physics Maxwell's Equations and Electromagnetic Radiation vol 2

7/28/2019 physics Maxwell's Equations and Electromagnetic Radiation vol 2

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"W hen I p laced the pr im ary conduc tor i n one corner o f a l arge l ec ture room 14 me terslong an d 12 me ters wide , t he spark s [ in t he secondary] co u ld be perce i ved in t he far thes tpar t s o f the room; the who le room seem ed fi l l ed wi th t he osc il la t ions o f the electric force."

~Heinr ich Hertz ,discoverer of electromagn etic radiation, predicted in 1865 by Maxw ell (1888)"I t seemed to me tha t i f t he rad ia t ion cou ld be increased , deve loped , an d con tro ll ed, i two uld be poss ib l e t o s igna l across space for cons iderab le d i s tances . "

~Gugl ie lmo Marconi ,inventor of radio communications,in his later life reflecting on the idea he had in 189 4, at the age of 20

C h a p t e r 1 5M axw ell's Eq uat ion s andElectromagnetic Radiation

6 3 0

C h a p t e r O v e r v i e wSection 1 5.1 introduces the chapter . Sect ion 15.2 presents a br ief h is tory of m od-ern com mu nicat ions. Sect ion 15.3 launches into the technical aspects of ou r subject,w i th a d iscuss ion o f Maxwel t 's displacement current , wh ich serves as a ne w sourcefor the ma gne tic fie ld in Ampere's law. This com pletes the equa t ions descr ib ing thee lec tr omagne t ic f i e l d~ M ax w e l l ' s equ a t i ons ~ w h ic h hav e s o lu t ions tha t c o rr espondto e lec t romagnet ic waves . To prepare you fo r e lec t romagnet ic waves , the nex t twosect ions discuss waves on a s tr ing, Sect ion 1 5.4 obtain in g th e wave e qua t ion fo rthe s tr ing, and Sect ion 15.5 solv ing i t for both s tanding waves (as for a gui tar) andtravel ing wa ves (as on a long tau t s tr ing) . Sect ion 15.6 solves Ma xwel l 's equat ionsfor a par t icu la r ly s imp le fo rm o f e lec t romagnet ic wave, known as a p lane wave.Sec tion 1 5 .7 po in ts ou t tha t e lec t romagnet ic rad ia t ion covers a spec trum rang ing f romthe lo w-fre que ncy wave s associated wit h po we r l ines, on u p to success ively h igher fre-quencies in microwaves, v is ib le l ight, and gam ma rays. Sect ion 15.8 considers p ow erf low associated with a p lane wave, and Sect ion 15.9 considers the associated mo-me ntum f low. Sec tion 15 .10 considers po la r iza t ion , a consequence o f the fac t tha tin free space the ele ctrom agn etic f ie ld vectors E and B are normal to th e direct ionof p ropag ation. Sect ion 15.1 1 considers e lectrom agn etic s tan ding w aves, as in a mi-crowav e cav i ty . Sect ion 15.12 considers travel ing waves for m icrowav e waveg uidesand coaxia l cables. Sect ion 15.13 show s ho w the electr ic and m agn etic proper t ies ofmat te r can in f luence, th rough the d ie lec t r ic cons tan t and the magnet ic permeab i l i ty ,the ve loc i ty o f an e lec t romagnet ic wave. We then show how th is in f luences the phe-nomena of ref lect ion and refract ion. Sect ion 15.14 rev iews the cr i t ica l exper imentswhereby Hertz establ ished that, for a c i rcui t that produced microwave osc i l la t ions,i t a lso produce d radiat ion in spac e who se p roper t ies were muc h l ike that fo r v is ib lel ight. Subsect ion 15.15.1 dea ls wi th energy f low for travel ing waves on a s tr ing, andSubsect ion 15.1 5.2 dea ls wi t h comp ress ional waves.


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15.2 A Brie f History of Com munications 631

15,1 I n t r o d u c t i o nW e a r e n o w p r e p a r e d t o t a k e t h e f i na l s t e p i n o u r s t u d y o f e l e c tr o m a g n e t i s m , a n dt r ea t e l ec t rom agne t i c r ad ia t ion . Befo re ge t t ing to the phys ics , w e w i ll r em ind yo uo f w h a t t h e w o r l d w a s l ik e b e f o r e t h e r e w e r e c o m m u n i c a t i o n s b a s e d u p o n e l e c tr i -ca l s igna l ing , e i the r a long w i res o r th rou gh s pace i t se l f. We w i l l d i scus s ma t t e r s o fh i s to ry and cu l tu re , ma t t e r s tha t s ome t ake fo r g ran ted , and o f w h ich o the r s a r ec o m p l e t e l y u n a w a r e . F o r t h o s e w h o s ti ll t h i n k t h a t e l e c t r o m a g n e t i c r a d i a t i o ni s mag ic , w e a re abou t to s ha t t e r tha t i l lu s ion . F o r thos e w ho no longer be l i evei t i s mag ic , w e w ou ld l ike to encourage you to r ega in a s ens e o f aw e , a f ee l ingtha t pe rha ps th e re i s ju s t a l i t tl e b i t o f mag ic a t w or k w h en r ad io and t e l ev i s ions igna l s a r e genera ted , t r ave r s e the ea r th , and then a r e de tec ted and ampl i f i edfo r ou r bus ines s o r p leas u re . E lec t rom agne t i c r ad ia t ion , o f cour s e , is no t mag ic .N ever the les s , i t i s an amaz ing f ac t tha t in a mere 100 yea r s s ince i t s d i s covery ,h u m a n s h a v e h a r n e s s e d i t s p o w e r a n d n o w a r e c o m p l e t e l y d e p e n d e n t o n i t .

15o2 A B r ie f H is t o r y o f C o m m u n i c a t i o n sW h a t f o ll o w s is a b r i e f h is t o r y o f c o m m u n i c a t i o n s , e s p e ci a ll y c o m m u n i c a t i o n sus ing e lec t r i ca l and e lec t romagne t i c s igna l s .T o da y , w e t a k e i t f or g r a n t e d t h a t w e w i l l b e a b le t o c o m m u n i c a t e w i t h o n eano th e r u s ing phone s th a t u s e pho ne l ines o f copp er w i r e o r op t i ca l fibe r, andmay a l s o inc lude mic row ave t r ans mis s ion and s a te l l i t e l inkups . The l a t e 1990ss a w th e f ir st w i d e s p r e a d u s e o f e -m a i l a n d t h e W o r l d W i d e W e b . A l l o f o u rc o m m u n i c a t io n s m e t h o d s u s e so m e p a r t o f th e s p e c t r u m o f e l e c tr o m a g n e t icr ad ia t ion , bas ed u pon the p r inc ip les l a id dow n in the p rev iou s chap te r s , and onem o r e ~ d u e t o M a x w e l l ~ t h a t w e s h a l l d e v e l o p s h o r t l y .O n ly in 183 7 , a f te r a num be r o f yea r s o f inven to r s t ry in g to t r ans m i t e l ec t r i cs ig n al s o v e r a w i r e t o c a u s e m e c h a n i c a l m o v e m e n t , d i d M o r s e p u t t h e f i r st c o m -merc ia l t e l eg rap h in to opera t ion . P rev ious to the e lec t r i c t e l eg raph , s mo ke s igna l s( ine f f ec t ive in w inds ) , l an te rn - f l a s h ing ( ine f f ec t ive in c loudy w ea the r ) , ca r r i e rp igeons ( ine f f ec t ive in w inds , c loudy w ea the r , and du r ing mat ing s eas on) , andmes s enger s (d i t to ) had been employed . N ever the les s , P a r i s and L i l l e , s epa ra tedby 150 mi les , cou ld convey mes s ages w i th in tw o m inu tes , u s ing a f lag s em aph ores y s t e m w i t h m a n y i n t e r m e d i a t e f l a g m e n . B y 1 8 6 1 , t h e o v e r l a n d t e l e g r a p h h a di m p r o v e d t o t h e p o i n t w h e r e N e w Y o r k a n d S a n F r a n c i s c o w e r e i n c o n t a c t w i t heach o the r .

15~2oI Telephony and TelegraphyIn the mid -1850s , t e l eg raph cab le w as f i r s t l a id ac ros s the A t lan t i c~ and f a i l ed ,b e c a u s e t h e t r i e d - a n d - t r u e e m p i r i c a l m e t h o d s t h a t t e l e g r a p h e n g i n e e r s h a dd e v e l o p e d f r o m n e a r l y 2 0 y e a rs o f e x p e r i e n c e w i t h o v e r l a n d r o u t e s w e r e i n -a d e q u a t e t o t h e u n d e r w a t e r e n v i r o n m e n t . I t w a s t h e n t h a t t h e y o u n g W i l l i a mT h o m s o n ( 1 8 2 4 - 1 9 0 7 ) a p p l i e d h i s p r o d i g i o u s s c i e n t i f i c t a l e n t s t o t h e A t l a n t i cc a b le p r o b l e m , b o t h d e v e l o p i n g a t h e o r y f o r i t a n d d e v e l o p in g m o r e a c c u r a t e a n dmore s ens i t ive me thods to genera te and de tec t t e l eg raph s igna l s . (The baud r a tefo r the o r ig ina l A t lan t i c cab le w as pe rha ps 1 b i t / s. I n the yea r 2000 , co m pu te r


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6 3 2 Chapter 15 m Maxwell 's Equations and Electromagnetic Radiation

Recal l that w hen a signal is sent along a wire, the wire is not in equi l ibr ium, and can notbe considered to be an equipoten t ial . Dif feren t parts of the wire are at di f feren t voltages,and thus a current f lows in the wire, according to Ohm's law. Moreover, that voltageis due to charge, according to Volta's law. From these ideas, in 1853 Kelvin developeda theory of the telegraph. The main reason signal ing by single-wire telegraph cable ismore d if f icult u nde rwa ter than in air is that the capacitance is much greater in water.This is because charge on a wire in water polar izes the water, and due to the largedielectr ic constant of water (K ~ 80), a nearly equal and opposite amount of charge inthe water surrounds the charge on the ( insulated) wire. Hence, at a given charge ona wire, for a wire in air the voltages along the wire are much greater than for a wirein water. Faraday discovered this di f ference in capacitance at the beg inning of 1853,about the same t ime as Kelvin's theoret ical work.

A secondary reason signal ing by single-wire telegraph cable is more dif f icult under-wa ter tha n in air is that the effect ive cable resistance is greater w ith water. The t ime-varying electr ic currents induced in the water by the signal along the cable lead toaddit ional Joule heating and increased dissipat ion of energy. This corresponds to anincreased effective resistance for the cable.

m o d e m s a t ta i n e d n e a r ly 5 6 , 0 0 0 b i t s /s , a n d I n t e g r a te d S e r v ic e D i g it a l N e t w o r k ~I S D N ~ l i n e s c o m m o n l y a t t a i n e d n e a rl y 1 2 8 , 0 0 0 b i t s/ s. ) F o r h is c o n t ri b u t io n s t ot el eg r a p h y, T h o m s o n b e c a m e r ic h , a n d w a s m a d e a b a r o n a n d t h e n a l or d : L o r dK e lv in , t h e n a m e b y w h i c h h e is m o r e c o m m o n l y k n o w n t o d a y . T h e t e l e p h o n e ,w h i c h t o r e p r o d u c e s p e e c h r e q u i r e d m u c h m o r e f a i t h fu l r e p r o d u c t i o n o f e le c -t r ica l s igna ls than the te legraph, was invented in 1875.Ke l v i n ' s p rac t i c a l work i n t he f i e l d o f communi ca t i ons was t o be supe rsededby t ha t o f M ax we l l on e l ec t romagne t i c r ad i at i on . Al t hou gh p robab l y t he f ir s tpe r son wi t h whom M axwe l l sha red h i s e l ec t romagne t i c t heory , Ke l v i n i n i t i a l l yd i d no t a ccep t M axw e l l 's p red i c t i on o f e l ec t romag ne t i c r ad i a ti on . Un for t una t e l y ,M ax we l l , wh o d i ed i n 1879 a t t he age of 48 , d i d no t l i ve t o see h is 1865 pre -d i c t i ons ve r i f i ed by He r t z , i n 1887 . Ke l v i n wro t e t he pre face t o t he Engl i sht rans l a ti on of He r t z ' s book , Elec t r ic Waves .

