Chapter 2: The Derivation of Maxwell Equations and the form of the boundary value problem

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1 Chape : The eiaion of Mawell quaions and he fom of he bounda alue poblem In moden ime, phsics, including geophsics, soles eal-wold poblems b appling fis pinciples of phsics wih a much highe capabili han meel he analical soluions fo simple, classic poblems. This is done wih gea assisance of sophisicaed laboao epeimens and poweful compue echniques, including boh he hadwae and he sofwae. Neeheless, all mahemaical analsis sill deepl oos wih fundamenal mahemaical phsics. The mahemaical phsical pinciples o ule he elecomagneic poblems ae he Mawell equaions. ames Clek Mawell , Figue. eleganl inegaed he elecic, magneic, and he eleco-magneic inducion heoies pio o his ea and fomed a se of diffeenial equaions. This inegaion has been known as he Mawell equaions heeafe. Figue.. ames Clek Mawell The ne subsecion gies he majo deiaion of he Mawell equaions. The inegaed he Ampee s law, he Faada s law and wo mahemaical-phsical heoems and fomed a se of fou paial diffeenial equaions.

2 . The Consiuie elaions imila o he consiuie elaions in coninuous medium mechanics, hee ae also consiuie elaionships in elecomagneics. Consiuie elaions descibe he medium s popeies and effecs when wo phsical quaniies ae elaed. I cane be iewed as he descipion of esponse of he medium as a ssem o ceain inpu. Fo eample, in coninuous medium mechanics, he esponse of a linea-elasic medium o sain can be descibed b he ooke s law, and he esulan is he sess. The elaion beween sess and sain is he ooke s law. In anohe wod, ooke s law is he consiuie elaions fo linea elasici. In elecomagneics, hee ae fou fundamenal consiuie elaionships o descibe he esponse of a medium o a aie of elecomagneic inpu. Two of hem descibe he elaionship beween he elecic field and he conducie cuen, and he elecic displacemen, and he ohe wo descibe he elaionship beween he magneic field and he magneic inducion, and he magneic polaiaion M. Quaniaiel, hese fou consiuie elaionships ae σ ε M χ whee σ is he elecic conducii, ε he dielecic pemiii, he magneic pemeabili, and χ he magneic suscepibili. We will deoe he enie Chape 3 o discuss he elecomagneic popeies of eah maeial in ems of hese fou paamees. I is noewoh ha he fis elaion is he well-known Ohm s law in a micoscopic fom. These fou paamees eclusiel descibe he elecomagneic popeies of a maeial. I is necessa o poin ou ha some of hem ae ine-elaed. To undesand he behaio of hese elecomagneic paamees ae he cenal piece o undesand he geophsical esponse when geophsical sues ae emploed o sole an engineeing, eploaion, and enionmenal poblems. We will deoe he enie Chape 3 o discuss hese maeial pope paamees in deails.. The eiaion of Mawell quaions In his secion we deie he Mawell equaions based of he diffeeniaion fom of a numbe of phsical pinciples. Fis, we discuss he Ampee s law. Ampee s law descibes he fac of ha an elecic cuen can geneae an induced magneic field. I saes ha in a sable magneic field he inegaion along a magneic loop is equal o he elecic cuen he loop enclosed. Mahemaicall, Ampee s law can be epessed as:

3 L dl j Le us ake a simple case o illusae he Ampee s law, as shown in Figue.. Recall ha he cul of a eco field is defined as an e book on eco field, and see Appendi I cula A lim L A dl n eplace A b he magneic field and conside he case of ha he magneic field is on he plane of he pape and he elecic cuen is flowing ou fom he pape wih he cuen nomal o he pape we can hae dl L j cul lim n whee is he cuen densi in an infiniesimal aea. Meanwhile, if he elecic field is no sable, i.e., aing wih espec o ime, and he aiaion fequenc is high enough and eends ino he ada fequenc, hee will be anohe cuen in he medium known as he displacemen cuen and is popoional o he aiaion of he elecic fined, and he popoional faco is he dielecic pemiii ε. Thus, hee will be anohe conibuo, d/d, o induce he magneic field. The displacemen cuen woks eacl he same wa as he conducie cuen, so ha he oal cuen should be d/d; pu boh conibuos ino he aboe equaion ends up wih he fis equaion of he Mawell equaions:

4 Figue.. Illusaion of he Ampee s law a and he Faada s law b. econd, we ake a look of he Faada s law. Faada s law saes ha a moing magne can geneae an alenaing elecic field. Mahemaicall, he moing magne can be epesened b he aiaion of a eco magneic poenial ψ and he Faada s law can be mahemaicall epessed as Ψ b aking cul o coss poduc of boh sides of he equaion we hae Ψ Ψ Ne, le us ake a look of anohe equaions oiginaed fom a mahemaical heo he Gaussian heoem. Fom he eco field heo see Appendi I we hae leaned ha he diegence of a eco field is defined as dia A lim A ds This elaion acuall comes fom he well-known Gaussian heoem ha saes he following elaionship:

