Maxwell's Equations in Media and Their Solution *

Transcription

1 Maxwell's Equatios i Media ad heir Solutio * ao Zhag College of Nuclear Sciece ad echology, Beijig Normal Uiversity, Beijig 1875 taozhag@bu.edu.c Abstract Magetic field magetizatio, polarizatio ad iduced magetizatio are aalyzed. It is show that the existig Faraday's law of electromagetic iductio is ot reasoable i media. A modified Faraday's law of electromagetic iductio i media is put forward ad used to revise existig Maxwell s Equatios i media. A solutio of the revised Maxwell s Equatios is preseted. Key words Faraday s Law of iductio i media, Maxwell s Equatios, electromagetic waves, total electric field, total magetic field 1. Itroductio Maxwell s electromagetic theory leads to the discovery of electromagetic waves (EWs). Maxwell s Equatios i vacuum have bee verified with may experimets. Besides EWs i vacuum, the iteractio betwee media ad EWs is also a importat subject. Recet related researches iclude millimeter wave imagig ad metamaterial etc. he cotradictios betwee the existig theory ad some ew experimetal results idicate that the iteractio betwee media ad EWs is a complicated subject. I this paper, propagatio of the EWs i the media is studied, ad some viewpoits are preseted. For simplicity, "media" i this paper meas isulatio, homogeeous, isotropic ad ifiite media, ad the media are i quiescet state. "Electromagetic waves (EWs)" meas oly the refractio parts of EWs, ad does ot iclude the lost parts of EWs due to reflectio, absorptio ad scatterig etc. 2. Recetly-proposed viewpoits about refractio mechaism of EWs 2.1 he electro cloud coductors Ref. [1] idicates that each electro cloud i the molecules of the media ca be see as a tiy coductor. he electro-cloud coductors are very small, so that the electric fields ad the magetic fields of the EWs ca peetrate them. Whe the alteratig magetic field B of the EWs exists i the electro-cloud coductor, db/dt iduces a electromotace i the electro-cloud coductor, ad there may be a iduced curret o the electro-cloud coductor. he iduced curret is actually a statistical result of the electro s motio, ad is a additive directioal motio superposed o the origial motio of the electro i its electro cloud. he magetizatio by the iduced curret existig i the electro-cloud coductor is called the iduced magetizatio [2]. For distiguishig, the traditioal magetizatio is called the magetic field magetizatio. * Beijig Sciece echology New Star Program (Grat No ). he Chiese versio of this paper ca be foud o 1

2 Hece, i additio to the magetic field magetizatio M ad the polarizatio P, there may be the iduced magetizatio M F i the media uder the EWs. M caused by the magetic field. M F caused by variatio of the magetic field, ad has a differet phase with the magetic field. M F is related to db/dt, ot B [2]. M is related to B, ot db/dt. 2.2 Refractio mechaism of EWs i media Refractio of the EW i the media is caused by the iduced magetizatio, ot by the polarizatio ad the magetic field magetizatio, because the eergy of the iduced magetizatio is ot a part of the eergy of the EW, while the eergies of the magetic field magetizatio ad the polarizatio are parts of the eergy of the EW itself [1]. his ca be explaied by the phase differece Δφ betwee the magetic field of the EW ad the magetic field of the iduced magetizatio (i.e. the magetic field of the iduced curret o the electro-cloud coductor). Δφ =π, see Fig.1. Δφ =π/2 was preseted i our previous articles, but ow we believe Δφ =π. 1. (a).5 (b) (c) ω t Figure 1 Phases of some variables. (a) B, i.e. the magetic field of the EW; (b) the iduced electromotace i the electro-cloud coductor by db/dt; (c) the magetic field of the iduced curret o the electro-cloud coductor. Sice the magetic field of the iduced magetizatio is i differet phase with B, it is ot ivolved i the mutual trasformatio process of electromagetic fields of the EW, ad the eergy of the iduced magetizatio should ot be regarded as a compoet of the eergy of the EW, although it comes from the EW. Ref. [1] calls the eergy of the magetic field of the iduced magetizatio the refractive eergy, ad believes that durig propagatio of the EW i the media the EW exchages the refractive eergies bac ad forth with the electros i the media (the electros have mai cotributio to the exchages, ad the cotributios of protos etc. ca be omitted). It is the eergy exchages that cause the EWs slowig dow i the media (EWs 2

