequency, what maximum voltage amplitude can it have if the maximum capacitor voltage is not exceeded?

25. In a series R–L–C circuit, L = 0.200 H, C = 80.0 μF, and the voltage amplitude of the source is 240 V. (a) What is the resonance angular frequency of the circuit? (b) When the source operates at the resonance angular frequency, the current amplitude in the circuit is 0.600 A. What is the resistance R of the resistor? (c) At the resonance frequency, what are the peak voltages across the inductor, the capacitor, and the resistor?

26. In an R–L–C series circuit, R = 300 Ω, L = 0.400 H, and C = 6.00 × 10-8 μF. When the ac source operates at the resonance frequency of the circuit, the current amplitude is 0.500 A. (a) What is the voltage amplitude of the source? (b) What is the amplitude of the voltage across the resistor, across the inductor, and across the capacitor? (c) What is the average power supplied by the source?

27. A 125 Ω resistor, an 8.50 μF capacitor, and an 1.2 mH inductor are all connected in parallel across an ac voltage source of variable frequency. (a) At what angular frequency will the impedance have its maximum value, and (b) what is that value?

28. For the circuit in Figure 22.23, R = 300 Ω, L = 0.500 H and C = 0.600 μF. The voltage amplitude of the source is 120 V. (a) What is the resonance frequency of the circuit? (b) Sketch the phasor diagram at the resonance frequency. (c) At the resonance frequency, what is the current amplitude through the source? (d) At the resonance frequency, what is the current amplitude through the resistor? Through the inductor? Through the branch containing the capacitor?

29. For the circuit in Figure 22.23, R = 200 Ω, L = 0.800, and C = 5.00 μF. When the source is operated at the resonance frequency, the current amplitude in the inductor is 0.400 A. Determine the current amplitude (a) in the branch containing the capacitor and (b) through the resistor.

30. (a) Use the phasor diagram for a parallel R–L–C circuit (see Figure 22.21) to show that the current amplitude I for the current i through the source is given by I = √(IR2 + (IC – IL)2) (b) Show that the result of part (a) can be written as I = V/Z with 1/Z = I = √(1/R2 + (ωC – 1/ωL)2).

31. A coil has a resistance of 48.0 Ω. At a frequency of 80.0 Hz, the voltage across the coil leads the current in it by 52.3°. Determine the inductance of the coil.

32. A large electromagnetic coil is connected to a 120 Hz ac source. The coil has resistance 400 Ω and at this source frequency the coil has inductive reactance 250 Ω. (a) What is the inductance of the coil? (b) What must the rms voltage of the source be if the coil is to consume an average electrical power of 800 W?

33. A parallel-plate capacitor having square plates 4.50 cm on each side and 8.00 mm apart is placed in series with an ac source of angular frequency 650 rad/s and voltage amplitude 22.5 V, a 75.0 Ω resistor, and an ideal solenoid that is 9.00 cm long, has a circular cross section 0.500 cm in diameter, and carries 125 coils per centimeter. What is the resonance angular frequency of this circuit?

34. At a frequency ω1 the reactance of a certain capacitor equals that of a certain inductor. (a) If the frequency is changed to ω2 = 2ω1, what is the ratio of the reactance of the inductor to that of the capacitor? Which reactance is larger? (b) If the frequency is changed to ω3 = ω1/3 what is the ratio of the reactance of the inductor to that of the capacitor? Which reactance is larger?

35. Five voltmeters, calibrated to read rms values, are connected as shown in Figure 22.22. Let R = 200 Ω, L = 0.400 H, and C = 6.00 μF. The source voltage amplitude is V = 30.0 V. What is the reading of each voltmeter if (a) ω = 200 rad/s (b) ω = 1000 rad/s?

36. Consider the circuit sketched in Figure 22.22. The source has a voltage amplitude of 240 V, R = 150 Ω and the reactance of the capacitor is 600 Ω. The voltage amplitude across the capacitor is 720 V. (a) What is the current amplitude in the circuit? (b) What is the impedance? (c) What two values can the reactance of the inductor have?

