Summary 843

SUMMARY The goal of Chapter 25 has been to understand the nature of electromagnetic induction and electromagnetic waves.

GENERAL PRINCIPLES

Electromagnetic Induction The magnetic flux measures the amount of magneti c field pass ing through a surface:

<I> ~ ABcose

Loop of area A

I s

lenz's law specifics that there is an induced current in a closed conducting loop if the magneti c flux through the loop is changing. The direc­ti on of the induced current is such tballhe induced mag­netic field opposes the change in flu x.

Currcnt mcter

IMPORTANT CONCEPTS

Faraday's law speci­fies the mag nitude of lhe induced emf in a closed loop:

E ~ I ~<~ I Multiply by N for an N-turn coil.

The size of the induced current is

E I ~ ­

R

Motional emf The photon model The

Electromagnetic Waves An elec tromagnetic wave is a sel f-sustaining osc illation of electric and magnetic fields.

The wave is a transverse wave with E, B, and v mutuall y perpe ndicular.

The wave propagates with speed

I Vcm = C = ---= 3.00 X 1011 m/s ~

The wavele ngth, freque ncy, and speed are related by

The amplitudes of the fi elds are re lated by

Eo = cBo

Photoll ellergy Wave lellgth

iJ Electromagnetic electromagnetic IOV) Gam ma rays (Ill)

X +++ X X

X X X

X X X /IV

X

F~ x V x

X X X

X X X

The motion of a conductor through a magneti c field produces a fo rce on the charges. The separation of charges leads to an emf:

E ~ vlB

APPLICATIONS

A changing nux in a solid conductor creates eddy currents.

--=-"

waves appear to be spectrum

made of discrete Electromagneti c units called photons. waves come in The energy of a pho- a wide range of ton of frequency f is wave lengths and

E ~ lif photon energies .

This photon view becomes increasingly important as the photon energy increases.

The plane of the electric field of an electro­magnetic wave defi nes its polarization. The intensity of polari zed li ght transmitted through a polarizing filter is given by Malus 's law:

1 = locos2(} 10

where (} is the angle between the e lectri c fi eld and the polarizer ax is.

~ l 1 x 104 X 10- 10

" X rays :g 100 X 10 •

0 Uhraviolct Visible ~

" x 10- 6

" :< "c 0.0 1 Illfrared 10 ' • 0 X < 0- s:.

I x IO- .f :<' Microwaves om 0

il" ~

I X 10- 6 " < FM radio/TV Q.

I X [Q - s 100 AM rad io

Thermal radiation has a peak wave­length that depends on an object's temperature according to Wien'slaw:

Intens ity

+-',,~----Wavelength A~al<

2.9 X 10' nm • K ApC:J.k(in nm) = T

844 CHAPTER 25 Electromagnetic Induction and Electrom agnetic Waves

tMP)TM For homewo~k assig~ed on MasteringPhysics, go to

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Problem difficulty is labeled as I (straightforward) to 11111 (challenging) .

QUESTIONS

Conceptual Questions

I. The world 's strongest magnet can produce a steady field of 45 T. If a c ircular wire loop of rad ius 10 cm were held in (his mag­netic fi eld, what curren t would be induced in the loop?

2. The rap id vibrat ion accompanying the swimming motions of BID mayfl ies has been measured by gluing a small mag net to a

swimming mayfly and recording the emf in a small coil of wire placed nearby. Explain how this technique works.

3. Parts a through f of Figure Q25.3 show one or morei:netal wires sliding on fi xed metal rail s in a magnetic field. For each, deter­mi ne if the induced current is clockwise, counterclockwise, or is zero.

(a) • • • • (b)

• • • •

• •

('J

(oj (n

X X 8=0

-------x x X X X X

X X X X X X

FIGURE 025.3

4. Figure Q25,4 shows four different loops in a magneti c field. The numbers indicate the lengths of the sides and the strength of the fi eld. Rank in order the magneti c flu xes ¢ I through ¢ 4,

from the largest to the smallest. Some may be equal. Explain.

• • • 8 = 0

• • · c=J • • • • • • 8=1 8=1 8 = 2 • • •

• • • • • • • • • • • • 8=2 • • • Loop I: Loop 2: Loop 3: Loop 4: 2 X 2 ] X] lX' 2XI

FIGURE 025.4

Problems labeled INT integrate significant material from earlier

chapters; BID are of biological or medical interest.

5. Figure Q25.5 shows four different c ircular loops that are per­pendicular to the page. The radius of loops 3 and 4 is twice that of loops I and 2. The mag netic field is the same for each. Rank in order the magnetic fluxes ¢ I th rough ¢4, from the largest to

the smaUesl. Some may be equal. Explain. - -- - -t ----- =e--. B ------- ------Loop I : Loop 2: Loop 3: Loop 4:

r = l r = 1 r = 2 r = 2

FIGURE 025.5

6. A circul ar loop rotates at constant speed about an axle through the center of the loop. Figure Q25.6 shows an edge view and defi nes the angle ¢, which increases from 0 0 to 3600 as the loop rotates.

Ii

•

7.

8.

a. At what angle or angles is the mag­net ic flux a maximum?

b. At what angle or angles is the mag­net ic flux a minimum?

FIGURE 025.6

c. At what angle or angles is the magnetic flux changing most rapidly?