3 refractio). his mechaism will be discussed further i Sectio 4.1. Accordig to above refractio mechaism ad the priciple of eergy coservatio, a calculatio method of the refractive idex is deduced [1]. he refractive idices of helium, eo, argo, air ad the alcohol solutios have bee calculated with the method. he results are listed as i ables 1 ad 2. able 1 he calculated ad the measured refractive idices of some substaces substaces helium eo argo air calculated refractive idices( -1) [3] [4] [5] -4 [1] measured refractive idices(-1) [6] [6] [6] -4 [7] able 2 he calculated ad the measured refractive idices of the alcohol solutios alcohol cotet/wt% calculated refractive idices [1] measured refractive idices [8] he calculatio results of refractive-idex are i good agreemet with the measured values i the hadboos. 2.3 Quatum characteristics of the refractive eergy, ad iflueces of frequecy of EWs o refractio ability of media he refractive eergies are holed by the electros i the molecules. Sice the electros are cofied i the molecules, accordig to the theory of quatum mechaics, the refractive eergies must be of quatizatio. herefore, as the EW iteracts with the electro i the molecules, oly if the electromagetic iductio is strog eough, i.e. the eergy quatum of the EW is large eough, to mae the electro absorb the refractive eergy from the EW. his situatio is similar to the photoelectric effect [9]. Usig above viewpoit to aalyze the relatioship betwee frequecy of the EWs ad the refractive abilities of the molecules, we ca aticipate [1] : (1) I the rage of light frequecy, the electromagetic iductio betwee light ad the electros i the molecules is strog eough, ad all the molecules have eough refractive abilities; (2) As the frequecy decreasig, firstly the ier electros ad the the outer electros i the molecules gradually do ot absorb refractive eergies, ad the molecules lose their refractive abilities gradually; (3) I the rage of microwave frequecy, the electros i most molecules do ot absorb refractive eergies, ad the molecules ad the media made of them i ature have little refractive abilities. Researches o millimeter-wave imagig utilizig refractive dielectric leses attract a lot of attetios at preset. his techology has may potetial applicatios, such as i security chec, military, medicie ad trasportatio. But its imagig quality is ot satisfactory so far. he dielectric leses are mostly made of polymer materials. Ref. [1] believes that most of the polymer materials have wea refractive abilities to the millimeter waves. hat should be oe of the reasos for the low quality of the imagig. 3

4 2.4 Method for improvig materials refractive abilities to millimeter waves he refractive idices of the traditioal artificial materials have ot reached the expected values [11]. Ref. [1] believes that the coductors i the traditioal artificial materials have wea electromagetic iductios whe they meet the millimeter waves, ad that the cage-shaped graule of coductor (CGC) should have a strog electromagetic iductio as it meets the millimeter waves, because CGC icludes closed loops of coductor. he millimeter waves ca pass through the closed loops ad produce strog electromagetic iductios with CGC. So strog iductio currets form o CGC, ad that maes CGC possess eough refractive ability to the millimeter waves. he experimetal results show that CGC materials have cosiderable refractive abilities to the millimeter wave, while the polymer materials have little refractive ability to the millimeter wave [1]. I a word, it is the iduced magetizatio that results i refractio of the EWs i the media. his viewpoit is ot cosistet with the traditioal oe. he traditioal viewpoit is that the magetic field magetizatio ad the polarizatio result i refractio of the EWs i the media. Sice the former opiio is supported by several good verificatios of the experimetal data i ables 1 ad 2, we have good reaso to doubt the ratioality of the latter opiio. 3. Problem with Maxwell s Equatios i media Ref. [12] believes that the existig Maxwell Equatios i media are ot quite reasoable. he existig Maxwell s Equatios i the media are [13] E =, (1) E =, (2) B =, (3) P E = μ + μ M + με, (4) B where E is the electric field itesity, B the magetic iductio, P the polarizatio ad M the magetic field magetizatio i the media. Equatio (2) is Faraday s Law of iductio. It ca be expressed as E dl = ds. (5) Equatios (2) ad (5) do ot iclude the polarizatio term ad the magetic-field-magetizatio term. herefore they do ot apply to usig i the media. he followig example [12] is offered to illustrate its failure i the media. Suppose that there are a toroid made of the medium ad a varyig magetic field B passig through the toroid (B is i vacuum), see Fig.2. Let B be perpedicular to plae S which is circled by ceter lie l of the toroid, ad the symmetry axis of the magetic field B coicide with that of the toroid, as show i Fig.2. hus the iduced electric field caused by d S coicides with the ceter lie l of the toroid, ad the absolute value of E is the same oe everywhere o the ceter lie l. he iduced electric field caused by 4