37. In a series R–L–C circuit, the components have the following values: L = 20.0 mH, C = 140 nF, and R = 350 Ω. The generator has an rms voltage of 120 V and a frequency of 1.25 kHz. Determine (a) the power supplied by the generator; and (b) the power dissipated in the resistor.

38. (a) Show that for an R–L–C series circuit the power factor is equal to R/Z. (b) Show that for any ac circuit, not just one containing pure resistance only, the average power delivered by the voltage source is given by Pav = I2rmsR.

39. In an R–L–C series circuit the magnitude of the phase angle is 54.0°, with the source voltage lagging the current. The reactance of the capacitor is 350 Ω and the resistor resistance is 180 Ω. The average power delivered by the source is 140 W. Find (a) the reactance of the inductor; (b) the rms current; (c) the rms voltage of the source.

40. In a series R–L–C circuit, R = 300 Ω, XC = 300 Ω, and XL= 500 Ω. The average power consumed in the resistor is 60.0 W. (a) What is the power factor of the circuit? (b) What is the rms voltage of the source?

41. In a series R–L–C circuit, the phase angle is 40.0°, with the source voltage leading the current. The reactance of the capacitor is 400 Ω and the resistance of the resistor is 200 Ω. The average power delivered by the source is 150 W. Find (a) the reactance of the inductor, (b) the rms current, (c) the rms voltage of the source.

42. A 100 Ω resistor, a 0.100 μF capacitor, and a 300.0 mH inductor are connected in series to a voltage source with amplitude 240 V. (a) What is the resonance angular frequency? (b) What is the maximum current in the resistor at resonance? (c) What is the maximum voltage across the capacitor at resonance? (d) What is the maximum voltage across the inductor at resonance? (e) What is the maximum energy stored in the capacitor at resonance? in the inductor?

43. Consider the same circuit as in the previous problem, with the source operated at an angular frequency of 400 rad/s (a) What is the maximum current in the resistor? (b) What is the maximum voltage across the capacitor? (c) What is the maximum voltage across the inductor? (d) What is the maximum energy stored in the capacitor? in the inductor?

44. What is the dc impedance of the electrode, assuming that it behaves as an ideal capacitor?

A. 0 B. Infinite C. √2 × 104 Ω D. √2 × 106 Ω

45. If the electrode is oscillated between two points 20 μ apart with a frequency of (5000/π) Hz, what is the impedance of the electrode?

A. 0 B. Inﬁnite C. √2 × 104 Ω D. √2 × 106 Ω

46. The signal from the oscillating electrode is fed into an amplifier, which reports the measured voltage as an rms value, Vrms. However, the number of interest for analyzing currents driven by the cell is the peak-to-peak voltage difference (Vpp), that is, the voltage difference between the two extremes of the electrode’s excursion. What is the value of Vpp in terms of Vrms?

A. Vrms/√2 B. Vrms/2√2 C. √2Vrms D. 2√2Vrms

**Chapter 23**

**Multiple-Choice Problems**

1. Light having a certain frequency, wavelength, and speed is traveling through empty space. If the frequency of this light were doubled, then

A. its wavelength would remain the same, but its speed would double.

B. its wavelength would remain the same, but its speed would be halved.

C. its wavelength would be halved, but its speed would double.

D. its wavelength would be halved, but its speed would remain the same.

E. both its speed and its wavelength would be doubled.

2. Unpolarized light with an original intensity passes through two ideal polarizers having their polarizing axes turned at 120° to each other. After passing through both polarizers, the intensity of the light is

A. (√3/2)Io B. (1/2)Io C. (√3/4)Io D. (1/4)Io E. (1/8)Io

3. Light travels from water (with index of refraction 1.33) into air (index of refraction 1.00). Which of the following statements about this light is true?