1 5 , 2 . 2 M a x w e l l 's C o n t ri b u ti o nFor a nu m ber o f yea rs , M axw e l l wo rked t o expre ss, i n m a t hem a t i ca l fo rm, t hephysica l ideas of Faraday. O ne of the m i s tha t e lec t r ic charge i s assoc ia ted wi the l ec tr i c f ie l d l i ne s ~G au ss ' s l aw. A second is t he l aw t ha t t he re is no t rue mag-ne t i c cha rge ( so t he f l ux of /~ t h rou gh any c l osed sur face is al ways ze ro) . A t h i rdi s t ha t t he emf produced i n a c i rcu i t i s p ropor t i ona l t o t he ra t e a t whi ch mag-ne t i c f l ux li nes c ross t he c i rcu i t ~ Fa ra day ' s l aw, f ir s t g i ven ma t hem a t i ca l fo rm b yF . Neumann i n 1845 . In add i t i on , M axwe l l t r i ed t o deve l op an ana l ogy due t oFaraday, be tw een e lec t r ic curre nts and m agn et ic f ie lds . This ap pears in a l e t te rt o Ke l vi n , da t ed 1861 . In t he end , M axw e l l d ro pp ed t h i s anal ogy , bu t he pu t an-o t he r i n i t s p l ace . He found t ha t , a l ong wi t h cur ren t (bo t h f ree and Amp~r i an) ,t h e r e s h o u l d b e a n e w s o u r c e te r m o n t h e r i g h t - h a n d s i d e o f A m p e r e ' s l a w .T h e l aw s o f e l e c tr o m a g n e t i s m ~ c a l l e d M a x w e l l' s e q u a t i o n s ~ a p p e a r t o b ec o m p l e x . H o w e v e r , it s h o u l d b e k e p t i n m i n d t h a t M a x w e l l u s e d a v er y d i ff e r e n t


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1 5 . 3 M a x w e l l 's N e w T e r m ~ T h e D i s p la c e m e n t C u r re n t 633

a n d e v en m o r e c o m p l e x - l o o k i n g ~ b u t e q u i v a l e n t ~ f o r m u l a t i o n o f t h e e q u a ti o n so f e l e c t r o m a g n e t i s m t h a n w e u s e t o d a y . W h a t w e t o d a y c a ll M a x w e l l ' s e q u a t i o n sa c t u a l ly w e r e f i r st p u b l i s h e d b y H e a v i s i d e i n 1 8 8 5 a n d , i n d e p e n d e n t l y , b y H e r t z ,i n 1 8 9 0 .

W e n o w t u r n t o M a x w e l l ' s g r e a t e s t s c i e n t i f i c d i s c o v e r y , t h e o n e o n w h i c hw i r e l e s s t e l e c o m m u n i c a t i o n s i s b a s e d . I t i s a l so t h e b a s i s o f a ll c a b l e c o m m u n i -c a ti o n s, i n c l u d i n g b o t h m e t a l w i r e a n d f ib e r o p t ic s . I n d e e d , H e a v i s i d e s t u d i e dM a x w e l l ' s t h e o r y i n o r d e r t o a p p l y i t t o t e l e g r a p h y a n d t e l e p h o n y .

15 3 M a x w e l l 's N e w T e r m - - T h e D i s p l a c e m e n t C u r re n tI t i s r e a s o n a b l e t o i n q u i r e , " If , i n F a r a d a y ' s l aw , a t i m e - v a r y i n g m a g n e t i c f i e ldc a n p r o d u c e a c i r c u l a t i n g e l e c t r i c f ie l d, w h y c a n ' t , i n A m p ~ r e ' s l aw , a t i m e -v a r y i n g e l e ct r i c f ie ld p r o d u c e a c i r c u l a ti n g m a g n e t i c f ie ld ? " O n c e t h i s q u e s t i o ni s a s k ed , t h e is s u e b e c o m e s t o d e t e r m i n e h o w m u c h c i r c u l a ti n g m a g n e t i c f i el di s p r o d u c e d , a n d in w h i c h d i r e c t io n . H o w e v e r , d i r e c t o b s e r v a t i o n o f t h e e f f e c tb y , f o r e x a m p l e , t h e r a p i d d i s c h a r g e o f a c a p a c i t o r , i s d i f f ic u l t, e v e n w i t h m o d e r ne q u i p m e n t . I n d e e d , e v e n a f t er M a x w e l l d e t e r m i n e d , b y t h e o r e t i c a l m e a n s , h o wl a rg e t hi s t e r m is, i t t o o k o v e r 2 0 y e a r s b e f o r e t h e m o s t i m p o r t a n t e f f e c t iti m p l i e d ~ e l e c t r o m a g n e t i c r a d i a t i o n ~ w a s d e t ec t ed , by H e r tz .

T h e k e y e l e m e n t f o r M a x w e l l w a s t h e r e a l i z a t io n t h a t a n e w t e r m w a s n e e d e di n o r d e r t o m a k e A m p ~ r e ' s l a w c o n s i s t e n t w i t h c o n s e r v a t i o n o f c h a r ge . W ep r e s e n t t w o j u s t i f i c a t i o n s , o n e v e r y g e n e r a l a n d o n e u s i n g t h e s p e c i f i c e x a m p l eo f t h e c h a r g i n g o f a c a p a c i t o r .

Ge neral A rgume ntW e a l r e a d y h a v e t h r e e o f t h e e q u a t i o n s t h a t d e s c r i b e t h e e l e c t r o m a g n e t i c f ie ld .T o r e p e a t , t h e y a r e G a u s s ' s l a w , o ri i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i ! i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i ii i i i i i i ii i i i i i i i i i i! i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i ii i ! i i i i i i i i i i i i i i i i i i i i i i i ! i l l! i i i i i i ! i i i i i i i i i i i i i i i i i i i i i i i i i i i ! t h e n o - m a g n e t i c - p o l e l aw , o r

i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i ii i i i i i i i i i i i i ii i i i i i i i i i i i i i i i i i i i i r

a n d F a r a d a y ' s l a w , o r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . - . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . , : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . - : . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . ~ . . . . . . . .. . . . . . . . .. . . . . . . .. . . . . . . . .. . . . . . . .. . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . .. . . . . . . .. . .: : ......................... ..... : .............

iiiiiii!i! !ii!iiiiiiiiii!i~iiiiiiii,iiiiiiiiiil ii ! ~iiilii~iiiiiiiiii::ii:iiiiiiiiiiiiiiiiii:illi!ii:~iiliiiiliililiii!iiiiiiiiiiiiiilii::i:iiiiiiiiiiiiiiiiiii!illi!ii!iiiiiiiiiiiiiiiiiiiiii!iiiiiliiiiiii!~ii~iliiliiiiii~ililllii iiiiiliiiiiiiiiiii!iiiiilililiiiiiiiiiiiiii!!i iii i!iiiiiliiii!iiiiiiiiiiiiii!ilii!iiiilli:iiiilliiiiiii!iiiiiiiiiiiililiiiiiiliii!!iiiiiiiiiiiiiiiiiiiii!iiiiii:!iiiiiiiiiiiiiiiiii!iiiiiiiii!iii!iliiiiiiiii!iiiiiiiiiiiiiiii:iliiiiiiiiiiiiiiiiiiiiiiiiiiili!iiiili!liiliiiiiiiiii!!iiiiiiiiiiii!iliiiiiiiiiiiiiiiiiiiiii!ii!iiii!iiii!iiiiiiiiiiiiiiiiii,:illiiiiiiiiiiiiiiiiiiii!iiiiiii!iiiiiiiiiiiT h e f o u r t h l a w w o u l d a p p e a r t o b e A m p e r e ' s l aw , b u t t h e r e i s s o m e t h i n g

m i s s i n g f r o m i t ( as M a x w e l l n o t i c e d ) , r e l a t e d t o c h a r g e c o n s e r v a t i o n . C h a r g e


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634 Chapter 15 9 Maxw ell 's Equations and Electromagnetic Radiation

cons e rva t ion s t a t es tha t the f lux o f e l ec t r ic c u r r en t J l eav ing a clos ed s u r facem us t r e s u l t in a dec reas e o f the cha rge enc los ed . M athemat ica l ly , th i s takes thefo rm

f J . d f i l - dQ .en~ (15.1)d t "Tak ing the t ime de r iva t ive o f (4 .8 ) and us ing (15 .1 ) l eads to the r es u l t tha t

j J d A = l f d4 n k - ~ . h d A . (15.2)N o te tha t thes e a r e bo th in teg ra l s over closed surfaces.N o w c o n s i d e r A m p ~ r e ' s l a w ,

J B . d-~ - 4zrkm f J . d/t , (15.3)

w h e r e t h e i n t e g r a l o v e r d i l l - ~ d A i s over an open surface, and d-d - ~ds and dfi~a re r e la ted by the c i r cu i t -no rmal r igh t -hand ru le .The in teg ra l on the r igh t -hand s ide o f (15 .3 ) i s ve ry l ike the in teg ra l on thele f t -hand s ide o f (15 .2 ) , excep t tha t in (15 .3 ) the s u r f ace i s no t c lo s ed . M axw el lmo d i f i ed the r igh t -h and s ide o f (15 .3 ) by add ing in a t e rm re la ted to the r igh t -hand s ide o f (15 .2 ) . H is mod i f i ed A mp6re ' s l aw r eads

j ; ~ a t j(M a xw e l l s mo d i f i ed ~ p e r e s [ig i:]~ ); :i

o r

J( f f c t o f d is p la c e m e n tc u n O i i sJ D w a s c a l l e d b y M a x w e l l the displacem ent current dens ity . T h e a c t u a l c u r r e n tdens i ty J , l ike the e lectr ic f ie ld E, has sources an d s inks (such as cap aci tor p la tes ) .H ow ever , r ea r r ang ing (15 .2 ) an d us ing (15 .5 ) r e vea l s tha t J + J D has no s ou rceso r s in ks : f J . d A = 0 - - - f f D" d A = O, so f ( J + J D )" d A = 0 .Equa t ions (4 .8 ) , ( 9 .21 ) , ( 12 .4 ) , and (15 .4 ) a r e co l lec t ive ly kno w n asMaxwell ' s equat ions . F or o u r pu rpos es (k in /k ) -1 = 9 x 1018 m2/s 2 , w h ich i sessent ia l ly c 2, w h ere c is the ve loc i ty o f l igh t in f r ee space . I t t hus s hou ld com eas no s u rp r i s e tha t l igh t i s s omehow re la ted to e l ec t r i c i ty and magne t i s m.


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15.3 MaxweU'sNew Term--The Displacement Current 6 3 5

Figure 15.1 (a) Wire carrying current I to charge a pair of capacitor plates. Theopen surfaces $1, $2, and $3 all have the same c ircuit C as their boundary. (b) O nthe same figure we represent, for C, a circuit element d~ and a normal fi, andwithin the capacitor , the directions of dE/dt and/3 .1 5 o 3 o 2 Disp lacement Cu r rent w i th i n a Capac i to r

T h e c o m b i n a t i o n f + J D g u a r a n t e e s t h a t , d e s p i t e t h e a m b i g u i t y i n d e t e r m i n i n gw h i c h a r e a t o u s e f o r f fidA, t h e r i g h t - h a n d s id e o f (1 5 . 4 ) i s u n i q u e l y d e t e r m i n e d .L e t u s see h o w th i s wo r k s o u t f o r an e l ec t r i c c i r cu i t co n s i s t i n g o f a l o n g wi r e t h a t i sex t e r n a l ly d i sch a r g in g th e p l a t e s o f a c ap ac i to r . A t a g iv en in s t an t , l e t t h e cu r r e n tb e l . C o n s i d e r a n A m p e r i a n c i r c u it C t h a t is a c o n c e n t r i c c i rc l e s u r r o u n d i n g t h ewi r e . I t h a s a c i r cu l a t i o n g iv en b y th e l e f t - h an d s id e o f ( 1 5 .4 ) . T o co m p u te t h er ig h t - h an d s id e o f ( 1 5 .4 ) , co n s id e r t h r ee su r f ace s a s so c i a t ed w i th C . Su r f ace $ 1i s d i sk sh ap ed , co r r e sp o n d in g to t h e c i r c l e , t h r o u g h wh ich a cu r r en t I p a sse s . SeeFigure 15 .1 (a ) .T h e e l ec t r i c f i eld is z e r o o n $1 , so t h e r i g h t - h an d s id e o f ( 1 5 .4 ) i s g iv en b y4rrkm f J . d ft + -~ d f ~ . d A = 4JrkmI + O - (15.6)

Su r f ace $2 i s o b t a in ed b y d e f o r m in g th e d i sk , a s i f i t we r e co m p le t e ly ex t en -s ibl e , wh i l e l e av in g th e p e r i m e te r o f t h e c i r c le i n p l ace. T h e cu r r e n t I i n t e r sec t s$2 , as fo r $1 , and the e lec t r ic f ie ld i s zero on $2 , as fo r $1 , so the r igh t -hand s ideof (15 .4) i s g iven by (15 .6) , as fo r $1 .An ev en m o r e ex t en d ed v e r s io n o f $2 i s su r f ace $ 3, wh ich h a s p a r t o f it ssu r f ace c r o ss b e tw een th e p l a t e s o f t h e cap ac i to r. I n t h a t c a se , n o cu r r en t c r o sse sth e cap ac i to r , b u t t h e e l ec t r i c f i e ld i s n o n ze r o w i th in t h e cap ac i to r . T h e e l ec t r i cf i e ld t h a t c r o sse s $ 3 co m es ex c lu s iv e ly f r o m th e ch a r g e o n th e cap ac i to r p l a t een c lo sed b y $3 . T h u s w e can a r ti f ic i a ll y c lo se t h e su r f ace $3 , an d th e n u se G au ss ' sl aw . T h u s f o r $3 th e r i g h t - h an d s id e o f ( 1 5 .4 ) i s g iv en b y

d d km d4 rrkm / J . d f t + ~ j~ r E . d A - O + ~ - ~ r E . d d ~ - ~ -~ r E . d Akm dk d t--4zrkO ,~nc = 4JrkmI. (15.7)

Hence the r igh t -hand s ide o f (15 .4) i s the same for $3 as i t i s fo r $1 and $2 .