5 d S maes the medium (the toroid) polarized. Suppose the polarizatio o the ceter lie l is P. P is i the same directio as the iduced electric field because of the symmetry. Let eep uchaged durig a period of time ( ), the d S i plae S eeps uchaged also. hus the iduced electric field by d S does ot vary with time, ad P does ot vary either. herefore the polarizatio curret i the medium is durig this period of time, i.e., the polarizatio of the toroid does ot ifluece the magetic field B. Differet ids of the media have differet P values, ad P iflueces E [13]. So the macro electric field E ad E d l o the ceter lie l are differet for differet media. his meas that E d l is ot l correct oly i vacuum. always equal to d S. Hece Eqs. (2) ad (5) are ot correct i the media. hey are symmetry axis of magetic field B coicides with that of toroid l ceter lie of toroid B toroid S cross sectio of toroid E P Figure 2 A varyig magetic field B passig through the toroid iduces a electric field o the ceter lie l of the toroid, ad this electric field produces a polarizatio P i the medium (the toroid). Sice P reduces the electric field, the fial macro electric field E o the ceter lie l of the toroid is differet from medium to medium. Usually, μ μ holds for most of the media, therefore the media have little ifluece o the excitig magetic field B. While differet media ofte show quite differet ε values, ad have quite differet electric fields i the media uder the same excitig electric field. So, if i Fig. 2 both the electric field ad the magetic field are i the media, it is also easy to illumiate that Eqs. (2) ad (5) are ot correct i the media. I the ext sectio, the existig Faraday's law of electromagetic iductio is modified so as to mae it applicable i the media. With this modificatio a revised Maxwell s Equatios i the media are obtaied. 5

6 4. Maxwell s Equatios i media 4.1 Without the iduced magetizatio i media Suppose that there are the polarizatios ad the magetic field magetizatios, but ot the iduced magetizatio i the media uder the EW. I the followig text, E ad B deote the electric field ad the magetic field i the media respectively, E ad B deote the electric field ad the magetic field i vacuum respectively. Firstly, chage Eq. (4) ito ( E + P/ ε ) ( B - μm ) = με, (6) Eqs. (4) ad (6) are reasoable i the media. hey reflect the relatioship betwee the chage of the electric field (E+P/ε ) ad the curl of the magetic (B μ M) [14]. hey are i agreemet with Maxwell s electromagetic theory: the chage of a electric field produces a magetic field, ad the chage of a magetic field produces a electric field. Similar to the relatioship betwee (E+P/ε ) ad (B μ M) i Eq. (6), accordig to the chage of a magetic field produces a electric field, coversely Faraday's law of electromagetic iductio i the media should reflect the relatioship betwee the chage of (B μ M) ad the curl of (E+P/ε ): ( B μm ) ( E + P/ ε) =. (7) he differece betwee Eq. (7) ad Eq. (2) is that Eq. (7) icludes the polarizatio term ad the magetic-field magetizatio term. Equatio (7) shows that [12] B, M, E ad P are all ivolved i the process of mutual trasformatio betwee the electric field ad the magetic field of the EW. heir eergies should be regarded as the compoets of the eergy of the EW. he propagatio of the EW i the media is the process of mutual trasformatio betwee varyig (E+P/ε ) ad varyig (B μ M). Ref. [12] calls E =E+P/ε the total electric field, ad B =B μ M the total magetic field. he meaigs of the total electric field E ca be explaied with Eq. (6) as follows: Chage of E with time produces the curret ε E/, ad chage of P/ε with time produces the curret P/. hese two currets idepedetly exist, ad each produces its ow magetic field. hese two magetic fields have the same phase, ad together they form the magetic field of the EW [15]. Sice chages i both E ad P/ε cotribute to the formatio of the magetic field of the EW, E =E+P/ε is called the total electric field. B =B μ M has a similar meaigs as E. Note that the expressios of E ad B give here are oly applicable i the media of this paper. Propagatio of the EW i the media is the process of the mutual trasformatio betwee E ad B. hus, i order to express the EWs i the media, we ca simply replace E ad B i Maxwell s Equatios i vacuum with E ad B ad obtai Maxwell s Equatios i media: E = ρ / ε, (8) Τ E =, (9) 6