A. The light has the same frequency in the air as it does in the water.

B. The light travels faster in the air than in the water.

C. The light has the same wavelength in the air as it does in the water.

D. The light has the same speed in the air as in the water.

E. The wavelength of the light in the air is greater than the wavelength in the water.

4. If a sinusoidal electromagnetic wave with intensity has an electric field of amplitude E, then a wave of the same wavelength will have an electric field of amplitude

A. 4E B. 2√2E C. 2E D. √2E

5. A plane electromagnetic wave is traveling vertically downward with its magnetic field pointing northward. Its electric field must be pointing

A. toward the south.

B. toward the east.

C. toward the west.

D. vertically upward.

E. vertically downward.

6. Suppose that a reflective solar sail (see Example 23.5) is deployed not perpendicular to the sun’s rays but at some other angle. In what direction will the sail accelerate?

A. In the direction the sun’s rays are moving.

B. Perpendicular to the surface of the sail.

C. At an angle somewhere between that of the sun’s rays and the perpendicular to the surface of the sail.

D. The sail will not accelerate unless it is perpendicular to the sun’s rays.

7. The index of refraction, n, has which of the following range of values?

A. n ≥ 1 B. 0 ≤ n ≤ 1 C. n ≥ 0

8. A ray of light going from one material into another follows the path shown in Figure 23.47. What can you conclude about the relative indexes of refraction of these two materials?

A. na ≥ nb B. na > nb C. na < nb D. na ≤ nb

9. Which of the following statements about radio waves, infrared radiation, and x rays are correct?

A. They all have the same wavelength in vacuum.

B. They all have the same frequency in vacuum.

C. They all have exactly the same speed as visible light in vacuum.

D. The short-wavelength x rays travel faster through vacuum than the long-wavelength radio waves.

10. Two lasers each produce 2 mW beams. The beam of laser B is wider, having twice the cross-sectional area as the beam of laser A. Which of the following statements about these two laser beams are correct?

A. Both of the beams have the same average power.

B. Beam A has twice the intensity of beam B.

C. Beam B has twice the intensity of beam A.

D. Both beams have the same intensity.

11. A ray of light follows the path shown in Figure 23.48 as it reaches the boundary between two transparent materials. What can you conclude about the relative indexes of refraction of these two materials?

A. n1 ≥ n2 B. n1 > n2 C. n1 < n2 D. n1 ≤ n2

12. A light beam has a wavelength of 300 nm in a material of refractive index 1.5. In a material of refractive index 3.0, its wavelength will be

A. 450 nm B. 300 nm C. 200 nm D. 150 nm E. 100 nm

13. A light beam has a frequency of 300 MHz in a material of refractive index 1.5. In a material of refractive index 3.0, its frequency will be

A. 450 MHz. B. 300 MHz. C. 200 MHz. D. 150 MHz. E. 100 MHz.

14. You are sunbathing in the late afternoon when the sun is relatively low in the western sky. You are lying flat on your back, looking straight up through Polaroid sunglasses. To minimize the amount of light reaching your eyes, you should lie with your feet pointing in what direction?

A. north. B. east. C. south. D. west. E. some other direction.

15. A beam of light takes time t to travel a distance L in a certain liquid. If we now add water to the liquid to reduce its index of refraction by half, the time for the beam to travel the same distance will be

A. 2t. B. √2t C. t/√2 D. t/2

**Problems**

1. When a solar flare erupts on the surface of the sun, how many minutes after it occurs does its light show up in an astronomer’s telescope on earth?

2. TV ghosting. In a TV picture, faint, slightly offset ghost images are formed when the signal from the transmitter travels to the receiver both directly and indirectly after reflection from a building or some other large metallic mass. In a 25 inch set, the ghost is about 1.0 cm to the right of the principal image if the reflected signal arrives 0.60 μs after the principal signal. In this case, what is the difference in the distance traveled by the two signals?

3. a) How much time does it take light to travel from the moon to the earth, a distance of 384,000 km? (b) Light from the star Sirius takes 8.61 years to reach the earth. What is the distance to Sirius in kilometers?

4. A geostationary communications satellite orbits the earth directly above the equator at an altitude of 35,800 km. Calculate the time it takes for a signal to travel from a point on the equator to the satellite and back to the ground at another point on the equator exactly halfway around the earth.