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636 Chapter 15 m Maxwell 's Equations and Electromagnetic Radiation

~ pp l i c a t i o n 1 5 . 'In free space , whe re th ere i s no t rue cur ren t , i f the /~ f ield is in to the page andstar ts to decrease , then the d isp lacement cur ren t , by (15 .5) , wi l l po in t ou t o fthe page . Let us app ly Ma xwel l ' s ex tension of Am pere ' s law, (15 .4) , to inc luded isp lacem ent cur ren t , and use Oers ted ' s r igh t -hand ru le. Then a /~ f ield wi l lc i rcu la te counterc lockw ise fo r the Am per ian c i rcu i t in Figure 15 .1 (b ) . Moreformal ly, s ince c lockwise d~ cor responds to h in to the page (and thus posi t ivecu r r en t i n to t h e p ag e ) , b o th t h e d i sp l acem en t cu r r en t ( i n t eg r a t ed o v e r t h ecross-sec tion) and th e m agnet ic c i rcu la t ion are negat ive, as expected .

H e r t z , i n t h e i n t r o d u c t i o n t o h i s Electric Wav es, n o t e d t h a t t h e r e a r e a t l e a s tf o u r way s t o t h in k ab o u t e l ec t r i c i t y : ( 1 ) a c t i o n a t a d i s t an ce ( Ch ap te r 3 ) ; ( 2 )th e e l ec t r i c f i e ld p r o d u ced b y d i s t an t f r ee ch a r g e ( t h i s i s l i t t l e m o r e t h an ac t io na t a d i s t an ce , a s in C h ap te r 4 ) ; ( 3 ) t h e su m o f t h e e l ec t r i c fi e ld s p r o d u c ed b yd i s t an t f r ee c h a r g e an d b y lo ca l ly n eu t r a l p o l a r i z a t i o n ch a r g e ( e.g. , in a d i e lec t r ic ,a s i n Ch ap te r 7 ); ( 4 ) t h e e l ec t r i c fi e ld d e sc r ib ed so l e ly b y th e l o ca l p o l a r i z a t i o n o fs p a c e P , a s e m b o d i e d i n t h e p h y s i c a l p i c t u r e o f fl u x t u b e s, a n d t h e m a t h e m a t i c s o ft h e d i s p l a c e m e n t v e c t o r D - s 0 /~ + / 5 , w h o s e o n l y s o u r c e is f r ee c h ar g e. H e r t zb e l i e v e d t h a t M a x w e l l t h o u g h t i n t h e f o u r t h f a sh i o n. M a x w e l l h i m s e l f n e v e rr ea l ly t o ld u s . He ju s t l e f t u s t h e eq u a t io n s .T h e p r e se n ce o f t h e d i sp l a cem en t cu r r e n t i s f e l t co n s t an t ly ; i t is e ssen t i a l t oe l e c t r o m a g n e t i c r a d i a t i o n . I n s h o r t , w i t h o u t M a x w e l l ' s n e w t e r m , t h e r e w o u l db e n o r ad io , n o t e l ev i s io n , n o su n l ig h t~an d n o l i f e .

1 5 . 4 E q u a t io n o f M o t i o n f o r a S t r i n g u n d e r T en s io nF o r M a x w e l l , i m m e r s e d i n t h e p h y s i cs o f hi s t i m e , i t w a s s e c o n d n a t u r e t o r e c o g -n i ze a wav e eq u a t io n . A b eg in n in g s tu d e n t o f p h y s ic s m ay n o t b e ab l e to d o th i s .I t is i m p o r t a n t , t h e r e fo r e , t h a t y o u l e a rn h o w t o r e c o g n iz e a w a v e e q u a t i o n a n dt h a t y o u b e c o m e a w a r e o f s o m e o f it s pr o p e r ti e s . B e c a u s e a s tr in g u n d e r t e n s i o ns u p p o r t s w a v es , it s e q u a t i o n o f m o t i o n s h o u l d s u p p o r t w a v es . F o r t h a t r e a so n ,w e s t u d y t h e m o t i o n o f a s t ri n g.F i r s t , h o wev e r , n o t e t h a t a l l wav es sa t i s f y t h e r e l a t i o n sh ip

x f = v , (15 .8)wh e r e )~ i s t h e w av e len g th , f i s t h e f r eq u e n cy in Hz ( cy c le s p e r seco n d ) , an d v isth e v e lo c i ty o f p r o p ag a t io n . ( No ta t io n a l wo e : W e ea r l i e r u sed th e sy m b o l )~ f o rc h a r g e p e r u n i t l e n g th , s o w e a g a i n ha v e t h e p r o b l e m o f t o o m a n y q u a n t i t i e s t orepre sen t . In th is chap ter , )~ wi l l re fer exclusively t o w a v e l e n g t h . )W e n o w m a k e a d i s t in c t i o n b e t w e e n t w o c la s se s o f w a v e s.

15.4 .1 Nondispers ive and Dispers ive WavesNondispersive wav es h av e th e sam e v e lo c i ty f o r a l l f r eq u en c ie s . T h i s i n c lu d eswav es o n a u n i f o r m s t ri n g , so u n d wav es i n a ir , an d e l ec t r o m a g n e t i c wav es i n


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15.4 Equ ation of Motion for a Str ing under Tension 6 3 7

Seismic waves on the sol id surface of the earth are nondispersive. That is wh y dista ntmeasurements perm it us to locate the p osit ion and intens ity of distant earthquakes.For a g iven type of wave to be usefu l f rom the po int o f v iew of communicat ion, i tshould be as nondispersive as possible. If water waves were nondispersive, we mightbe able to communicate f rom ship to d is tant sh ip by us ing them. They aren ' t , sowe can ' t . We use our vo ices (sound w aves) whe n near , and rad io (e lectromagnet icrad ia t ion) whe n far.

f r ee s pace . F o r each o f thes e , th e re i s a cha rac te r is t i c ve loc i ty o f p ropaga t io ntha t w aves o f a ll fr equenc ies s at is fy . A s a cons equence , i f a com pl ica ted s igna l( l ike s om eone ' s vo ice ) r ad ia tes ou tw ard , w h en i t a rr ives a t the l is t ene r , a l l t hef r equenc ies w i l l ar r ive a t the s ame t im e ( i. e. , nond i s pe r s ive ly ) , and the s oundwil l be in te l l ig ib le .Dispers ive w aves have a ve loc i ty tha t va r ie s w i th f r eque ncy (or , equ iva len t ly ,v a ri e s w i t h w a v e l e n g t h ) . T h e m o s t c o m m o n e x a m p l e o f t h is is w a v e s o n t h es u r f ace o f a bod y o f w a te r . The s e a ll tr ave l a t d i f f e r en t ve loc it ie s , the h igh f r e -quenc ies t r ave l ing the f as tes t . A pebb le d ropped in to a poo l genera tes w aveso f m a n y f r e q u e n c ie s a n d , b y ( 1 5 . 8 ), c o r r e s p o n d i n g ly m a n y w a v e le n g t h s. H e n c e ,i f a s econd p ebb le i s d rop ped in to the poo l , a f ew s econds a f t e r the f i r st , t heh i g h - f r e q u e n c y , s h o r t - w a v e l e n g t h w a v e s f r o m t h e s e c o n d p e b b l e w i l l c a t c h u pto and ou t ru n the low - f r equency , long -w ave len g th w aves f rom the f ir s t pebb le .To a lesser extent , l ight in mater ia ls and sound are d ispers ive.

!5o4~2 Physical PictureW e beg in w i th s om e genera l cons ide ra t ions . F o r s imp l ic i ty, neg lec t g r avi ty . A t ta chthe ends o f a s t r ing o f mas s pe r u n i t l eng th # to t w o pos ts , s uch th a t th e s t r ing isund er t ens ion F . (N o ta t iona l w oes aga in : O f ten the s ymb o l T i s u s ed fo r tension,bu t cons i s t en t w i th ou r p rev ious us age , T w i l l be r e s e rved fo r the per iod . T h esymbol T is a lso used for t empera ture . ) The s t r ing then fo rms a s t r a igh t l ine .S ee F igu re 15 .2 (a ) . Ro ta t ing the s t r ing ( a change in s lope) does no t caus e i tto v ib ra te . S ee F igu re 15 .2 (b ) . L i f t ing the s t r ing up an d dow n as a w h o le ( aun i fo rm d i s p lacem en t ) does no t caus e i t t o v ib ra te . S ee F igu re 15 .2 (c ) . Thusthe acce le r a t ion o f any pa r t o f the s t r ing does no t dep end on e i the r the va lueo f the ve r t i ca l d i s p lace me n t y o f the s t r ing o r o f i ts s lope d y / d x , w here x i s thehor izon ta l d i s t ance a long the s t ring .

F igu re 15 .2 A string under tension, which, when straight, does notmove: (a) original configuration, (b) ro tated configuration, (c) verticallydisplaced configuration.


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638 Chapter 15 9 Maxw ell 's Equations and Electromagnetic Radiation

M o r e o v e r, i f t h e s t ri n g is w r a p p e d o n a c u r v e d f r am e , a n d t h e f r a m e s u d d e n l yi s r em o v ed , t h e s t r i n g w i l l n o t r e t a in i t s sh ap e : i t w i l l s t a r t t o m o v e . T h i s t e l l su s t h a t c u r v a t u re , w h i c h i s p r o p o r t i o n a l t o d 2 y / d x 2, can cau se t h e s t r i n g toacce lera te . Since acce lera t ion i s g iven by d 2 y / d t 2, w e e x p e c t t h a t t h e s e c o n dt im e d e r iv a t iv e i s p r o p o r t i o n a l t o t h e seco n d sp ace d e r iv a t iv e . I n o th e r wo r d s ,the accelera t ion i s propor t ional to the curva ture .N o w c o n s i d e r t h e p r o p o r t i o n a l i t y c o n s t a n t . T h e a c c e l e r a t io n d 2 y / d t 2 s h o u l db e p r o p o r t i o n a l t o t h e f o rc e t h a t a c ts o n i t ( a n d t h is f o r c e s h o u l d b e p r o p o r t i o n a lt o t h e t e n s i o n F ), a n d i n v e r se l y p r o p o r t i o n a l t o t h e s t r in g ' s m a s s ( w h i c h s h o u l db e p r o p o r t i o n a l t o t h e m a s s p e r u n i t l e n g t h # ) . T h u s , i f w e a re l u c k y a b o u t t h ed i m e n s i o n a l i t y o f F a n d # , a n d d o u b l y l u c k y a b o u t t h e a r b i t r a r y c o n s t a n t , w em i g h t g u e s s t h a td 2 y F d 2 y

ndt 2 # dx 2" (15.9)