7 B =, (1) E B = με. (11) Substitutig E = E+P/ε ad B = B μ M ito Eqs. (8) (11) we have ( E + P/ε ε, (12) ) = ρ / ( 7 ( B μ M ) E + P/ ε) =, (13) ( B μ M) =, (14) ( E + P / ε) ( B μm ) = με. (15) E, B ad M are of curl fields [13], ad there is o et charge i the media (ρ=). So E= P = B= M= [13]. herefore Eqs. (12) (15) ca be chaged ito E =, (16) M E = + μ P/ ε, (17) B =, (18) E P B = μ ε + μ + μ M. (19) Comparig Eqs. (16) (19) with Eqs. (1) (4), we ca see that Eqs. (16), (18) ad (19) are idetical to Eqs. (1), (3) ad (4) respectively, except for Eq. (17), which is derived from Eq. (13). Eq. (17) ad Eq. (13) show that the chage of B ad the chage of μ M together produce a electric filed, the this electric filed produces the polarizatio P ad the compositive field E. Eq. (17) ad Eq. (13) are ot oly applicable i vacuum but also i the media. he itegral form of Eq. (13) is ( B μ M ) ( E + P/ ε) dl = ds. (2) Eq. (2) accords with the experimet i Fig. 2: I the ceter l, E+P/ε is equal to E [16] which is the field whe the toroid does ot exist, i.e. E is the field i vacuum. B μ M is equal to i B which is the magetic field i vacuum (because M=). So Eq. (2) becomes E dl = ds. Obviously, it is correct. l I a word, Faraday's law of electromagetic iductio i the media should reflect the relatioship betwee the field (B-μ M) ad the field (E+P/ε ). It ca write as Eq. (7), Eq. (9), Eq. (17) or Eq. (2). hese equatios have the same meaigs, just i differet forms. 4.2 With the iduced magetizatios i media I this sectio the followig situatio will be cosidered: Uder the EW, there are ot oly the magetic field magetizatio ad the polarizatio, but also the iduced magetizatio i the

8 media. As metioed i Sectio 2.2, the magetic field of the iduced magetizatio has a differet phase with the magetic field of the EW, so the magetic field of the iduced magetizatio is ot a compoet of the magetic field of the EW itself. It is well ow that, whe describig the EW, the static fields should ot be cosidered i Maxwell's Equatios because they do ot belog to the fields of the EW itself. For the similar reaso, we believe that the field ad the curret of the iduced magetizatio should ot be icluded i Maxwell's Equatios whe describig the EW. hus Maxwell's Equatios i the media i this sectio are idetical to those obtaied i the last sectio. I short, whether i vacuum or i the media, ad regardless whether there is the iduced magetizatio i the media, Maxwell's Equatios ca write as Eqs. (8)-(11) or Eqs. (16)-(19) whe they are used to describe the EWs. 5. Solutio of Maxwell's equatios Ispired by the solutio of Maxwell's equatios i vacuum which ca be foud i may related boos, we obtai a simple-harmoic-wave solutio to Eqs. (8) - (11): E B = E = B m m, (21) where E m ad B m are the amplitudes of E ad B respectively, ω is the agle speed, is the wave umber, is the referece idex, φ is the iitial phase. Propagatio directio of the EW is alog the z directio of the rectagular coordiate system, ad E ad B poit to the x directio ad the y directio respectively. is also the ratio of the eergy of the EW i vacuum over that i the media, or the ratio of the volume of the EW i vacuum over that i the media [1]. Eq. (21) is the solutio of the EW ot oly i vacuum but also i the media. From Eqs. (8)-(11) ad the solutio Eq. (21) we have ω = ω ε μ =, (22) c or ω ω = =, (23) c / u where u=c/, c=1/(ε μ ) 1/2. u is propagatio velocity of the EW. If there is o other ow coditio, u or value caot be obtaied from Eqs. (8) - (11) ad the solutio. u or is the udetermied parameter of Eq. (21), just lie ω ad φ. u or ca be obtaied from measuremets or calculatios. It ca be see that is ot directly related to ε r ad μ r. I other words, refractio of the EWs i the media is ot directly related to the polarizatio ad the magetic field magetizatio. he polarizatio ad the magetic field magetizatio ifluece value idirectly [1]. From Eqs. (8)-(11) ad the solutio Eq. (21) we also obtai E = cb, (24) 8