5. Consider electromagnetic waves propagating in air. (a) Determine the frequency of a wave with a wavelength of (i) 5.0 km, (ii) 5.0 μm (iii) 5.0 nm. (b) What is the wavelength (in meters and nanometers) of (i) gamma rays of frequency 6.50 × 1021 Hz (ii) an AM station radio wave of frequency 590 kHz?

6. Most people perceive light having a wavelength between 630 nm and 700 nm as red and light with a wavelength between 400 nm and 440 nm as violet. Calculate the approximate frequency ranges for (a) violet light and (b) red light.

7. The electric field of a sinusoidal electromagnetic wave obeys the equation E = -(375 V/m)sin[(5.97 × 1015 rad/s)t + (1.99 × 107 rad/m)x] (a) What are the amplitudes of the electric and magnetic fields of this wave? (b) What are the frequency, wavelength, and period of the wave? Is this light visible to humans? (c) What is the speed of the wave?

8. A sinusoidal electromagnetic wave having a magnetic field of amplitude 1.25 μT and a wavelength of 432 nm is traveling in the +x direction through empty space. (a) What is the frequency of this wave? (b) What is the amplitude of the associated electric field? (c) Write the equations for the electric and magnetic fields as functions of x and t in the form of Equations (23.3).

9. Visible light. The wavelength of visible light ranges from 400 nm to 700 nm. Find the corresponding ranges of this light’s (a) frequency, (b) angular frequency, (c) wave number.

10. Ultraviolet radiation. There are two categories of ultraviolet light. Ultraviolet A (UVA) has a wavelength ranging from 320 nm to 400 nm. It is not so harmful to the skin and is necessary for the production of vitamin D. UVB, with a wavelength between 280 nm and 320 nm, is much more dangerous, because it causes skin cancer. (a) Find the frequency ranges of UVA and UVB. (b) What are the ranges of the wave numbers for UVA and UVB?

11. Medical x rays. Medical x rays are taken with electromagnetic waves having a wavelength around 0.10 nm. What are the frequency, period, and wave number of such waves?

12. Radio station WCCO in Minneapolis broadcasts at a frequency of 830 kHz. At a point some distance from the transmitter, the magnetic-field amplitude of the electromagnetic wave from WCCO is 4.82 × 10-11 T. Calculate (a) the wavelength, (b) the wave number, (c) the angular frequency, and (d) the electric-field amplitude.

13. A sinusoidal electromagnetic wave of frequency travels in vacuum in the +x-direction. The magnetic field is parallel to the y-axis and has amplitude (a) Find the magnitude and direction of the electric field. (b) Write the wave functions for the electric and magnetic fields in the form of Equations (23.3).

14. Consider each of the electric- and magnetic-field orientations given next. In each case, what is the direction of propagation of the wave? (a) E in the +x direction, B in the +y direction. (b) E in the –y direction, B in the +x direction. (c) E in the +z direction, B in the –x direction. (d) E in the +y direction, B in the -y direction.

15. An electromagnetic wave has a magnetic field given by B = (8.25 × 10-9 T) sin[(ωt + 1.38 × 104 rad/s)x], with the magnetic field in the +y direction. (a) In which direction is the wave traveling? (b) What is the frequency f of the wave? (c) Write the wave function for the electric field.

16. Laboratory lasers. He–Ne lasers are often used in physics demonstrations. They produce light of wavelength 633 nm and a power of 0.500 mW spread over a cylindrical beam 1.00 mm in diameter (although these quantities can vary). (a) What is the intensity of this laser beam? (b) What are the maximum values of the electric and magnetic fields? (c) What is the average energy density in the laser beam?

17. Fields from a lightbulb. We can reasonably model a 75 W incandescent lightbulb as a sphere 6.0 cm in diameter. Typically, only about 5% of the energy goes to visible light; the rest goes largely to nonvisible infrared radiation. (a) What is the visible light intensity (in W/m2) at the surface of the bulb? (b) What are the amplitudes of the electric and magnetic fields at this surface, for a sinusoidal wave with this intensity?