I n f ac t, t h i s eq u a t io n i s co r r ec t . A ch e ck th a t i t h a s t h e co r r ec t d im en s io n a l i t yc a n b e d o n e b y o b s e r v i n g t h a t , b e c a u s e o f t h e s p a c e a n d t im e d e p e n d e n c e s i nt h e d e n o m i n a t o r s o f ( 1 5 . 9 ) , F/ I~ s h o u l d h a v e t h e s a m e d i m e n s i o n a s t h e s q u a r eo f a v e lo ci ty. T o v e r i fy t hi s , n o t e t h a t F h a s t h e sam e d im en s io n a s a f o rce , o r N ,a n d t h a t # h a s t h e s a m e d i m e n s i o n a s a m a s s p e r u n i t l e n g th , o r k g / m . T h u s F / #h a s t h e s a m e d i m e n s i o n a s N - m / k g = J / k g = ( m / s ) 2, w h i c h is i n d e e d t h e s q u a r eof a ve loc i ty .T h i s d i scu ss io n n eg lec t s m o t io n in an d o u t o f t h e p ag e ( i. e. , a lo n g z) . Becau seg r av i ty h a s b een n eg lec t ed , m o t io n a lo n g z w i l l b e s im i l a r t o m o t io n a lo n g y .T h u s m o t i o n a l o n g z c a n b e d e s c r i b e d b y ( 1 5 . 9 ) w i t h t h e y 's r e p l a c e d b y z's.A c o m m e n t a b o u t n o t a t i o n : T h e v a r i ab l e y d e p e n d s o n b o t h x a n d t . In ( 1 5 . 9) ,t h e d / d x ' s a r e t ak en a t f ix ed t an d th e d / d t ' s a r e t ak en a t f i x ed x. I t is co n v en t io n a lt o u s e a n o t h e r s y m b o l f o r t h e s e par t ia l der iva t i ves , w h e r e o n ly o n e i n d e p e n d e n tv a r i ab l e is ch an g ed . ( T h in k o f i n d ep e n d en t v a r i ab l es l ik e t a s i n p u t , an d d ep en -d e n t v a r i ab l e s l ik e x as o u t p u t . ) T h u s , t h e m a t h e m a t i c a l c o g n e s c e n t i w r i t e a / a xor ax ins tead of d / d x , a n d a / a t or Ot i n s t e a d o f d / d t . W e wi l l b e a b i t s l o p p y inw h a t f o l lo w s , b u t a t le a s t w e h a v e p a i d l ip s e r v ic e to t h e m a t h e m a t i c a l c o n v e n -t i o n s t h a t y o u wi l l s ee i n m o r e ad v an ced co u r se s .

15.4~3 Der i va ti on o f the Wave E qua t i onL e t u s n o w d e r iv e ( 1 5 .9 ) . I t ap p l i e s i n t h e l im i t wh e r e t h e s lo p e d y / d x is small .Co n s id e r a c lo seu p d i ag r am o f t h e s t ri n g . See F ig u re 1 5 .3 .

I n a sp a t i a l r eg io n o f sm a l l l en g th d x, t h e l en g th o f s t r i n g is d s =v / d x 2 + d y 2 - d x v / 1 + ( d y / d x ) 2 ~ d x . T h u s d x h as m ass d m - # d s ~ , ~ d x . T h em ass t im es t h e acce l e r a t i o n in t h e y - d i r ec t io n i s t h u sd 2 y ~, # d x d 2 yd m - d ~ d t 2" (15.10)

H o r i z o n t a l m o t i o n o f t h e s t ri n g w o u l d c a u s e s ig n i fi c an t e x t e n s i o n o r c o m -p r e ss i o n, w h i c h w o u l d c h a n g e t h e t e n s i o n F . H o w e v e r , s i n c e t h e m o t i o n o f t h es t r i n g is e s sen t ia l l y v e r ti c a l , su ch t en s io n - ch an g in g e f f ec ts a r e n eg l ig ib le . I n t h a t


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15 .5 W av e s on a S tr i ng 639

Figure 15.3 Closeup o f s t r ing under t ens ion F . Inthe absence of constraining forces, a curved str ingwil l mov e, e ven if it in i t ial ly is at rest .c a s e, a s c a n b e s e e n i n F i g u r e 1 5 . 3 , t h e n e t f o r c e d F y o n d m i n t h e y - d i r e c t i o nis g i v e n b y t h e d i f fe r e n c e i n t h e y c o m p o n e n t s o f t h e n e a r l y c o n s t a n t t e n s i o n F .S i n c e a t x t h e a n g l e O(x) t h a t t h e s t r i n g m a k e s t o t h e h o r i z o n t a l is s m a ll , w e c a ns e t s i n 0 ~ t an 0 - d y / d x . T h e n

Fy = F [ s in O(x + dx) - s i n 0 ( x ) ] ~ F [ t an O(x + dx) - t a n 0 ( x ) ][ dy dy ] .~ F d2y dx= F d x x+dx d x x ~ " (15.11)

T h e l a s t a p p r o x i m a t e e q u a l i t y o f ( 1 5 . 1 1 ) u s e d t h e s t ra i g h t -l i n e a p p r o x i m a t i o n( v a l i d f o r s m a l l d x) t h a t f (x + dx ) - f (x) ~ (d f /d x) dx, w i t h f ( x ) = d y / d x . B yN e w t o n ' s s e c o n d l a w o f m o t i o n , w e e q u a t e ( 1 5 . 1 0 ) t o ( 1 5 . 1 1 ) , t h u s o b t a i n i n g( 1 5 . 9 ) .

C e r t a in l y , w e b e l i e v e t h a t ( 1 5 . 9 ) h a s w a v e s o l u t i o n s b e c a u s e s t r in g e di n s t r u m e n t s ~ l i k e v i ol in s a n d v io l as a n d c el lo s a n d g u it a rs a n d b a s s e s ~ s u p p o r tw a v e m o t i o n . L e t u s u s e t h a t t o g u i d e u s in f in d i n g w a v e s o l u t i o n s t o ( 1 5 . 9 ) . A sa s t a r t, r e w r i t e ( 1 5 . 9 ) a s

F . . . . .v = - - ( w a v e e q u a t i o n f o r a g t r i ~ ( I5 . 12 ~i~t2 ~x 2 ' lz iw h e r e v h as t h e d i m e n s i o n s o f a v e lo c i ty .

W e w i l l d is c u ss t w o t y p e s o f so l u t io n s t o ( 1 5 . 1 2 ) , s t a n d in g w a v e s a n d t rave l ingwav e s .

1 5 . 5 W a v e s o n a S t r i n gS t a n d i n g w a v e s o c c u r w h e n t h e r e a r e f ix e d b o u n d a r i e s , a s w i t h s t r in g e d i n s t r u -m e n t s . T r a v e li n g w a v e s o c c u r w h e n t h e r e a r e o p e n b o u n d a r i e s , o r w h e n w e a r ef a r f r o m t h e b o u n d a r i e s , a s w h e n w e s h o u t f r o m t h e c e n t e r o f a n im m e n s e a u d i -t o r i u m .

15o5~I Stand ing Waves and the Theory o f S t r i nged Ins t rumentsN o w , w h a t d o e s a w a v e f o r a s t r i n g e d i n s t r u m e n t l o o k l ik e ? I f i ts e n d s a r e a t x = 0a n d x - L, t h e n w e m a y c h o o s e c o o r d i n a t e s t h a t m a k e y = 0 a t t h e s e e n d s. S e eF i g u r e 1 5 . 4 .


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640 Cha pter 15 u Maxw ell 's Equations and Electroma gnetic Radiation

W h e n a s t r i n g o n a s t r i n g e d i n s t r u m e n t i s p l u c k e d o r p i c k e d o r p u l l e d o rbow ed , i t ma kes a cha rac te r i s t i c s ound . A na lys i s o f th i s s ound s h ow s tha t i t i sdo m ina ted b y a s ing le f requency . T here fo re , l e t u s t ry a s o lu t ion o f the fo rm

y ( x , t ) - g ( x ) sin (o) t + ~b). (15.13)H e r e t h e r e a r e m a n y u n k n o w n s " t h e f r e q u e n c y co i n ra d i a n s p e r s e c o n d , t h es h a p e f u n c t i o n g ( x ) , and the phas e q~ .(Reca l l tha t co , the f r equency in r ad iansper s econd , equa l s 2~r t im es f , t he f i e -L ~ q u e n c y i n c y c le s p e r s e c o n d . ) T h e f u n c -t ion g (x ) m us t s a t i s fy the b o u n d a r y c o n d i -

t ions t h a t g ( 0 ) - 0 a n d g ( L ) - O , i n o r d e rFigure 15.4 S tring that is f ixed a t i ts to m ak e y - 0 a t th e ends . Su bs t i t u t in g~nds. (15.13) in to (15.12) yields_o)2 g _ v2 d2 g v2 = --,F (15.14)d x 2 ' #

w h e r e w e h a v e f a c t o r e d o u t s in (~ o t + ~ ) .Th i s is a d i s gu i s ed ve r s ion o f the ha rm on ic os c i l l a to r equa t ion , (14 .11 ) . W i thK t h e s p r i n g c o n s ta n t , M t h e m a s s, a n d X t h e p o s i t i o n c o o r d i n a te , t h e h a r m o n i cos c i l l a to r equa t ion i s

- K X - M ddtX . (15.15)I n ( 1 5 . 1 5 ) , t is t h e i n d e p e n d e n t v a r ia b l e a n d X i s t h e d e p e n d e n t v a r i a bl e .Eq ua t ion (1 5 .1 5) has as i ts so lu t io n

X (t) - A sin(f2 t + ~b0), Kn = ~ , (]5 .1 6)w h e r e ~b0 a n d A a r e d e t e r m i n e d b y w h a t a r e c a ll e d th e i n i t i a l c o n d i t i o n s : th ein i t i a l va lues o f X and d X / d t . [ D o n ' t b e b o t h e r e d b y t h e u s e o f s2 i n ( 1 5 . 1 6 ) ,in s tead o f o90 as in (14 .12 ) ; w e a l r eady have u s ed th e looka l ike w in (15 .14 ) . ]C o m p a r i s o n s h o w s t h a t t h e s e t ( K , X , M , t ) of (1 5 .1 5) is jus t l ike th e set(0)2 , g , v 2 , x) of (1 5 .14) . S ince the solut io n t o (1 5 .1 5) is (1 5 .1 6) , w e can obt aint h e s o l u t i o n t o ( 1 5 . 1 4 ) b y s u b s t i t u t i n g t h e a p p r o p r i a t e q u a n t i t i e s . R e p l a c i n gs2 - v / K / M b y q - v/o )2/v i - co~v, t h e s o l u t i o n f o r t h e s h a p e f u n c t i o n g ( x ) o f(1 5 .14) is g iven by

g ( x ) - A sin(qx + ~b0), o)q - - . ( 15.17)vW e c a l l q ( y e t a n o t h e r n o t a t i o n a l w o e ~ q i s n o t t o b e m i s t a k e n f o r a c h a r g e ! )t h e w a v e n u m b e r . In (15 .17) , q p lays the sam e role as ~2 does in (15 16) , and as~o0 d i d i n ( 1 4 . 1 2 ) . ( W e r e a l ly h a v e a p r o b l e m w i t h t o o m a n y q u a n t i t i e s f o r t h e


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15.5 Wa ves on a Str ing 6 4 1

n u m b e r o f s y m b o l s a t o u r d i sp o s al . O t h e r s y m b o l s o f te n u s e d f o r w a v e n u m b e ra r e Q , K , an d k . E ach o f t h e m h as it s n o t a t i o n a l d i f f icu l ti e s ! )E q u a t io n ( 1 5 .1 7 ) s t i l l i sn ' t a so lu t io n to o u r p r o b lem ev en th o u g h i t s a t i s f i e sth e d i f f e r en t i a l eq u a t io n ( 1 5 .1 4 ) . T h e p r o b lem i s t h a t i t d o e sn ' t y e t sa t i sf y t h ebo un da ry con di t ion s g (0) = 0 = g (L ) . To sa t i sfy g (0 ) = 0 , se t ~b0 = 0 in (15 .17) .To sa t i sfy g ( L ) = 0 , se t s i n q L = 0 i n ( 1 5 .1 7 ) , w h ich im p l i e s t h a t