9 I the uiform ad ifiite media, which is the precoditio of this paper, D=ε E [17], where D=ε E+P ad D is electric displacemet [13]. So E = E+P/ε = D/ε = E, (25) ad B = B. (26) herefore E m =E m ad B m =B m, where E m ad B m are respectively the amplitudes of the electric field ad the magetic field of the EW i vacuum. Usually, EWs itesities declie due to reflectio, absorptio ad scatterig as they pass the media. At the Itroductio it is stated that we oly cosider the refractio parts of EWs, ad do ot cosider the lost parts of EWs due to reflectio, absorptio ad scatterig etc. So Eq (26) oly deals with the refractio parts of EWs. Hece i both vacuum ad the media, regardless whether there is the iduced magetizatio i the media, aother form of solutio of Eqs. (8)-(11) is E B = E = B m m, (27) I Eq. (21) or Eq. (27) value i vacuum is differet from that i the media. 6. Coclusio he existig Faraday's law of electromagetic iductio E dl = ds is ot reasoable i media because it does ot iclude the polarizatio item ad the magetic field magetizatio item. I isulatio, homogeeous, isotropic ad ifiity media, regardless of whether there is the iduced magetizatio i the media, Maxwell's equatios for describig the EWs are E =, Τ E =, B =, E B = με. hese equatios are also applicable i vacuum. Propagatio of the EWs i the media is the process of mutual trasformatio betwee varyig (E+P/ε ) ad varyig (B μ M). A simple-harmoic-wave solutio to the above equatios is 9

10 E B = E = B m m. is the udetermied parameter of the above solutio, just lie ω ad φ. value i vacuum is differet from that i the media. Refractio of the EWs i the media is ot directly related to the polarizatio ad the magetic field magetizatio. Refereces [1]Zhag ao. Refractio of light i media. Sciece i Chia G 5(5) 591-6, 27 [2]Zhag ao. A possible mechaism of curret i medium uder electromagetic wave. Chiese Physics, 15(8): , 26 [3]Zhag ao. Meaig ad calculatio of equivalet volume of electro cloud Mar. 27 [4]Zhu Mi, Liu Wei, Zhag ao. Calculatio of Electro Wave Fuctios ad Refractive idex of Ne. Sciece i Chia G, 51: [5]XIE Guo-Qiu,MA Ku,HUANG Shi-Zhog. heoretical calculatio of the refractive idex for Argo. Joural of Atomic ad Molecular Physics, 26(5): (i Chiese), 29 [6]Zhag Xiag-Yu. Hadboo of Chemistry (i Chiese). Beijig: Natioal Defece Press, pp ,1986 [7]Su He, Wag Zhi-Liag. Hadboo of Fudametal Physics (i Chiese). Hohehot: Ier Mogolia Press, p544, 1981 [8]Yao Yu-Bi, Xie ao, Gao Yig-Mi. Hadboo of Chemistry ad Physics (i Chiese). Shaghai: Shaghai Sciece ad echology Press, pp424-44, 1985 [9]Zhag ao. Electromagetic iductio betwee light ad electro Feb. 29 [1]Zhag ao. Material Made of Artificial Molecules ad Its Refractio Behavior uder Microwave, 25 Sep. 214 [11]Colli R E. Field heory of Guided Waves. New Yor: McGraw-Hill Boo Co.,196 [12]Zhag ao. Maxwell s Equatios i Medium Mar. 26 [13] see may boos o electromagetics [14]Che Big-Qia, Shu You-Sheg, Hu Wag-Yu. Moographic Study o Electromagetics (i Chiese). Beijig: Higher Educatio Press, p656, 21 [15] same as [14], p661 [16]Cheg Shou-Zhu, Jiag Zhi-Yog. Commo Physics (volume 2) (i Chiese). Beijig: Higher Educatio Press, p115, 1998 [17] same as [16], p124 1