The power lines that run through your neighborhood carry a/lemming ClIrrcnts that reverse direction 120 times per second. As the current changes, so does the magnet ic field around a line. Suppose you wanted to put a loop of wire up near the power line to ex tract power by "tapping" the magnetic field. Sketch a picture of how you would orient the coil of wire next to a power line to develop the maximum emf in the coil. (Note that thi s is dangerous and illegal, and not something you should try. ) The magnetic flux pass ing through a coil of wire varies as shown in Figure Q25.8. During which time interval(s) will an induced curren t be present in the coil ? Ex plain.

<],

FIGURE 025.8 FIGURE 025.9

9. There is a counterclockwise induced curren t in the conducting loop shown in Figure Q25.9. Is the magne ti c field inside the loop increasi ng in strength, decreasing in strength, or steady?

10. Cars on the "Tower of Doom" carniva l ride are dropped from a great he ight, giving riders a few seconds of free fall To stop the cars, strong permanent magnets attached to the bottoms of the cars pass very close to aluminum vanes stick ing up from the bottom of the track. Explain how thi s braking system works.

II. A magne t dropped th roug h a clear plast ic tube acce lerates as expected in free fall. If dropped through an al uminum tube of exac tl y the same length and diameter, the magnet falls much more slowly. Explain the behavior of the second magnet.

12. The conducting loop in Figure Q25. 12 is moving into the region between the magnetic poles shown. a. Is the induced current (viewed from above) clockwise or

counterc lockwise? b. Is there an attractive magneti c force that tends to pull the

loop in , li ke a magnet pulls on a paper clip? Or do you need to push the loop in against a repulsive force?

()

FIGURE 025 .12 FIGURE 025.13

13. Figure Q2s.13 shows two concentri c, conducting loops. We wi ll ddi ne a counterclockwise current (v iewed from above) to be positive, a clockwi se current to be negati ve. The graph shows the current in the outer loop as a function of time. Sketch a graph that shows the induced cunent in the inner loop. Explain.

14. Two loops of wire are stacked ~ verti ca ll y, one above the other, C ? IIV as shown in Figure Q2s .14. C _ Does the upper loop have a FIGURE 025.14 clockwise current, a counter-clockwise curren t, or no current at lhe following times? Explain your reason ing. a . Before the switch is closed b. Immediately after the switch is closed c. Long after the switch is closed d. Immediate ly after the switch is reopened

IS. A loop of wire is hori zontal. A bar magnet is pushed toward the loop from below, along the ax is of the loop, as shown III

Figure Q25.15. a. In what direction is the curre nt in the loop? Explain. b. Is there a magnetic force on the loop? I.f so, in which direc­

tion? Explain. Hint: Recall that a current loop is a magnetic dipole. c. Is there a magnetic force on the magnet? If so, in which

direct ion?

S

FIGURE 025 .15

C_--"te---:::::> Is " NI

FIGURE 025.16

Questions 845

16. A bar magne t is pushed toward a loop of wire, as shown in Fi gu re Q2s.16. Is there a current in the loop? If so, in which direct ion? If not, why not?

17. A conduct ing loop around a region of strong magnetic field con­tains two light bulbs, as shown in Figure Q2s. 17. The wires con­necting the bulbs are ideal. The magnetic field is increas ing rapidly. a. Do the bulbs glow? Why or why not? b. If they glow. which bulb is brighter? Or are they equally

brigh t? Expla in.

x x x x x x

---- x x x x x x , x x , , , , , x x x x x x ' x x x x , ii

~~ ~ x x : ~ ~B x x x x x x X X , ,

x X x x x x , , x x X I x x x x X

Fixed rail 'Wire

FIGURE 025.17 FIGURE 025.18

18. A metal wire is resting on a U-shaped conduct ing rail , as shown in Figure Q2s .18. The rail is fix ed in position, but the wi re is free to move. a. If the magnet ic field is increas ing in strength , whal does the

wire do? Does it remai n in place? Or does it move to the right or left, or up or down, or out of the plane of the page (break­ing contact with the rail )? Does it rotate clockwise or cou n­terclockwise? Does it both move and rotate? Explain.

b. If the magnetic fi eld is decreasing in slrength, what does the wire do?

19. Though sunlight is unpo\arized, the li ght that re flects from smooth surfaces may be partially polarized in the direction par­allel to the plane of the renecting surface. How should the long axis of the polarizing molecules in polarized sunglasses be or iented-vert ically or hori zon tally-to reduce the glare from a hori zonta l surface such as a road or a lake?

20. Two polarizers are orien ted with axes at 90°, so no light passes through the pair. A piece of plastic is placed between the two. If the plastic is stressed, by being squeezed, light that passes through the first polarizer and the plastic now passes through the second polarizer. The dark and light li nes allow the pattern of stress to be de termined. What e ffect does the stressed plastic have on the polarization of light pass ing through it?

With polarizing filters Without polarizing filte rs

2 1. Old-fashioned roof-mounted te lev ision antennas were designed to pick up signals across a broad frequency range. Explain why these antennas had metal bars of many di ffer­ent lengths.

846 CHAPTER 25 Electromagnetic Induction and Electromagnetic Waves

22. An AM radio detects the oscillating magnet ic field of the radio wave with an antenna cons isting of a co il of wire wrapped around a ferrite bar, as shown in Figure Q2S.22. Ferrite is a magnet ic material that "amplifies" the magnetic field of the wave. a. Explain how the radio antenna detects the magnet ic field of

the radio wave. b. If a radio stat ion is located due north of you, how must the

ferrite bar be oriented for best reception? Assume that the stat ion broadcasts with a vertical antenna like the one shown in Figure 25.37.