/ 4 7 rq - ~ . ( n a n o n ze r o in t eg e r ) ( 1 5.18 )LH e n c e , t h e b o u n d a r y c o n d i t i o n s r e s t r ic t t h e a l l o w e d v a lu e s o f t h e w a v e n u m b e r q .I t i s u n iv e r sa l ly t h e ca se t h a t b o u n d a r y co n d i t i o n s im p o se r e s t r i c t i o n s . SeeF ig u r e 1 5 .5 fo r t h e m o d e s co r r e sp o n d in g to n = 1 an d n = 2 .L e t ' s n o w f in i sh t h in g s u p . P l ac in g ( 1 5 .1 7 ) i n to ( 1 5 .1 3 ) y i e ld s t h e sp ace an dt im e v a r i a t i o n o f t h e d i sp l a cem en t y o f a s t r i n g t i ed d o w n a t x = 0 an d x = L .It is

~~~~~i~i~ii~~i~i~~~i~i~i~i~i~i~~~~i~i~~~i~i~i~ii i i i ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~!i i i i ~ ~ ~ ~ e ~ i i ~ ~ !i ii i i i i i i i i i i i i i ! i l i i ! i ~ i i ~ i i

wh er e , f r o m ( 1 5 .1 7 ) an d ( 1 5 .1 8 ) , ~o o n ly t ak es o n th e v a lu e siiiiiii~iiiii~iii!iiiiiiiiiiii]iiiiiiiiii~iiiiiii~i~ii~iii~i~iiii~i~iiiiiiiiiiii~iiiiiiiiiiii~iiiii~i~!iiiii~ii!iiiiiii~iiiiiiiiiiiiiiiiiiii~i~i~ii!ii~ii~iiiii~ii~iii~iiii~iiiiiiiiii~iiiiii!~i~iiiiiii~ii~iiiiiiii~i~i~i~~iii i i ) i i i i i i i i i

i i i i i i i i i i i i ! i i i i i i i i i l i i i ! i! i i ! i l i i i i i l i ! i i ii i i i i i i i ! i ! i i i i i i i l i i l ! ! i i i i i ! i i ! i i i i i i i i i i i ! i ! i i i i iT h i s eq u a t io n g ive s t h e n a tu r a l v ib r a t i o n a l f r eq u en c ie s , o r h a r m o n i c s of a s t r ingu n d e r t en s io n . T h e n = 1 f r eq u e n cy i s c a l l ed t h e f u n d a m e n t a l , o r f i r s t h a r m o n i c .T h e n = 2 f r eq u en cy i s c a l l ed t h e seco n d h a r -monic , o r f i r s t over tone . E q u a t io n ( 1 5 .2 0 ) a l soc o n t a i n s t h e t h e o r y o f t h e t u n i n g o f s t r i n g edi n s t r u m e n t s . F o r a gi v en i n s t r u m e n t t h e l e n g t hL i s f i x ed ; t h u s t h e sh o r t e r i n s t r u m en t s w i l lh av e th e h ig h e r f r eq u en c ie s . Fo r a g iv en l en g than d t en s io n , t h e l i g h t e r t h e s t r i n g th e h ig h e rFigure 1 5 . 5 (a) Fundamenta l th e f requenc y . (No te : Al l s t r ings on a g ivenm o d e o f a u n if o rm s tr in g . ( b) i n s t r u m en t h av e ab o u t t h e sam e t en s io n ; o th -Secon d h a rm o n ic o f a u n if o rm e r wi se , so m e su p p o r t s wo u ld b e b u i l t s t r o n g e rs tr in g, t h a n o th e r s . ) Fo r a g iv en l en g th an d m ass p e ru n i t l e n g t h , t h e h i g h e r t h e t e n s i o n t h e h i g h e rt h e f r e q u e n c y . A n y m o t i o n o f t h e s t ri n g is a s u p e r p o s i t i o n o v e r a p r o p e r a m o u n t( w i t h t h e p r o p e r p h a s e ) o f e a c h o f t h e m o d e s .W av es sa t i s f y in g ( 1 5 .1 9 ) a r e c a l l ed s t an d in g wav es b ecau se t h ey r e t a in t h e i rsh ap e ( g iv en b y s in q x ) a l t h o u g h t h e i r a m p l i t u d e c h a n g e s w i t h t i m e . W e s a y t h a tth e s t r i n g r e s o n a t e s a t t h e r e s o n a n t f r e q u e n c i e s g iv en b y ( 1 5 .2 0 ) . T h e aco u s t i cr e s o n a n c e s o f a r o o m a n d t h e e l e c t r o m a g n e t i c r e s o n a n c e s o f a m i c r o w a v e c a v it ya r e an a lo g o u s t o t h e r e so n an ce o f a s t r i n g .


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6 4 2 C h ap t e r 15 ~ M ax w e l l 's E q u a t i o n s an d E l ec t r o m ag n e t i c R ad i a t io n

~ Tuning a guitar stringA g u i t a r s t r i n g i s 6 0 cm l o n g an d h a s a m as s p e r u n i t l en g t h o f 2 .2 g / m .F i n d t h e t en s i o n i t s h o u l d b e g i v en s o t h a t i t s t h i r d h a r m o n i c h a s a f r eq u en cyf - 6 9 0 H z .Solution: Since ~o - 2zr f = 4335 s -1 , L - 60 cm, and n - 3 , the f i rs t par t o f(15.20 ) gives v = o ) L / n ; r - 276 m/s . The second par t o f (15 .20) g ives F = #v 2 =167 .6 N . Th is w ould l i ft a mass m - F / g = 17 .0 kg under the ear th ' s g r av ity .

Traveling WavesT r a v e l in g w a v e s , a s s e e n b y a n o b s e r v e r a t r e st , d o n o t r e p e a t i n ti m e . T h e y r a d i a t ee n e r g y a w a y f r o m t h e r e g i o n i n w h i c h t h e y a r e g e n e r a t e d . T h e r e i s s o m e t h i n gs p e c ia l a b o u t t h e w a v e e q u a t i o n g i v en b y ( 1 5 . 1 2 ) : i ts tr a v e l in g w a v e s d o n o tc h a n g e s h a p e. ( T h i s is b e c a u s e t h e v e l o c i t y is i n d e p e n d e n t o f t h e w a v e l e n g t h ; i tis n o n d i s p e r s i v e . ) T h u s , f o r a n o b s e r v e r m o v i n g w i t h t h e t r a v e li n g w a ve , t h e w a v ew i l l a p p e a r t o b e a t r e s t . I n t h i s s e c t i o n , w e d e r i v e ) ~ f = v , a n d w e s h o w t h a t( 1 5 . 1 2 ) h a s s o l u t i o n s t h a t c o r r e s p o n d t o w a v e s w i t h v e l o c i t y v .

C o n s i d e r a p o s s ib l e s o l u t io n t o ( 1 5 . 1 2 ) w i t h t h e f o r my ( x , t ) - g ( x - v t ) . (15 .21)

C l e a r ly , i n ( 1 5 . 2 1 ) , d e r i v a t i v e s o f g w i t h r e s p e c t t o x a re p r o p o r t i o n a l t o d e r i v a -t i v e s w i t h r e s p e c t t o t . I n d e e d , d y / d t - - v ( d y / d x ) , a n d d 2 y / d t 2 - v 2 ( d 2 y / d x 2 ) ,

x0

Figure 15.6 A r i g h t w a r d - m o v i n gtravel ing wave.

w h i c h i s ( 1 5 . 1 2 ) . H e n c e ( 1 5 . 2 1 ) i s i n d e e da s o l u t i o n t o ( 1 5 . 1 2 ) . M o r e o v e r , g ( x - v t )r e p r e s e n t s a w a v e t h a t t r a v e ls t o t h e r i g h ta t v e l o c i t y v b e c a u s e t h e x = 0 , t = 0 v a l u eg ( 0 ) i s a t x = v t a t t i m e t . S e e F i g u r e 1 5 . 6 .

N o t e t h a t , i n a w a v e , t h e m a t e r i a l d o e s n ' tt r a v e l ri g h t w a r d , o n l y t h e p o s i t i o n o f t h ep e a k ( a n d , m o r e g e n e r a l l y , t h e s h a p e ) o ft h e w a v e . A c h a l k m a r k n e a r t h e l e ft e n d o f

t h e s t ri n g d o e s n ' t t r a v e l t o t h e r i g h t w i t h a r i g h t w a r d w a v e , b u t r a t h e r i t m o v e sv e r t i c a l l y . S i m i l a r l y , w e d o n ' t g e t c a r r i e d t o s h o r e w h e n w e a r e f l o a t i n g o n t h ew a t e r a n d a w a t e r w a v e g o e s by .

N o w c o n s i d e r , f a r f r o m t h e w a l l s , t h e s p e c i f i c w a v e f o r mg ( x - v t ) - s i n [ q ( x - v t ) ] , (15 .22)

w h e r e t h e w a v e n u m b e r q i s a r b i t r a r y . ( W e c o u l d a l s o a d d a n a r b i t r a r y p h a s e~b, o r u s e a c o s in e f u n c t i o n . ) W e c a n r e l a t e q t o b o t h t h e f r e q u e n c y c o a n d t h ew a v e l e n g t h X .

F i rs t , i f a t f i x e d x w e w a i t a t i m e p e r i o d t = T = 1 / f = 2 zr /c o, t h e n b y d e f i -n i t i o n , i n ( 1 5 . 2 2 ) t h e r e w i l l b e a d e c r e a s e i n p h a s e o f 2 z r. E x p l ic i t ly , i n ( 1 5 . 2 2 )t h i s d e c r e a s e i n p h a s e i s q v T = q v ( 2 r r / c o ) = 2Jr, so

q v - o 2 - 2 J r f . (15 .23)


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1 5 . 6 E l e ct r o m a g n e ti c W a v e s 643

Also, i f a t f ix ed t we d i sp l ace o u r se lv e s b y a wa v e len g th k , t h en , b y d e f in i ti o n ,in ( 1 5 .2 2 ) t h e r e w i l l b e an in c r ea se i n p h ase o f 2 J r . E x p l i ci t ly , i n ( 1 5 .2 2 ) t h i sincreas e in pha se i s q )~ - 2J r , soiiiiiiiiiii!iiiiilli!iiii!!ili!iiiiiiiiiiiiiiii!ii iii !ii !!iiiii i i i i i i i ii ! i i ! iii iiiil iiiiiiiiiiiiiiii!ii!l iiii!!ill iiiiiiiiiiiiiii!iiiii!i!i!iiiii i i ! i i i iiiiiiiiiiiiiiii!iiiiiiii!iiiiiii!iiiiliiiiiiii:iiiiiiiii!i!iii::i!iiiiiiiiiii!iiiiii!iiiiiiiii!iiii!i!iiiiiii!!T ak in g th e r a t i o o f ( 1 5 .2 3 ) an d ( 1 5 .2 4 ) y i e ld s

As in d i ca t ed ea rl ie r , t h i s r e l a t i o n sh ip i s sa t is f i ed b y a l l wav es . I f v i s i n d ep en d en to f f r eq u en cy , a s f o r n o n d i sp e r s iv e wav es , t h e f r eq u en cy an d wav e len g th a r e i n -v e r se ly p r o p o r t i o n a l t o e ach o th e r .N o t e t h a t t h e p h y s i c a l m o t i o n o f t h e s t ri n g is a lo n g y , w h e r e a s t h e w a v ep r o p a g a t e s a l o ng x . H e n c e t h e m o t i o n o f t h e s t ri n g is t r a n s v e r s e t o t h e d i r e c t i o no f p r o p ag a t io n . I n co n t r a s t , s o u n d wav es i n a i r , i f t h ey t r av e l a lo n g x , p r o d u c ea d i s p l a c e m e n t o f t h e a ir a lo n g x, a n d t h u s t h e m o t i o n is l o n g i t u d i n a l t o t h ed i r e c t io n o f p r o p a g a t i o n .~ From requency to wavelength

Take the ve loc i ty o f sound in a i r and in water to be V a i r - 3 4 0 m / s , an dVwater - - 1500 m/s . For the gu i ta r s tr ing o f the p rev ious example , v = 276 m/s .For a f requency f = 300 Hz, f ind the cor respo nding waveleng ths .Solution: Using (15.25), f -3 0 0 Hz yields wavelengths k a i r = 1.133 m and)~water= 5.0 m, and ) ~ g u i t a r - - 0.92 m.Y o u a re n o w e x p e r t s o n t h e p r o p e r t i e s o f w a v e e q u a t i o n s, a n d y o u k n o w h o wt o re c o g ni z e a w a v e e q u a t i o n ~ ( 1 5 .1 2 ) ~ w h e n y o u s ee on e. W e c a n n o w r e t u r nt o t h e p r o b l e m o f e l e c t r o m a g n e t i c w a v es .

t 5 o 6 E lec t rom agn e t ic W avesN o w c o n si d er , i n t h r e e - d i m e n s i o n a l s p ac e , a n e l e c t ro m a g n e t i c w a v e t h a t h a s n oy o r z d ep en d en ce . T h i s i s c a l l ed a p l a n e w a v e , s in ce f o r an y p o in t o n an y p l an ex - c o n s t a n t , t h e w a v e h a s t h e s a m e a m p l i t u d e .