18. Threshold of vision. Under controlled darkened conditions in the laboratory, a light receptor cell on the retina of a person’s eye can detect a single photon (more on photons in Chapter 28) of light of wavelength 505 nm and having an energy of 3.94 × 10-19 J. We shall assume that this energy is absorbed by a single cell during one period of the wave. Cells of this kind are called rods and have a diameter of approximately 0.0020 mm. What is the intensity (in W/m2) delivered to a rod?

19. High-energy cancer treatment. Scientists are working on a new technique to kill cancer cells by zapping them with ultrahigh-energy (in the range of pulses of light that last for an extremely short time (a few nanoseconds). These short pulses scramble the interior of a cell without causing it to explode, as long pulses would do. We can model a typical such cell as a disk in diameter, with the pulse lasting for 4.0 ns with an average power of 2.0 × 1012 W. We shall assume that the energy is spread uniformly over the faces of 100 cells for each pulse. (a) How much energy is given to the cell during this pulse? (b) What is the intensity (in W/m2) delivered to the cell? (c) What are the maximum values of the electric and magnetic fields in the pulse?

20. At the floor of a room, the intensity of light from bright overhead lights is 8.00 W/m2. Find the radiation pressure on a totally absorbing section of the floor.

21. The intensity at a certain distance from a bright light source is 6.00W/m2. Find the radiation pressure (in pascals and in atmospheres) on (a) a totally absorbing surface and (b) a totally reflecting surface.

22. A sinusoidal electromagnetic wave from a radio station passes perpendicularly through an open window that has area 0.500 m2. At the window, the electric field of the wave has rms value 0.0200 V/m. How much energy does this wave carry through the window during a 30.0 s commercial?

23. Two sources of sinusoidal electromagnetic waves have average powers of 75 W and 150 W and emit uniformly in all directions. At the same distance from each source, what is the ratio of the maximum electric field for the 150 W source to that of the 75 W source?

24. Radiation falling on a perfectly reflecting surface produces an average pressure p. If radiation of the same intensity falls on a perfectly absorbing surface and is spread over twice the area, what is the pressure at that surface in terms of p?

25. A sinusoidal electromagnetic wave emitted by a cellular phone has a wavelength of 35.4 cm and an electric field amplitude of 5.40 × 10-2 V/m at a distance of 250 m from the antenna. Calculate: (a) the frequency of the wave; (b) the magnetic-field amplitude; (c) the intensity of the wave.

26. Two plane mirrors intersect at right angles. A laser beam strikes the first of them at a point 11.5 cm from their point of intersection, as shown in Figure 23.49. For what angle of incidence at the first mirror will this ray strike the midpoint of the second mirror (which is 28.0 cm long) after reflecting from the first mirror?

27. Three plane mirrors intersect at right angles. A beam of laser light strikes the first of them at an angle θ with respect to the normal. (a) Show that when this ray is reflected off of the other two mirrors and crosses the original ray, the angle α between these two rays will be α = 180° - 2θ (b) For what angle θ will the two rays be perpendicular when they cross?

28. Two plane mirrors A and B intersect at a 45° angle. Three rays of light leave point P (see Figure 23.51) and strike one of the mirrors. What is the subsequent path of each of the following rays until they no longer strike either of the mirrors? (a) Ray 1, which strikes A at 45° with respect to the normal. (b) Ray 2, which strikes B traveling perpendicular to mirror A. (c) Ray 3, which strikes B perpendicular to its surface.

29. Prove that when a ray of light travels at any angle into the corner formed by two mirrors placed at right angles to each other, the reflected ray emerges parallel to the original ray (see Figure 23.52).

30. A light beam travels at 1.94 × 108 m/s in quartz. The wavelength of the light in quartz is 355 nm. (a) What is the index of refraction of quartz at this wavelength? (b) If this same light travels through air, what is its wavelength there?

31. Using a fast-pulsed laser and electronic timing circuitry, you find that light travels 2.50 m within a plastic rod in 1.5 ns. What is the refractive index of the plastic?

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