Axis of ferrite bar

11"""111"""111111\

FIGURE 025.22

23. Three laser beams have wavelengths AI = 400 nm, A2 =

600 nm, and A] = 800 nm. The power of each laser beam is I W. a. Rank in order, from largest to smallest, the photon energies

EI> £2' and £] in these three laser beams. Explain. b. Rank in order, from largest to sma l.l est, the number of pho­

tons per second N[ , N2, and N3 delivered by the three laser beams. Explain.

24. The intensity of a beam of light is increased but the light's fre­quency is unchanged. As a result, which of the fol.lowing (per­haps more than one) are true? Explain. A. The photons travel faster. B. Each photon has more energy. C. The photons are larger. D. There are more photons per second.

25. The frequency of a beam of light is increased but the light 's intens ity is unchanged. As a resu lt, which of the following (per­haps more than one) are true? Explain. A. The photons travel faster. B. Each photon has more energy. C. There are fewer photons per second. D. There are more photons per second.

26. Arc welding uses electric current to make an extremely hot electric arc that can melt metal. The arc emits ultravi­olet light that can cause sun­burn and eye damage if a welder is not wearing pro­tective gear. Why does the arc give off ultraviolet light?

Multiple-Choice Questions

27. I A circular loop of wire has an area of 0.30 m2• It is lilted by

45 0 with respect to a uniform OAO T magnetic field. What is the magnetic flux through the loop? A. 0.085 T' m' B. 0.12 T· m2

C. 0.38 T' m' 0.0.75 T· m2

E. 1.3 T· m'

28. I In Figure Q2S.28, a sq uare loop is x rotating in the plane of the page around an x x axis through its center. A uniform mag-netic field is directed into the page. What x

is the direction of the induced current in x x

the loop? x x A. Clockwise. x x~ x x B. Counterclockwise. C. There is no induced current. FIGURE 025.28

29. I A diamond-shaped loop of wire is pulled at.a constant veloc­ity through a region where the magnetic field is directed into the paper in the left half and is zero in the ri ght half, as shown in Figure Q2S.29. As the loop moves from left to ri ghI , which graph best represents the induced current in the loop as a func­tion of time? Let a clockwise current be posit ive and a counter­clockwise current be negative.

A. '--__ L/ ___ _ B. ,--_~A,----."--__

, C. L-_-LD_ L-__

D. '----"'""""7---

>? E. L-----L-\V,---FIGURE 025.29

30. II Figure Q2S.30 shows a trian­gular loop of wire in a uniform magnetic field. If the field strength changes from 0.30 to 0.10 T in 50 ms, what is the induced emf in the loop? A. 0.08 Y B. 0.12 Y C. 0.16 Y

0 .0.24 Y E. 0.36 Y

x x x x x x x x x x x x x x x x ii )( x x )(

O.20m x x x x x x x x x O.20m x x x

FIGURE 025.30

3 1. II A device call ed aj7ip coil can be used to measure the earth's magnet ic field. The coil has 100 turns and an area of 0.010 m2

.

It is oriented with its plane perpendicular to the earth 's mag­netic field , then flipped 1800 so the field goes through the coil in the opposite direction. The earth 's magnetic field is 0.050 mT, and the coil flips over in 0.50 s. What is the average emf induced in the coil during the flip? A. 0.050 mY B. 0.10 mY C. 0.20 mY D. 1.0 mY

32. I The electromagnetic waves that carry FM radio range in fre­quency from 87 .9 MHz to 107.9 MHz. What is the range of wavelengths of these radio waves? A. 500--750 nm B.0.87-9 1.08m C. 2.78- 3.4 1 m 0.278- 341 m E. 234-410 km

33 . I A spacecraft in orbit around the moon measures its altitude by refl ec ting a pulsed 10 MHz radio signal from the surface. If the spacecraft is 10 km high, what is the time between the emis­sion of the pulse and the detection of the echo? A. 33 ns B. 67 ns C. 33 p.s D. 67 p.s

34.

35.

I A 6.0 mW ve rti caJiy polari zed laser beam passes through a polariz ing fi lter whose axis is 75° fro m ve rtical. What is the laser-beam power after pass ing through the fi lter? A. DAD mW B. 1.0 mW c. 1.6 mW D.5.6 mW I Conununicati on with submerged submarines via rad io waves is diffi cult because seawater is conductive and absorbs electromag­net ic waves. Penetration into the ocean is greater at longer wave­lengths, so the United States has radio install ati ons th at tra nsmit

PROBLEMS

Section 25.1 Induced Currents

Section 25.2 Motional emf

1. I A potent ial d ifference of 0 .050 V is deve loped across the IO-em-long wire in Figure P25. 1 as it moves through a magneti c field at 5.0 m/s. The magnetic ri eld is perpe ndicular to the ax is of the wire. What are the d irect ion and strength of the magnetic fie ld?

IOem

I! -I I so m/'

FIGURE P25.1

2. II A scalloped hamme rhead shark swims at a steady speed of BID 1.5 Ill/s with its 85-cm-wide head perpendic ul ar to the earth 's

50 J..LT magnet ic field. What is the magnitude of the emf induced between the two s ides of the shark's head?

3. 11 A IO-em-long wire is pulled along a U-shaped conducti ng rail in a perpend icular magnetic fie ld. The total resistance of the wire and ra il is 0.20 n. Pulli ng the wire with a force of 1.0 N causes 4.0 W of power to be d iss ipated in the c ircui l. a. What is the speed of the wire when pulled with a force of 1.0 N? b. What is the strength of the magnet ic field?