Ou r d e r iv a t io n f o r p l an e wav e e l ec t r o m ag n e t i c r ad i a t i o n c lo se ly f o l lo ws th ed e r i v a ti o n o f th e l a s t c h a p t e r f o r t h e s k in d e p t h . I t e m p l o y s t h e s a m e r e c t a n g u l a rc i r cu i t s , an d i t em p lo y s t h e sam e l aws , Am p ~r e ' s l aw an d Fa r ad ay ' s l aw , ex cep tt h a t n o w w e e m p l o y M a x w e l l ' s v e rs i o n o f A m p ~ r e ' s l aw . T h e r e s u l t f o r F a r a d a y' sl aw i s e x a c t l y t h e sam e a s i n t h e p r ev io u s ch ap te r . I f y o u a r e f am i l i a r w i t h t h a tsec t io n , d o n ' t b o th e r t o r ead th e su b sec t io n o n Fa r ad ay ' s l aw . T h e r e su l t f o rA m p ~ r e ' s l a w c h a n g e s , h o w e v e r, b e c a u s e n o w , i n s t e a d o f t h e r e a l c u r re n t , t h e r eis t h e d i s p l a c e m e n t c u r r e n t .C o n s i d e r e m p t y s p a c e f o r t h e r e g i o n x > 0 . L e t t h e x = 0 p l a n e be a s h e e t o fc o n d u c t o r t h a t p r o v i d e s a t i m e - v a r y i n g c u r r e n t a l o n g t h e y - d i r e c t i o n . H e n c e , a t


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6 4 4 Chapter 15 ~ Maxwell 's Equations and Electromagnetic Radiation

leas t w i th in the s hee t , by O hm ' s l aw the re i s an e lec t r ic f ie ld a long the y -d i r ec t ion .W e t h e r e f o r e a s s u m e t h a t t h e e l e c tr i c fi e ld p o i n t s a l o ng t h e y - d i r e c t i o n . M o r e -o v er , f ro m A m p e r e ' s r i g h t - h a n d r u l e, w e e x p e c t a m a g n e t i c f i e ld n e a r t h e s h e e tt h a t p o i n t s a l o n g t h e z -a x is . W e t h e r e f o r e a s s u m e t h a t t h e m a g n e t i c f i el d p o i n t sa long the z -d i r ec t ion . A p lane w ave w i th th i s fo rm i s s a id to be linearly polar-ized becaus e as t ime goes by the e lec t r i c f i e ld po in t s a long o r aga ins t the s amel inea r d i r ec t ion in s pace ( and s imi la r ly fo r the magne t i c f i e ld ) . I n con t r as t , l i gh tf r o m a l ig h t b u l b o r f r o m t h e s u n h a s e l e c tr i c a n d m a g n e t i c f i e ld v e c t o r s w h o s ed i r e c t i o n i n sp a c e c h a n g e s n e a r l y ra n d o m l y ( a n d r a p i d l y ) w i t h t i m e . S u c h l i g h tw aves a r e s a id to be unpolarized.O u r g o a l is t o d e t e r m i n e t h e t w o e q u a t i o n s d e s c r i b in g h o w t h e e l e c t r ic a n dmagne t i c f i e ld s Ey and Bz v a r y i n s p a c e a n d i n t i m e . W e a s s u m e t h a t t h e o n l ys p a t i a l d e p e n d e n c e c o m e s f r o m x .

U s e o f F a r a d a y "s L a wBecaus e the magne t i c f i e ld i s a long z, i t p roduces a magne t i c f lux a long z . Le t u st h e r e f o r e a p p l y F a r a d a y 's la w t o a s m a l l r e c t a n g u l a r c i r c u i t w h o s e n o r m a l h i isa long z, w i th d ime ns ion h l a long y and d x a long x . S ee F igu re 15 .7 .B y t h e c i r c u i t -n o r m a l r i g h t - h a n d r u l e, d ~ l m u s t b e t a k e n t o c i r c u la t e a s i n t h ef igu re . T hen the e lec t r i c c i r cu la t ion i s g iven by

f -. dE yE . d -g l - [ E y ( x + dx) - Ey( x) lh l ,~ -d-~x (h l dx) . (15.26)

T h e a s s o c ia t e d m a g n e t i c f lu x is

~ B - / B " d f t - l B . h ~ d A - B z ( h ld x ). (15.27)

T h e n e g a t i v e r a te o f c h a n g e o f t h e a s s o c ia t e d m a g n e t i c f lu x is

d ~ B = d B~ (h l d x) .d t d t (15.28)

dx /Fo r Faraday's aw- Ihi

f l l

=dx~For Am pere -Maxw ell law

Figure 1 5 . 7 Geometry describingelectroma gnetic radiation flow ing tothe right, caused by an electriccurrent on the x - 0 plane,oscillating alon g y. Tw o imag inarycircuits are draw n, one for use withFaraday's law (involving ma gneticflux), and the o ther for use with theAm pere-Maxw ell law ( involvingelectric flux).


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15.6 ElectromagneticW av e s 6 4 5

Usi ng Fa raday 's l aw t o equ a t e (15 .26) and (15 .28) y i e l dsd E y d B z= (15.29)d x d t "

1 5 , 6 , 2 U s e o f Am p ~re's L aw, as M od i f i e d b y M axw e l lBecause the e lec t r ic f ie ld i s a long y , i t p roduces an e l ec t ri c f l ux a long y . Le t ust he re fore app l y Ampere ' s l aw t o a sma l l r e c t angul a r c i rcu i t whose norma l ~2 i sa long y , wi t h d i mens i on h2 a l ong z and d x a l ong x . B y t he c i rcu i t -norm a l r i gh t -hand ru l e , we mus t t ake d~2 such t ha t i t c i r cu l a t e s a s i n F i gure 15 .7 . Then t hemag ne t i c c i rcu l a ti on is g iven by

J B . d-~2 - [ - B ~ ( x + d x ) + B~(x)lh2 d B z ( h 2 d x ) (15.30)~ x "For empt y space , t he t rue e l ec t r i c cur ren t i s z e ro , bu t t he d i sp l acement cur ren ti s nonzero. From (15.5) , i t i s given by

1 d E y ( h 2 d x )f / D " d f i - f J D" f t 2 d A - J D y ( h 2 d x ) - 4 rr k d t " ( ]5.31)Us i ng Ampere ' s l aw t o re l a t e (15 .30) and 4rc k m t i mes (15 .31) y i e l ds

dBz kmd E y d E yd x k d t - # ~ 1 7 6d t (15.32)

1 5 o 6 , 3 The E lectrom agnet ic W ave E quat ionTaki ng t he t i m e de r i va t i ve of (15 .29) and t he x-d e r i va t ive of (15 .32) , w e cane l i mi na t e d 2 E y / d t d x - d 2 E y / d x d t , t o ob t a i n

d 2Bz km d 2 B~ d 2 Bzdx ~ = k d t 2 = t z ~176 t 2 9 (15.33)A s i mi l a r equa t i on can be de r i ved for Ey . Not on l y a re E and/ ~ norma l t o e achot he r, t hey a re norm a l t o t he d i rec t i on o f p ropo ga t i on , wh i ch i s a l ong x .C o mp ar i son t o (15 .12) show s t ha t (15 .33) i s a wave equa t i on , w i t h ve l oc i ty

i i i i ~ i i ~ ~ i i i i ~ i i ~ i i i i i ii ~ i i ii ii ~ i ~ i ii i i l i i i i i i i ii~i i i i i i i i i i ii i i i il l l i i i i i i i l i i i i ' i i ~ i i ! i i iii i i i l i i i i iii!i !i i iii iiiiiil l i !i iiiiiii iiiiiiiiiiiiiii!Thus M axwe l l ' s equa t i ons have , i n f ree space , a so l u t i on cor re spondi ng t o acoupl ed wave i nvol v i ng bo t h t he e l ec t r i c and magne t i c f i e l ds , whi ch propaga t ea t a speed i den t i c a l to t h e speed of li gh t c in vacuum ! Sure l y th i s i s a h i n t t ha tl i gh t i s a fo rm of e l ec t romagne t i c r ad i a t i on ; i ndeed , t he e l ec t romagne t i c na t ureof l i gh t ha s been borne ou t by expe r i ment s and prac t i c a l app l i c a t i ons fo r ove r100 years.


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646 Cha pter 15 ~ Maxw ell 's Equat ions and Electromag net ic Radiat ion

15,6~ Pro perties o f E lectromagn etic WavesC o n s i d e r a t r a v e li n g w a v e o f t h e f o r m

B z ( x , t ) - A si n ( q x - o ) t ) . (15.35)Su b s t i t u t i o n i n t o (1 5 . 3 2 ) y i e l d s , w i t h (1 5 . 3 4 ) ,

d E y k d B zd t k m d x = - c 2 A q cos ( q x - o)t). (15.36)

T h i s i n t eg ra t e s t oEy - c 2 A q sin ( q x - o ) t ) = c A si n ( q x - c o t ),o) (15.37)

w h e r e w e h a v e u s e d ( 1 5 . 2 3 ) w i t h v - c . C o m p a r i s o n o f ( 1 5 . 3 5 ) a n d ( 1 5 . 3 7 )g ivesE y - c B z , o r I E I - c l B I , o r E - c B , (15.38)

f o r t h i s r i g h t w a r d - t r a v e l i n g w a v e . A l e f t w a r d - t r a v e l i n g w a v e w o u l d h a v e t h ew a v e f o r m c o s ( q x + o ) t ) , a n d Ey = -cBz.F r o m t h i s di s c u ss i o n w e c o n c l u d e t h a t , i n a v a c u u m , e l e c t r o m a g n e t i c w a v e sh av e t h e fo l l o w i n g p ro p e r t i e s :1 . T h ey t r av e l w i t h t h e v e l o c i t y o f l i g h t c .2. T h e i r f r e q u en cy ca is r e l a t ed t o t h e i r w av en u m b e r q b y co - c q .3 . L i k e w av es o n a s t r in g , t h ey a r e tr an s v e r s e . T h a t i s, t h e q u an t i t i e s t h a t v a ry -

/~ a n d / ~ a r e n o r m a l t o t h e d i r e c t i o n o f p r o p a g a t i o n ~ . T h e n , j u s t as (~, j , k)i s a r i g h t -h an d ed t r i ad , s o is ( /~ , / ~ , ~ ) a r i g h t - h an d ed t r i ad . M o re s p eci fi c al ly ,E x B p o i n t s i n t h e d i r e c t i o n o f p ro p ag a t i o n ~ , s o

- - - c ~ x /~ , /~- - lc ) E . (15 .39 )c4 . I E I - c l B I . Propert ies of radiat ion

Con s ider an e l ec t rom agnet i c wave tha t p ro pagates a long 3? , wi th E ins tan ta-n eo u s l y p o i n t i n g a l o n g ~ . E h a s a max i mu m amp l i t u d e Em of 18 V/m. (a )Find the ins tan taneo us d i rec t ion o f /3 . ( -b ) F ind the m ax im um value o f I/~1.S o l u t i o n : (a) Because /~, /~, and the direction of propagation ~) are mutuallyperpendicular,/~ must point along +2. Specifically, B along +3c satisfies (15.39).(b) By (15.38), the maximum value Bm o f IBI is Bm -- Em/c = 6.0 10 -8 T.

F in al ly , n o t e t h a t b y F o u r i e r ' s t h e o r e m , a n y s h a p e c a n b e r e p r o d u c e d a s a s u mo v e r si ne s a n d c o si n es i f e n o u g h w a v e l e n g t h s a r e i n c l u d e d . H e n c e w e c a n d e c o m -p o s e , f o r ex am p l e , a r i g h t w a rd - t r a v e l i n g l o ca l i z ed p u l s e , in t o a s u m (o r i n t eg ra l )


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15.7 The Fu ll Electromagnetic Spectrum 6 4 7

over w aves w i th a l l w ave leng ths . N ow , s ince in a vacuum each w ave leng tht r ave l s a t the s ame ve loc i ty , tha t means tha t a loca l i zed pu l s e a s a w ho let r ave ls r igh tw ard a t tha t s ameDirection of propagation

v-

Figure 15 .8 Representation of an electromagneticwave that is traveling rightward. At any instant oftime, its electric field and ma gnetic field areuniform along any plane defined by a constantvalue of x. In this case,/~ po ints alo ng y an d/~points along z.

ve loc i ty , w i thou t any changein s hape . Th i s nond i s pe r s ivebehav io r a l s o ho lds fo r s oundin a i r and for waves on a s t r ing .As discussed ear l ier , that isw h y s o u n d a n d l i g h t ar e u se f u lf o r c o m m u n i c a t i o n s .F igu re 15 .8 s ummar izesour r e s u l t s . Th i s dep ic t s tw ot y p e s o f d e p e n d e n t v e c to r s,th e electr ic f ie ld vector /~ ,and the magne t i c f i e ld vec to r/~ , a t a g iven ins tant of t ime.Proper ly , E and /~ , havin gd i f f e r en t un i t s , do no t r ea l lyco -ex i s t in s ame coord ina tes pace ; the x -ax i s r ep res en t sthe real space x-axis , the y-axis represents Ey space, and the z-axis represents

Bz s pace . N ever the les s , w e p res en t F igu re 15 .8 to s how how bo th /~ and /~propa ga te toge th e r in t ime a t the s peed o f l ight . A s t ime goes by , the cu rvesr e p r e s e n t i n g E a n d / ~ w o u l d m o v e r i g h t w a r d w i t h v e l o c i t y c .

15o7 The Fu l l E lec t romagnet ic Spect rumEqu a t ion (15 .34 ) i s c r it i ca l . I t says tha t a ll e l ec t rom agne t i c w aves in fr ee s pacemo ve w i th the s ame ve loc ity . I t a l so says tha t e l ec t rom agne t i c r ad ia t ion canoccur , in pr inciple , a t inf in i te ly h igh and inf in i tes imal ly low f requencies , wi thc o r r e s p o n d i n g s h o r t a n d l o n g w a v e l e n gt h s . A t t h e l o w - f r e q u e n c y e n d i s a c p o w e r(60 H z) . S ucces s ive ly h igher f r equency and s ho r te r w ave leng th g ive long r ad iow a v es , s h o r t ra d i o w a ve s, U H F , V H F , m i c r o w a v e s ( p r o d u c e d b y m i c r o w a v e t u b e sand by molecu la r ro ta t ions ) , in f r a r ed (p roduced by l a s e r s and molecu la r v ib ra -t ions ) , op t i ca l (p rodu ced by low -en ergy e lec t ron t r ans i tions ) , u l t r av io le t (p ro -duc ed by h igh -energy e lec t ron t r ans i tions ) , x - r ays (p rod uced b y ve ry h igh -ene rgye lec t ron t r ans i t ions ) , and gamma rays (p roduced by h igh -energy nuc lea r t r ans i -t i o ns ) . B e c a u s e o f t h e D o p p l e r e f fe c t, w h e r e b y t h e f r e q u e n c y o f a w a v e i n c re a s es(dec reases ) w h en the s ou rce app roaches ( r ecedes f rom) the obs e rve r , thes e fo rmsof e l ec t rom agne t i c r ad ia t ion a r e equ iva len t to one ano the r , exc ep t fo r the i r d i f -f e r en t f r equenc ies . S ee F igu re 15 .9 .Tab le 15 .1 s ummar izes the w ave leng ths a s s oc ia ted w i th v i s ib le e l ec t romag-n e t i c r a d i a t i o n ~ l i g h t .

Tab le 15 .1 Wavelengths of colorsi ! i i l ii i l l i i i . . . . . . . . . . .. . .. . .. . .. . .. .

400-440 nm 440-480 nm 480-560 nm 560-590 nm 590-630 nm 630-700 nm


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6 4 8 Chapter 15 ~ Maxwel l ' s Equat ions and E lec t romagnet ic Rad ia t ion

F r e q u e n c y f ( H z )

10 24 10 21H z I I I I I IG a m m a r a y s

E H z1018

I !

P H z1015

I I I I( u v )V i s i b l e

X - r a y sm I I I I I I I I I I I I

1 0 - ] 5 1 0 - 1 2 1 0 - 9 1 0 - 6f m p m g mm

T H z G H z M H z k H z H z1 0 1 2 1 0 9 1 0 6 1 0 3 1 0 ~

I I I I I i I I I I I I I I I

( IR) A m a t e u rr a d i o b a n d

M i c r o w a v e s T V F M A M L o n g r a d i o w a v e sI ! I I ! t I ! I I I t I I I1O - 3 10 ~ 10 3 10 6 10 9

m m m k l n

W a v e l e n g t h ; l ( m )Figure 15 .9 A r epresen ta t ion o f the e lec t rom agnet ic spec t rum . The up per l ine g ivesf r equenc y f , and the low er l ine g ives the cor r espond ing w aveleng th )~, wh ere )~ f - c .

1 5 , 8 E l e c t r o m a g n e t i c E n e r g y a n d P o w e r F l o wA n e n e r g y d e n s i t y i s a s s o c i a te d w i t h a n e l e c t r o m a g n e t i c w a v e , a n d i t t r av e l s a tt h e v e l o c i ty c o f t h e w a v e . T h e t o t a l e n e r g y d e n s i t y u i s t h e s u m o f t h e e l e c t ri ca n d m a g n e t i c e n e r g y d e n s it ie s , s o f r o m p r e v i o u s c h a p t e r s,

E 2 B 2U - - U E nu / A B - - { (15 .40)8zrk 8zrk~"

F o r t h e p l a n e w a v e o f S e c t i o n 1 5 6, E 2 2 a n d B 2 29 = E y - B ~ . U s e o f ( 1 5 . 3 4 ) t h e ny i e l d s c / k - ( km c ) - 1 , s o w i t h ( 1 5 . 3 8 ) , E q u a t i o n ( 1 5 . 4 0 ) b e c o m e s

H - - E B c E B E B E B E B8 z r k ) 8 r r c k m = 8 z r c k m r 8 z r c k m = 4 z r c k m " (15 .41)

W e n o w c o n s i d e r t h e f l o w o f e n e rg y , u si n g a n a r g u m e n t s i m i la r to t h e o n e w eu s e d i n C h a p t e r 7 w h e n w e c o n s i d e r e d t h e f lo w o f c h a rg e . I n s t e a d o f c h a r g ed e n s i t y n e , w e n o w c o n s i d e r e n e r g y d e n s i t y u ; in s t e a d o f d r i ft v e l o c it y v a ,w e n o w c o n s i d e r t h e s p e e d o f li g h t c ; a n d i n s te a d o f e le c t r ic c u r r e n t p e r u n i ta r e a J , w e n o w c o n s i d e r t h e p o w e r p e r u n i t a r e a S .C o n s i d e r a sm a l l im a g i n a r y b o x o f a re a A n o r m a l t o x a n d t h i c k n e s s d x


20/49

15.8 Electromagnet ic Energy and P ower Flow 649

A m o r e g e n e r a l t r e a t m e n t s h o w s t h a t t h e e n e r g y f l o w a n d t h e m a g n i t u d e o ft h e i n t e n s i ty a r e b o t h g i v en b y t h e P o y n t i n g v e c t o ri i ! i ii i i i i i i ii i i i i i i i! i i i i ' i i i i i i i i i { i i i i i ! i i i i i i i i ' ! ii l i i i i i i i i i i i i i i ! i! i i i i i i i i ' i i i i ! i i i i i i i ! i i i i i i i i li i i i i i i i ! i i i i i i i i ii ! i i i i i ii i i i i i i i i ii i i i i i i i i i i i i l i ! i ! i i ! ! i i i i il i i i i i i i i! i i i i ! i i i i i i ! !i i i i i i i i i i':!~ ':' : ~ , i i i i i ! ~!',iii! ~ i i~ i i i i ! ii ll, i i i~ : ,iii iiilll i i iiii iiiii i i i ~ i 0 i i i i ~ i~ i~: ,ii i i i ! i i i i i i i i i i i ! i i i i i i i ! i i i i i i i i i i ii ~ i i i i i i i i i i i i i i i i { i ! i i i i i i i i i ! i i i i i l li: i ': ,i i i ! i ! i l i l i ! i i i i i i i i i i i i i ! i i i il i l li!i~:': iii i, i! ii i i i i i i i i i i i ii ii! i lli!i iliil i i i i i i i i i! i i i ! i i ; !,i!iiii i i i i i i i ii i i i i l li i i i i i i i ~ , ! i i i i i i i i i i i l,ii i l i i i i i i i i i i i ! i i i i i i i i i i i i i i i i i i i i i ii , i i ~ , i i i i i i ' : i i i i i i i i i i i ! ! i i i i i ! ! i i ! i i i i i i ii i i ii i i ! i ! i i i!!i ii ii il !i i l i i i i i i i i i i l iC le a r ly , $ i s t he v e c to r f o r m o f S, so S - IS I . T h e P o y n t i n g v e c t o r e x p l a i n s h o wp o w e r f l o w s i n t o a w i r e t h a t i s s u b j e c t t o J o u l e h e a t i n g . T h e e n e r g y e n t e r s t h es ide s o f t he w i r e , r a the r t ha n a long i t s a x is , f r om the e lec t romagnet ic f i e ld t o t h ew i r e . O f c o u r se , a v o l t a i c c e ll o r m e c h a n i c a l m o t i o n c o n v e r t e d t o a n e m f ( b yF a r a d a y ' s la w ) i s t h e u l t i m a t e s o u r c e o f e n e r g y f o r t h e e l e c t r o m a g n e t i c f ie ld . L ight f rom the the ear thu n at

V is ib l e r a d i a t i on f r om the sun ha s a n ave r a ge i n t e ns i t y S o f a bou t 1500 W /m 2a t t he e a r th ' s o r b i t , f o r w h ic h RE = 1 .50 x 1011 m. F ind the charac te r i s t icva lue o f t he m a x im um e l e c t r ic a nd m a gne t i c fi e ld s Em a n d Bm i n c i d e n t o nthe e a r th .Solution: Including a factor of on e-half f rom averaging over an oscil la tion (as forac c ircuits) , (15.42) and (15.38) give

S - E2m - E2m . (1 5. 44 )8J r km c 2 /x0cH e r e Em denotes a max im um e lec t ric f ie ld , ave raged over a l l r ad ia t ion f requen-cies. Solving for Em gives Em = v/2t . toc$, which eva lua tes to Em = 1060 V /m .Cor responding to th is is B m = E m / c = 3.54 x 10 -6 T.

Radia t ion by an iso t rop ic spher ica l source . A n i so t rop ic spher ica l source i s onef o r w h i c h t h e r a d i a t i o n i n t e n s i t y i s t h e s a m e i n a ll d i r e c ti o n s , a s f o r a l i g h t b u l ba n d f o r t h e s u n . I n th i s c a se , t h e a v e r a g e t o t a l p o w e r 75 i s o b t a i n e d b y m u l t i p l y i n gt h e a v e r a g e i n t e n s i ty S ~ a p o w e r p e r u n i t a r e a ~ b y t h e s u r f a ce a r e a 4z r R 2 o f as p h e r e o f r a d i u s R . T h u s

73 - $(4zc R2). (15.45 )S i n c e 73 i s i n d e p e n d e n t o f r ad i u s R , i t i s u s e f u l t o r e w r i t e ( 1 5 . 4 5 ) a s

$ - 4zr R 2" (15 .46)B y ( 1 5 . 4 4 ) , t h e i n t e n s i t y $ v a r i e s a s E ~ . H e n c e ( 1 5 . 4 6 ) i m p l i e s t h a t Em fa l l s of fi n v e r s e l y w i t h d i s t a n c e , a n d s i m i l a r l y f o r B m . L i g h t f r o m t h e s u n a t J u p i t e r

Find 72s, , , and S, Em and Bm a t Jupi te r . Take Rj = 1 .43 x 1 0 1 2 m .


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650 Ch ap ter 15 a Maxw el l' s Equat ions and Elec t romagnet i c Rad ia t ion

Solution: Using va lues o f S and R appropr ia t e to the ear th , f rom (15 .45) wefind that the sun radiates an average power 75sun = 4.18 X 1026 W 1. (W ith ou t thispower, there w ould be no l ife on the earth.) N ext , using Rj = 1.43 x 1012 m in(15.46), we dedu ce tha t , a t Jupiter , $ = 16.3 W /m 2. Final ly , using the values ofEm and Bm at the earth (Example 15.4), and the fact that Em a n d Bm fall off in-verse ly wi th radius , we deduce tha t Em = 111 .2 V /m and Bm - - 3.71 x 10 -7 T a tJup it e r. Only t e r res ti a l measu rem ents o f S , R E, and Rj were neede d to ob ta in th isinformat ion. (However, i f the raw data on S is taken on the earth, compensat ionm us t be m ade fo r absorp t ion and sca t t e ring by the ear th ' s a tmosphere . )