4. 1 Figure P25.4 shows a IS-em-long melall'Od pulled along two fr icti onless. conducti ng rails at a constant speed of 3.5 m/s . The rail s have neg li gible res is tance, bUl the rod has a res istance of 0.65 !l . a. What is the currenl induced in the rod? b. What force is required to keep the rod movi ng at a constant

speed?

x x x x x x x x

x x x x x x x X B = 1.4 T

x x x x x x x x , x x x x x x x x

x x x x x x x x

FIGURE P25 .4 X x x x x x x x

5. III A 50 g ho ri zon tal metal bar, 12 cm long, is free to sl ide up and down between two ta ll , ve rtica l metal rods that are 12 cm aparl. A 0.060 T magnet ic field is d irected perpendicular to the plane of the rods. The bar is ra ised to near the top o f the rods, and a 1.0 n resistor is connec ted across the two rods at the top . Then the bar is dropped. What is the termi nal speed at which the bar fall s? Assume the bar remains hori zontal and in contact with the rods at all times.

Problems 847

at 76 Hz for submarine communications. What is the approxi­mate wavelength of those extremely low-frequency waves? A. 500 km B. 1000 km C. 2000 kin D. 4000 km

36. II How many photons are emi tted duri ng 5.0 s of operat ion of a red laser po inter? The dev ice OUlputs 2.8 mW at a 635 nm wave length. A. 4.5 X 10 10

C. 4.5 X 10" B. 4 .5 X 10" D. 4.5 X 10"

6. II A deli very truck with 2.8-m-high aluminum sides is dri vi ng west at 75 kmlhr in a reg ion where the ea l1h's magnet ic fi eld is Ii = (5.0 X 10- 5 T, north). a. What is the potenti al d iffere nce between the to p and the bot­

tom of the truck's side panels? b. Will the tops of the panels be pos itive or negative relative to

the bottoms?

Section 25.3 Magnetic Flux

7. I Figure P25.7 is an edge-on view of a la-em-diameter c ircular loop rotating in a uni form 0.050 T magne ti c fi e ld. What is the magnetic flux through the loop when (J is 0°, 300, 60°, and 90°?

~~--l----' \

Circular loop

FIGURE P25.7

20 em

x x x x • ••• x x x x ••••

2.0T l.O T x x x x •••• x x x x • ••• X x x X ••••

20 em 20cm

FtGURE P25.8

8. II What is the magne ti c fl ux th roug h the loop shown in Figure P25.8?

9. III The 2.0-cm-diameter sole no id in Figure P25.9 passes throug h the center o f a 6.0-cm-diamete r loop. The magnet ic fi eld inside the solenoid is 0 .20 T What is the mag netic flux through the loop (a) when it is perpendicu lar to the soleno id and (b) whe n it is tilted at a 60° ang le?

(3) Solenoid

" (b) ~

1 11111111 WJl II~11111111 1

FIGURE P25.9

10. II At a typical locat ion in the Un ited States, the earth 's magnetic field has a magn ilUde of 5 .0 X 10- 5 T and is at a 65° angle from the horizontal. What is the flux through the 22 cm X 28 cm front cover of your tex tbook if it is fla t on your desk?

11. I The metal equ ilateral triangle in a.IO T x X I

Figure P25. 11 , 20 c m on each s ide, is halfway into a 0. 1 0 T mag net ic fie ld. a. What is the magneti c flux th ro ugh

the triang le? x b. If the magneti c fi eld strength x

dec reases, what is the d irect ion of x~x~x~x~':-";;"'» the induced curren t in the tri angle? FIGURE P25 .11

848 CHAPTER 25 Electromagnetic Induction and Electromagnetic Waves

Section 25.4 Faraday's Law

12. I Figure P25.12 shows a lO-cm-diameter loop in three differ­ent magnetic fields. The loop's res istance is 0.10 n. For each case, determine the induced emf, the induced curren t, and the direction of the current.

(a) H increasing lit 0.50 Tis

(b) fJ decreasing at 0.50 Tis

(e) B decreasing at 0.50 TIs

0·' Ox: X : :l: •• xx xxx

• • X X X X X

FIGURE P25 .12

x

X

X

X

13. II A loop of wire is perpendicular to a magnetic field. Rank, from greatest to least, the magnitudes of the loop 's induced emf for the following situations: A. The magneti c field strength increases from 0 to I Tin 6 s. B. The magnetic field strength increases from I T to 4 T in 2 s. C. The magneti c fi eld strength remains at 4 T for I min. D. The magnetic field strength decreases from 4 T to 3 T in 4 s. E. The magnetic fi eld strength decreases from 3 T to 0 T in I s.

14. II Patients undergo ing an MRJ occasionally report seeing BID flashes of light. Some practitioners assume that this results from

e lectric stimulation of the eye by the emf induced by the rapidly changing fields of an MRl solenoid. We can do a quick calcula­tion to see if this is a reasonable assumption. The human eye­ball has a diameter of approximately 25 mm. Rapid changes in current in an MRI solenoid can produce rapid changes in field, with fiBlfil as large as 50 TIs. What emf would this induce in a loop c ircling the eyeball? How does this compare to the 15 mV necessary to trigger an action potential?

15. III A 1000-turn coil of wire 2.0 cm in diameter is in a magnetic field that drops from 0.10 T to 0 Tin 10 ms. The axis of the coil is parallel to the field. What is the emf of the co il ?

16. II The loop in Figure P25.16 has an induced current as shown. The loop has a resistance of 0.10 n. Is the magne tic field strength increas ing or decreasing? What is the rate of change of the field. t>BI t>/?

~150mA

X X X X 20 f1 • • • • • •

x x x x • 8.0cm x x x x • · • • • • 9.0 v

x~ • • • • x x • • • • • • • • • • • •

• • • • • • • • •

8.0cl11 • • • • • • • • • FIGURE PZ5 .16 FIGURE PZ5 .17

17. II The circuit of Figure P25.l7 is a square 5.0 cm on a side. The magnetic field increases stead il y from 0 T to 0.50 T in 10 ms.

18. What is the curren t in the resistor during thi s time? III A 5.0-cm-diameter loop of B (T)

wire has resistance 1.2 n. A 1.5

nearby solenoid generates a uni-form mag netic field perpendic- 1.0

0.5 ular to the loop that varies with time as shown in Figure P25.1 8. Graph the magnitude of the cur­rent in the loop over the same time interval.

o +-~L-~_~_~ I (s) o 0.1 0.2 0.3 0.4

FIGURE P25 .18

Section 25.5 Induced Fields and Electromagnetic Waves

Section 25.6 Properties of Electromagnetic Waves

19. I What is the electric field amplitude of an electromagnetic wave whose magnetic field amplitude is 2.0 mT?

20. I What is the magnetic field amplitude of an electromagnetic wave whose electric field amplitude is 10 VIm?

21. II A microwave oven operates at 2.4 GHz with an intensity inside Ihe oven of 2500 Wltn"!. What are the amplitudes of the osc illating electric and magnet ic fields?

22. II The maximum allowed leakage of microwave radiation from a microwave oven is 5.0 mWlcm 2.lfmicrowave radiation out ­side an oven has the maximum value, what is the amplitude of the oscillating electric field?

23. III A typ ical helium-neon laser found in supermarket checkout scanners emits 633-nm-wavelength light in a 1.0-mm-diameter beam with a power of 1.0 mW. What are the amplitudes of the osc illating electric and magnet ic fie lds in the laser beam?

24. I The magnet ic field of an electromagnetic wave in a vacuum is B: = (3.0 JLT) sin« I.O X 107)x - 27f!1) , where x is in 111 and 1 is in s. What are the wave 's (a) wavelength, (b) frequency, and (c) electric field amplitude?

25. II The electric field of an electromagnetic wave in a vacuum is E,.- = (20 V/m)sin«6.28 X 108)x - 27f!I), where x is in m and 1 is in s. What are the wave's (a) wavelength, (b) frequency, and (c) magnetic field amplitude?

26. II A radio receiver can detect signal s with electric field ampli­tudes as small as 300 JL VIm. What is the intens ity of the small ­est detectable signal?

27. II A 200 MW laser pulse is focused with a lens to a diameter of INT 2.0/L1ll.

a. What is the laser beam's electric field amplitude at the focal point?

b. What is the ratio of the laser beam's electric field to the e lec­tric field that keeps the electron bound to the proton of a hydrogen atom? The radius of the electron's orbit is 0.053 nm.

28. III A radio antenna broadcasts a 1.0 MHz radio wave with 25 kW of power. Assume that the radiation is em itted uniformly in all directions.

29.

30.

a. What is the wave's intensity 30 k1l1 from the antenna? b. What is the electric tield amplitude at thi s di stance? II At what di stance from a 10 W point source of electro­magneti c waves is the e lectric fie ld amplitude (a) 100 Vim and (b) 0.010 Vim? I The intensity of a polari zed electromagnet ic wave is 10 W/m 2. What will be the intensity after passi ng through a polarizing filter whose axis makes the following angles with the plane of polari zat ion? (a) f) = 00 (b) e = 30° (c) e = 45° (d) e ~ 600 (e) e ~ 900.

3 1. II Only 25% of the intens ity of a polarized light wave passes through a polarizing filter. What is the angle between the elec­tric field and the axis of the tilter?

32. II A 200 mW horizontally polarized laser beam passes through a polariz ing filter whose axis is 25° from vert ical. What is the power of the laser beam as it emerges from the filter?

33. II The polari zation of a he lium-neon laser can change with time. The light from a 1.5 mW laser is initially horizontally polarized; as the laser warms up , the light changes to be verti ­cally polarized. Suppose the laser beam passes through a polar­izer whose axis is 30° from horizontal. By what percent does the light intensity transm itted through the polarizer decrease as the laser warms up?

Section 25.7 The Photon Model of Electromagnetic Waves

34. I What is the energy (in eV) of a photon of visible light that has a wave length of 500 nm?

35. I What is the energy (in eV) of an x-ray photon that has a wave length of 1.0 nm?

36. I What is the wavelength of a photon whose ene rgy is twice that of a photon with a 600 nm wavelength?

37. II One recent study has shown that x rays with a wavelength of BIO 0.0050 nm can produce mutations in human ce lls.

a. Calculate the energy in eV of a photon of radi ati on with this waVelength.

b. Assuming tha the bond energy holding together a water molec ule is lypical, use Table 25. 1 to estimate how many molecular bonds could be broken with thi s energy.

38. I Rod cell s in the retina of the eye de tect light using a pho­BID topigment called rhodopsin. 1.8 eV is the lowest photon energy

that can trigger a response in rhodopsin. What is the max imum wavelength of electromagnetic rad iation that can cause a transi­tion? In what part of the spectrum is this?