1 5 9 M o m e n t u m o f E le c tr o m a g n e t i c R a d i at io n ,R a d i a t i o n P r e s s u r eJ u s t as t h e l i g h t w a v e c a r r i es e n er g y , i t a ls o c a r ri e s m o m e n t u m ~ t h i s d e s p i t e t h el i g h t h a v i n g n o m a s s ! I f i t is a n y c o n s o l a t i o n , f o r li g h t t h e r a t io o f m o m e n t u m pt o en e rg y U is as l o w a s p o s s i b l e : p / U = 1 / c .T h e d i s c u s s i o n t h a t f o l l o w s s h o w s t h a t a c h a r g e d p a r t i c l e c a n a b s o r b e n e r g ya n d m o m e n t u m f r o m a l i n e a r l y p o l a r i z e d l i g h t w a v e , a n d t h u s t h a t l i g h t i t s e l fm u s t p o ss e ss b o t h e n e r g y a n d m o m e n t u m . M o r e o v e r, t h e r a t io o f e n e r g y a b so r p -t i o n to m o m e n t u m a b s o r p t io n is i n d e p e n d e n t o f th e s p e c if ic p a rt ic l e. H o w e v e r ,t h a t r a ti o is v e r y m u c h d e p e n d e n t o n t h e f a c t t h a t e n e r g y an d m o m e n t u m a r ea b s o r b e d f r o m l i g h t .C o n s i d e r a c h a r g e q , o f m a s s m , i n a p l an e e l ec t ro m ag n e t i c f i e ld lik e~ t h a ti n F i g u re s 1 5 . 7 an d 1 5 . 8 , t r av e l i n g to t h e r i g h t , w i t h E a l o n g y , a n d B a l o n gz . T h e e l e c t r i c f o r c e o n q i s a l o n g y , a n d t h e m a g n e t i c f o r c e i s n o r m a l t o z .E x p l i c i tl y , q f ee ls t h e L o ren t z f o r c e (1 1 . 3 5 ) , o r

F - q ( E + f i x B ) = q ( E y ~ - v x B ~ , + v yB z Y c). (15 .47)B e c a u s e E y - c B z , and ]vl


22/49

15.10 Index # Refraction an d Snell's Law of efraction 65 1

B e c a u s e B z = E y / c , c o m p a r i s o n o f ( 1 5 . 4 8 ) a n d ( 1 5 . 4 9 ) s h o w s t h a t Fx e q u a l s72/c. M ore exp l i c i tly , a t any in s tan t o f t im e7~ F y v y q E y v y E y= = = = c . ( 1 s . s o )Fx Fx q vyB z Bz

T h u s t h e e n e rg y a b s o r b e d ( d U = 7)dt) a n d t h e m o m e n t u m a b so r b ed [dpx =( d p ~ / d t ) d t = Fxdt] a r e p r o p o r t i o n a l , w i t h a c o e ff i c ie n t t h a t i s i n d e p e n d e n t o f t h eabsorber . Expl ic i t ly , (15 .50) g ives d U = D d t = c F x d t = c d p ~ . I n t e g r a t i o n o v e rt i m e t h e n g i v e sU = p c . ( e n e r g y U a n d m o m e n t u m p o f E M w a v e ) ( 15 .5 1)

( H e r e w e h a v e w r i t t e n p f o r Px.)Co ns ide r a s e t o f vanes , pa in ted b la ck on one s ide and s i lve r on the o the r ,t h a t a r e f r e e t o r o t a te . I n a h ig h v a c u u m , t h e y w i ll , u n d e r u n i f o r m i l lu m i n a t i o n ,a b s o r b m o r e m o m e n t u m f r o m t h e s il v e r s id e. T h i s is b e c a u s e i n t h e r e f l e c t io np r o c e s s t h e r e is a m o m e n t u m c h a n g e t h a t i s t w i c e a s g re a t a s t h e a c t u a l in c i d e n tm o m e n t u m . H o w e v e r , i n ex p e n s i ve l o w - v a c u u m r a d i o m e t e rs t u r n o p p o s it el y .T h i s is b e c a u s e t h e m o m e n t u m o f t h e r e l a ti v e l y m a s s i v e ai r m o l e c u l e s ( m a s s iv er e l a ti v e to l i g h t) d o m i n a t e s a t a t m o s p h e r i c p r e s s u re , a n d t h u s t h e h o t m o l e c u l e st h a t l e a ve t h e b l a c k s i d e o f t h e v a n e g i v e t h e v a n e m o r e o f a k i c k t h a n d o t h el igh t w aves tha t r e f l ec t o f f the s i lve red side .W e m i g h t t r y t o u s e t h e m o m e n t u m o f l i g h t t o d i r e c t a s p a c e c ra f t . T a ki n ga bu r s t o f ene rgy U = 10 W-hr s = 3 .6 x 104 J fo r a d i r ec ted l igh t s ou rce , thec o r r es p o n d i n g m o m e n t u m is p = 1 .2 x 1 0 . 4 kg-m/s . I f th i s l igh t com ple te lyre f l ec t s o f f a s pacec ra f t o f mas s 103 kg , the s pacec ra f t w ou ld ga in tw ice th i sm o m e n t u m , t h e r e b y i n c r e a si n g i ts v e l o c i ty b y 1 . 2 x 1 0 - 7 m / s . T h e m o m e n t u mof l igh t is low be caus e i t s ve loc i ty c i s h igh .S i n ce p r e s s u re P i s f o r c e p e r u n i t a r e a , o r r a t e o f c h a n g e o f m o m e n t u m p e ru n i t a r e a, b y ( 1 5 . 5 1 ) w e h a v e

s ince S is po w er p e r un i t a r ea . Th i s i s the rad i a t i on pre s sure of ligh t . The r ad ia t ionp r e s s u r e o f t h e s u n i s b e l i e v e d t o b e r e s p o n s i b le f o r p r o d u c i n g c o m e t t a il s. A tt h e s u r fa c e o f t h e s u n , S - ~ /4 r c R s~ n - 6 8 7 x 107 W /m 2, so (1 5 .52) yields ar ad ia t ion p res s u re th e re o f P rod = 0 .229 P a .

15.10 I n d e x o f R e f r a c t i o n a n d S n e l l ' s L a w o f R e f r a c t i o nI n a re a l m a t e r i a l, t h e e q u a t i o n s d e s c r ib i n g t h e e l e c tr i c a n d m a g n e t i c f i e ld m u s ti n c l u d e t h e e f f e c t o f e l e c t ri c a n d m a g n e t i c p o l a r i z a t i o n o f t h e m a t e r i a l. A s s h o w nb y ( 7 . 1 3 ) , t h e e l e c t r i c f i e l d w i t h i n a d i e l e c t r i c m e d i u m i s d e c r e a s e d b y t h e d i -e lec t r i c cons tan t (o r r e l a t ive pe rmi t t iv i ty ) x , due to e l ec t r i c po la r i za t ion . Thus


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652 Cha pter 15 ~ M axwell 's Equations and Electromagnetic Radiation

Ta b l e 15.2 Ta ble of index o f refraction (characteristic o f optical frequencies). . . . . . .(5 TP ) W a t e r C r o w n g l a s s F l in t g l a s s D i a m o n d ~ I ce B e n z e n e : L u c i t e S a lt

1.000293 1. 33 3 1.52 1.66 2.42 1.31 1.501 1. 49 1 1.544k = (4zre0) -1 -+ (4zrKe0) ~ . Al tern at ively , the pe rm it iv i ty E0 off ree space is m ul-t ip l i ed by K .S i m il a rl y , t o o b t a i n t h e m a g n e t i c f i el d w i t h i n a m a g n e t i c m e d i u m , t h e e f f e c to f t h e m a t e r i a l 's m a g n e t i c p o l a r iz a t i o n , a s s h o w n b y ( 1 0 . 2 3 ) a n d ( 1 0 . 2 4 ) , t h em a g n e t i c p e r m e a b i l i t y o f f r e e s p a c e # 0 i s m u l t i p l i e d b y t h e r e l a t i v e m a g n e t i cp e r m e a b i l i t y # r . F er r it e s , f o r exam ple , h ave # r ~" 10 4 a t m i c r o w a v e f r e q u e n c i e s.H o w e v e r , e v e n f o r m a g n e t i c m a t e ri a ls , # r ~'~ 1 a t opt ic al f requ encie s , b eca use thep r o c e s s e s r e s p o n s i b l e f o r m a g n e t i c p e r m e a b i l i t y , s u c h a s d o m a i n w a l l m o t i o n ,c a n n o t r e s p o n d a t o p t i c a l f r e q u e nc i e s , w h i c h a r e a r o u n d 1 01 5 H z . O n t h e o t h e rhand , s ince e lec t rons a r e r e s pons ib le fo r e l ec t r i c po la r i zab i l i ty , and they c a nr e s pond a t op t i ca l f r equenc ies , 6r i s no t un i ty a t op t i ca l f r equenc ies .

T h e e f f e ct o f n o n z e r o p o l a r i z a b i li t y a n d p e r m e a b i l i t y o n th e p r o p a g a t i o nv e l o c i ty o f e l e c t r o m a g n e t i c w a v e s is t h a t , w h e n c o m p u t i n g t h e v e l o c i ty o f li g h t,i n ( 1 5. 3 4 ) w e m u s t m a k e t h e r e p l a c e m e n t s k --+ k/6r (60 -+ 606r) and km -+ km#r( # 0 --+ # o l l r ) 9 T h i s l e a d s t o

C m C7J "-" V k m ~ ~ 6 r ~ r - - - ~ '

~:: ~ ! i; ~ i !i 84184il ! i~ / 6 r / ~ r , ( sp e e d o f I ig h t i ~ m a ~ } i i

w h ere n is ca l l ed the i n d e x o f r e f r ac t io n . I t d e p e n d s u p o n t h e d e t a i l e d p r o p e r t i e s o fa g iven mate r i a l . S ee Tab le 15 .2 . The o r ig in o f the w o rd re f rac t ion o n l y b e c o m e sc l e a r w h e n w e d i s c u s s w h a t h a p p e n s w h e n l i g h t p a s s e s f r o m o n e m a t e r i a l t oanother , as we do shor t ly .F o r e a c h m a t e r i a l t h e r e i s a c h a r a c t e r i s t i c f r e q u e n c y d e p e n d e n c e . S e eF igure 15 .10 .1. 7

Index ofrefract ion 1.6 -

1.5

Heavy flint glass

Light flint glass

Zinc crown glass1 I I I I I I I0.4 0.5 0.6 0.7 0.8

Wave leng th (nm)Figure 15.10 Index of refraction as a function of wavelength for threetypes of glass.


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1 5 . 1 0 I n d e x o f R e f r a c t io n a n d S n e ll 's L a w o f R e f r a c t io n 6 5 3

T h e d e p e n d e n c e o f n o n f r e q u e n c y l e a ds t o d i s p er s io n ; t h a t i s, t h e d i f f e re n tc o lo r s o f l i g h t i n t h e m e d i u m d o n o t t r a v e l w i t h q u i t e t h e s a m e v e l o ci ty . D i s p e r -s io n m u s t b e m in im ized f o r o p t i c a l f i b e r s a lo n g wh ich l i g h t s ig n a l s a r e sen t . I n aq u an t i t a t i v e sen se, t h is d i sp e r s io n o f l i g h t i n m a te r i a l s i s m in o r co m p ar e d to t h ed i sp e r s io n o f wa te r wav es .

15,10.1 H o w F r e q u en c y a n d W a v e l e n g t h B e h a v e o n R e fl e cti o nand Re f rac ti onCo n s id e r a p l an e wav e o f f r eq u en c y o21 im p in g in g o n th e a i r - wa te r su r f ace a t anan g le 01 wi th r e sp e c t t o t h e n o r m a l . See F ig u re 1 5 .1 1 ( wh ich f ea tu r e s a b r o w np e l i can ey e in g a r a th e r l a r g egoldf ish) .T o l e a r n h o w f r e q u e n c ych an g es o n c r o ss in g th e su r f ace ,co n s id e r a r e l a t ed p r o b lem , i n -v o l v i n g s o u n d t r a n s m i s s i o n a n dr e f l e c t i o n . L e t a d r u m m e r b e a tm o n o t o n o u s l y a t t h e r a t e o fo n c e p e r s e c o n d , w i t h t h e s o u n din th e a i r t r an sm i t t ed ac r o ss t h esu r f ace o f a f i sh t an k . T h e t im ei n t e r v a l b e t w e e n b e a t s i n b o t hth e a i r an d th e wa te r i s o n eb e a t p e r s e c o n d . M o r e o v e r , t h ef r e q u e n c y o f a n y s o u n d r e f l e ct e do f f t h e su r f ace is o n e b ea t p e r

seco n d . I n o th e r wo r d s , t h e f i e -Figure 15.11 Reflec tion and refrac tion of l igh t, que ncie s in each mate r ia l a re theA beam of l igh t f rom beh ind the pe l ican i s same.

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