39. III What is the energy of I mol of photons that have a wave­length of 1.0 J-Lm?

40. II The the rmal emission of the human body has maximum BID intens ity at a wave length of approx imately 9.5 J-Lm . What pho­

ton energy corresponds to thi s wave length? 41. II The intensity of e lectromagneti c rad iati on from the sun

reaching the earth 's upper atmosphere is 1.37 kW/m2• Assum­ing an average wavelength of680 nm for thi s radiation , find the number o f photons per second that strike a 1.00 m2 so lar pane l directl y facing the sun on an orbiting satellite.

42. III The human eye can barely de tect a star whose intensity at the BID earth 's surface is 1.6 X 10- 11 W /m2

. If the dark-adapted eye has a pupil di ameter of 7.0 mm, how many photons per second en te r the eye from the star? Assume the starlight has a wave­length of 550 nm.

Section 25.8 The Electromagnetic Speclrum

43. II The spectrum of a glowing fil ament has its peak at a wave­length of 1200 nIl1. What is the temperature of the filament. in °C?

44. III While using a dimmer switch to in vestigate a new type of incan­descent light bulb, you notice that the li ght changes both its spec­tral characteristics and its brightness as the voltage is increased. a. If the wavelength of maximum intensit y decreases from

1800 nm to 1600 nm as the bulb 's vo ltage is increased, by how many °C does the filament temperature increase?

b. By what factor does the total radiation from the fi lament increase due to thi s temperature change?

45. I The photon ene rgies Llsed in different types of medical x-ray BID imaging vary wide ly, depending upon the application. Single

dental x rays use photons with energ ies of about 25 keY. The photon energy used for x-ray microtomography, a process that a llows repea ted imaging in single planes at varying depths with in the sample , is 2.5 times greater. What are the wave­lengths of the x rays used for these two purposes?

General Problems

46. II A 10 cm x 10 cm square is bent at a 90° angle as shown in Figure P25.46. A uniform 0.050 T magne ti c field points down­ward at a 45° ang le. What is the magnetic flux throug h the loop?

Ii 5 em

~~<m '<m

FtGURE P2S .46

Problems 849

FIGURE P25.47

Magnelic field R

47. I What is the magneti c flux through the loop shown in Figure P25.47?

48. II a. A circular loop antenna has a diameter of 20 cm. If the plane of the loop is perpend iclliar to the earth' s 50 J-LT magneti c fi eld, what is the nux through the loop?

b. What is the flux if the loop is rotated by 30°? 49. II An 1.I-Il1-diame ter MRI so lenoid with a length of2.4 m has

a magneti c fi e ld of 1.5 T along its ax is. If the curren t is turned off in a time of 1.2 S, what is the induced emf in one turn of the solenoid 's windings?

50. II A magnet and a coil are ori­ented as shown in Figure P25.50. The magnet is moved rapidl y into tbe coil. he ld stat ionary in the coi I for a short ti me, and then rapidly pulled back out of the coil. Sketch a graph showing the FIGURE P25 .50

reading of the ammeter as a funct ion of time. The ammeter reg­isters a positi ve value when current goes into the "+" terminal.

51. III A wire loop with an area of 0.020 m2 is in a magnetic fie ld of 0.30 T'directed at a 30° angle to the plane of the loop. If the field drops to zero in 45 ms, what is the average induced emfin the loop?

52. III A I OO-lurn , 2.0-cm-diamcter coil is at rest in a hori zontal plane. A uniform magneti c field 60° away from verti cal increases from 0.50 T to 1.50 T in 0.60 s. What is the induced emf in the co il ?

53. 1111 A 25-turn, 10.0-cm-diameter coil is ori en ted in a vertical plane with its axis aligned cast-west. A magnet ic field pointing to the northeast decreases from 0.80 T to 0.20 Tin 2.0 s. What is the emf induced in the coil?

54. 1111 People immersed in strong unchanging magnetic fie lds occa­BKJ sionally report sens ing a metall ic taste. Some investigators suspect

that mot ion in the constant field could produce a changing flux and a resulting emf that could stimulate nerves in the tongue. We can make a simple model to see if this is reasonable by imagining a somewhat ex treme case. Suppose a patient having an MRI is immersed in a 3.0 T field along the ax is of hi s body. He then quickl y tips his head to the side. toward his right shoulder, tipping his head by 30° in the rather shOit time of 0.15 s. Estimate the area of the tongue; then calculate the emf that could be induced in a loop around the outside of the longue by thi s motion of the head. How does thi s emf compare to the approx imately 15 m V neces­sary to tri gger an action potenti al? Docs it seem reasonable to sup­pose that an induced emf is responsible for the noted effect?

55. 1111 A 20 cm length of 0.32-mm-diameter nichrome wire is INT we lded into a ci rcular loop. The loop is placed between the

poles of an e lectromagnet. and a fi eld of 0.55 T is switched on in a Lime of 15 ms. What is the induced current in the loop?

56. 1111 Currents induced by rapid fie ld changes in an MRI solenoid BIO can, in some cases, heat tiss ues in the body, but under normal INT c ircumstances the heating is

small. We can do a quick esti ­mate to show th is. Consider the ;; Ioop" of muscle ti ssue shown in Figure P25.56. This might be muscle circl ing the bone of

8.0cm

() ()

1.0 ern

FIGURE P25 .56

850 CHAPTER 25 Electromagnetic Induction and Electromagnetic Waves

your arm or leg. Muscle tissue is not a great conductor, but cur­rent will pass through muscle and so we can consider this a con­ducting loop with a rather high resistance. Suppose the magnet ic field along the axis of the loop drops from 1.6 T to a T in 0.30 s, as it might in an MRI solenoid. a. How much energy is dissipated in the loop? b. By how much will the temperature of the tissue increase? Assume that muscle ti ssue has resistivi ty 13 n· m, density 1.1 X J 0' kg/m ' , and specific heat 3600 J/kg , K.

57. III A lOa-turn, S.O-cm-diameter co il is made of 0.50-mm­

SOmis

• • • • • INT diameter copper wire. A magnetic field is

perpendicular to the coil. At what rate must B increase to induce a 2.0 A current

8 = 0.20T • • • • •

in the coil? • • • • • • • • • • • • • • •

5.Dem

58. II The loop in Figure P25.58 is being pushed into the 0.20 T magnetic field at 50 m/s. The resistance of the loop is 0.10 n. What are the direction and mag­nitude of the CUITent in the loop? FIGURE P25 .58

59. III A 20-cm-long, zero-res is­INT tance wire is pulled outward,

on zero-resistance rai ls, at a steady speed of 10 mls in a 0. 10 T magneti c field. (See Figure P25 .59.) On the oppo­site side, a 1.0 n carbon resis-

Zero-resistance wires

• • • • •

IOm/s

• • • tor completes the circuit by connecting the two rails. The FIGURE P25 .59

mass of the resistor is 50 mg. a. What is the induced current in the circuit?

\

• • • • l.On

• • • • O.IOT

• • • •

b. How much force is needed to pull the wire at thi s speed?

• • • • •

c. How much does the temperature of the carbon increase if the wire is pulled for to s? The speci fi c hea t of carbon is 710 J/kg' K. Neglect the rmal energy transfer out of the resistor.

60. 1111 A TMS (transcrani al magnetic st imulation) device creates BID very rapidly changing magnet ic fi elds. The field near a typical INT pulsed-field machine rises from 0 T to 2.5 T in 200 !-Is. Suppose

a technician holds his hand near the device so that the axis of hi s 2.0-cm-diameter wedding band is parallel to the field. a. What emf is induced in the ring as the field changes? b. If the band is gold with a cross-section area of 4.0 mml,

what is the induced curren t? Can you see why TMS technicians are adv ised to remove all jewelry?

61. II The I O-cm-wide. zero-resistance wire shown in Figure P25.61 is pushed toward the 2.0 n resistor at a steady speed of 0.50 m/s. The magnetic fi eld strength is 0.50 T. a. How big is the pushing force? b. How much power does the pushing force supply to the

wire? c. What are the direction and magnitude of the induced current? d. How much power is di ss ipated in the res istor?

• •

•

Zero-res istance wires \

• • • • • • Push 2.0 n

• • • • • 0.50 m/s 0.50 T

• • • • • •

FIGURE P25 .61

ii

Come;! , \

FIGURE P25.62

- 6.O-mm-diameter coil

62. 1111 Experiments to study vision often need to track the move­BID ments of a subjec t'S eye. One way of doing so is to have the

subject sit in a magnetic field while wearing special contact lenses that have a coi l of very fine wire circling the edge. A cur­rent is induced in the co il each time the subject rotates hi s eye. Conside r an experiment in which a 20-turn, 6.0-mm-diameter coil of wire c ircles the subject 's cornea while a 1.0 T magnetic field is directed as shown in Figure P25.62 . The subject begins by looking straight ahead. What emf is induced in the coil if the

63.

64. BID IN!

65.

subject shifts hi s gaze by 5.00 in 0.20 s? III The fi lament in the center of a 100 W incandescent bulb emits approximately 4.0 W of vis ible light. If you assume that aU of this light is emitted at a single wavelength , est imate the electric and magnetic field strength at the sUlface of the bulb. II A LASIK vision correction system uses a laser that emits 10-ns-long pulses of light, each with 2.5 mJ of energy. The laser is focused to a 0.85-mm-diametercircle. (a) What is the average power of each laser pulse? (b) What is the electric field strength of the laser light at the focus point? I When the Voyager 2 spacecraft passed Neptune in 1989, it was 4.5 X 109 km from the earth. Its radio transmitter, with which it sent back data and images, broadcast with a mere 21 W of power. Assuming that the transmitter broadcast equally in all directions . a. What signal intensity was received on the earth? b. What electric fi eld amplitude was detected? (The rece ived signal was slightly stronger than your result because the spacecraft used a di.rectional antenna.)

66. II A new cordless ehone emits 4.0 mW at 5.S GHz. The manufac­turer claims that the phone has a range of 100 feet. If we assume that the wave spreads out evenly with no obstructions. what is the electri c fi eld strength at the base unit 100 feet from the phone?

67. III 633-nm-wavelength light from a helium-neon laser is vert i­cally polarized. Suppose a laser beam passes through a po larizer with its axis 45° from the vertical. What is the ratio of the laser beam 's electri c field strengths before and after the polarizer?

68. II In reading the instruction manual that came with your garage-door opener, you see that the transmitter unit in your car produces a 250 mW signal and that the receiver unit is supposed to respond to a radio wave of the correct frequency if the elec­tri c fi e ld amplitude exceeds 0.10 VIm. You wonder if this is really true. To find Ollt, you put fresh batteries in the transmitter and start walking away from your garage while opening and clos ing the door. Your garage door finally fail s to respond when you' re 42 m away. Are the manufacturer's claims true?

69. III Unpolarized light passes through a vertical polari zing fiJter, emerg ing with an intens ity 10 , The light then passes through a hori zontal filter, which blocks all of the light; the int.ensity lransmined through the pair of filters is zero. Suppose a third polarizer with axis 45 0 from verti cal is inserted between the first two. What is the transmitted intensity now?

70. I a. What is the wavelength of a gamma-ray photon with energy 1.0 X 10- 13 J?

71. I

b. How many visible-light photons with a wavelength of 500 nm would you need to match the energy of this one gamma-ray photon?

Gamma rays with the very high energy of 2.0 X 10 13 eV are occasionally observed from distant astrophys ical sources. What are the wavelength and frequency corresponding to this photon energy?

72. I A 1000 kHz AM radio stat ion broadcasts with a power of 20 kW. How many photons does the transmitting antenna emit each second?

73. II Fireflies emit flashes of light to communicate , directly con­BID verting chemical energy into the energy of visible-light photons. A

firefly emits a 0.15-s- long pulse of light with an average wave­length of590 nm. During the flash, the emitted power is 40 p-W. a. What is the energy of one light photon havi ng the average

wave length? b. Assume that all photons have the average wavelength. How

many photons does the firefly emit during each flash? 74. III! The human body has a surface area of approximate ly 1.8 m2

,

BID a surface temperature of approx imately 30°C, and a typical emissjvity at infrared wavelengths of e = 0.97. If we make the approx imation that all photons are emitted at the wavelength of peak intensity, how many photons per second does the body emit?

75. III For rad io and microwaves, the depth of penetration into the BID human body is approximalely proportional to VA. If 27 MHz

rad io waves penetrate to a depth of 14 em, estimate the depth of penetration of2.4 GHz microwaves.

Passage Problems

The MetHi Detector

Metal detectors use ind uced currents to sense the presence of any metal- not just magnetic materials such as iron. A metal detec tor, shown in Figure P25.76, consists of two coil s: a transmitter coil and a receiver coil. A high-frequency oscillating current in the transmit­te r coil generates an osci llating magnet ic field along the ax is and a changi ng flux through the receiver coil. Consequentl y, there is an osc iUating induced current in the receiver coi I.

Induced current -_... Receiver coi l due to eddy curren t~ ~~

Eddy currcnt~ in the Metal metal reduce the induced

TrJnsmincr coil

FIGURE P25 .76

d \

Induced current due [0 the [r.msminercoi l

Stop to Think 25.1: E. According to the right- hand ru le, the mag­neti c force on a positive charge carrier is to the right.

Stop to Think 25.2: D. The fi eld of the bar magnet emerges from the north pole and points upward. As the co il moves toward the pole, the flux through it is upward and increasing. To oppose the increase, the induced fi eld must poin t downward. This req ui res a clockwise (negat ive) cunent. As the coil moves away from the pole, the upward flu x is decreas ing. To oppose the decrease, the induced field must point upward. This req uires a counterclockwise (positive) current.

Stop to Think 25.3: E. The right-hand rule requ ires jj to point into the page if the wave is to propagate upward.

Problems 851

If a piece of metal is placed between the transmitter and the receiver, the osc iUating magnetic field in Ihe metal induces eddy cur­rents in a plane parallel to the transmitter and receiver coils. The rece iver coil then responds to the superpos ition of the transmitte r's magnet ic fi eld and the magne ti c field of the eddy currents. Because the eddy currents attempt to prevent the flu x from chang ing, in accordance with Lenz's law, the net fi eld at the receiver decreases when a piece of metal is inserted between the co il s. Electronic cir­cuits detec t the currenl decrease in the rece iver coil and set off an alarm. 76. I The metal detector will not detect insulators because

A. Insulators block magnet ic fie lds. B. No eddy current can be produced in an insulator. C. No emf can be produced in an insulator. D. An insulator will increase the field at the receiver.

77. I A metal detec tor can detect the presence of metal screws used to repair a broken bone inside the body. This tells us that A. The screws are made of magnet ic materials. B. The ti ssues of the body are conducting. C. The magneti c fi e lds of the device can penetrate the (i ssues

of the body. D. The screws must be perfect ly aligned with the ax is of the

device. 78. I Suppose the magneti c field from the transmitter coil in

Figure P25.76 points toward (he receiver coil and is increasing with time. As viewed along thi s ax is, the induced currents are A. Clockwise in the meta l, clockwise in the receiver coil B. Clockwise in the metal, counterclockwise in the rece iver

coil C. Counterclockwise in the metal, clockwise in the recei ver

coil D. Counterclockwise in the metal , counterclockwise in the

receiver coi I 79. I Which of the following changes would nol produce a larger

eddy cunen t in the metal ? A. Increas ing the frequency of the osci ll ating current in the

transm itter coil B. Increas ing the mag ni tude of the osc illati ng current in the

transmitter coil C. Increas ing the res isti vity of the metal D. Decreasing the distance between the metal and the transmitter

Stop to Think 25.4: II) > IA > In = Ie- The intens ity depends upon cos20, where 0 is the angie be/ween the axes of the two filters. The filters in D have 0 = 00, so all light is transmitted . The two fil ­ters in both Band C are crossed (0 = 90°) and transm it no light at all.

Stop to Think 25.5: A. The photon energy is proportional to the rrequency. The photons of the 90.5 MH z station each have lower energy, so more photons must be emitted per second.

Stop to Think 25.6: D. A hotter objec t emits more rad iation across the entire spectrum than a cooler objec t. The 6000 K star has its max imum intens ity in the blue reg ion of the spectrum, but it st ill emi ts more red radiation than the somewhat cooler stars.