5 A d A ds Gaussian heoem saes ha he inegaion of he diegence of a eco field oe a ceain olume is equialen o he eco field iself inegaed oe he enie closed suface ha conains he olume. Appling he diegence and Gaussian heoem o he elecic field and he magneic field esuls in diffeen esuls. Le s ake a look of he elecic field fis. Fom he consiuie elaionships we knew ha ε, and we also knew he elecic field caused b an elecic chage q is q 4πε 3 Replacing A wih in he definiion of diegence and consideing he Gaussian heoem and ake he closed suface as a spheical shell ceneed a he locaion of he elecic chage we hae lim ds lim ε ds lim q ε 4πε 3 ds q ρ e whee ρ e is he chage densi in he infiniesimal olume confined b he enclosed suface. In conas, he magneic field is a compleel anohe so wih he fac ha he magneic field is a oaion field wih onl he dipole souce a a gula locaion and he souh and noh poles ae onl sepaaed b a infiniesimal disance. en when he enclosed suface conains he magneic souce, he ne magneic chage, in analog wih he elecic field, is sill eo. Thus, when appl he diegence and Gaussian heoem o he magneic inducion we hae lim ds lim ds upon he fac ha he ne magneic flu goes hough he closed suface is eo Figue.3.

6 Figue.3. The case of a oaion-fee eco field a and a souce-fee eco field b. In summa, he Mawell s equaions Goening he elecomagneic fields can be epessed in he following fom: ρ e The fis equaion is deied fom he Ampee s law, i denoes ha he elecical cuen boh he conducie cuen and he displacemen cuen d/d induces he magneic field. The second equaion is deied fom he Faada s law; i denoes ha he aiaion of he magneic field induces he elecical field. The hid and he fouh equaions ae deied fom he Gaussian Theoem, one fo he magneic field, and he ohe fo he elecical field. In soling he Mawell s equaion, eihe analicall fo simple poblems o numeicall fo moe complicaed poblems, i is necessa o wie up he Mawell equaions ino hee componens in accodance wih ceain gien coodinaes. The Mawell s equaions in Caesian coodinae can be epessed:

7 e ρ Mawell s equaions in he clindical coodinae can be epessed as: e ρ Mawell s equaions in he spheical coodinae ae:

8 e ρ.3 ielecic pemiii and magneic pemeabili We hae shown ha he consiuie equaions ae: M χ ε σ Fo a moe conenien mahemaical manipulaion and ug he I uni ssem, he magneic pemeabili and he elecic pemiii ε can be wien ino wo pas as: ε ε ε wih he paamees wih subscip eo defined as he magneic pemeabili and he elecic pemiii in acuum; and he ones wih subscip as he elaie pemeabili and elaie pemiii o he so-called dielecic consan. In acuum we hae / 4 / mee enies mee Faads π π ε

9 and i is eas o ecognie ha he speed of ligh is he elecomagneic wae eloci in acuum as c c ε 9 6 ol / kg 8 3 mee / sec ε The paamees of maeial pope shown in hese consiuie elaionships ae no oall independen wih one anohe. Thei ineelaionship is no consan oe he fequenc band. Fom he magneosaic poin of iew he magneic pemeabili and he magneic suscepibili ae elaed in he fom of and i is eas o see ha o χ, o χ M χ M χ χ Appaenl, suscepibili χ is a moe ininsic paamee han pemeabili o descibe he magneic pope of a maeial. I epesses how eas o how had a maeial can be magneied and how much i conibues o he magneic inducion..4 The fomaion of he elmhol quaions I is noiceable ha he fis wo equaions, i.e., he wo come ou fom he Ampee s law and he Faada s law, he elecic field and he magneic field ae coupled wih each ohe. The ae cleal saes he coupling naue of he elecomagneic inducion. Mahemaicall, howee, i is elaie moe cumbesome o sole and makes he phsical naue moe implici. In his secion we will de-couple hese wo equaions and make hei diffusion and wae popagaion naue moe eplici b some mahemaical manipulaions. Fis we need o ecall ha in he eco field heo we hae shown ha an gien eco field A can be epessed as A Φ Ψ i.e., A can alwas be epessed as he sum of he gadien of a scala poenial Φ and he cul of a eco poenial ψ. And he following wo equaliies ae alwas held.

10 Φ Ψ also we hae A A A we will pu hese elaionships ino he fis wo equaions of he Mawell equaions, i.e., he one deied fom he Ampee s law and he one fom Faada s law. Fis le us ge he cul of he fis equaion and we hae σ ε σ ε σ ε σ ε The elaionship of he second equaion of he Mawell equaions he Faada s law and he consiuie elaionship wee used in he aboe deiaion. ug he elaionship of qn., and ecall ha he magneic field is alwas onl he cul of a eco poenial magneic field is a souce-fee field, he lef hand side of he aboe becomes Thus eenuall we ge σ ε Ug he same agumen and simila appoach we can ge simila esuls fo he second equaion of he Mawell equaions he Faada s law and we ge a pai of he following equaions: σ ε σ ε

11 Obiousl he wo equaions hae been decoupled, i.e., onl one phsical quani field, eihe he elecic field o he magneic field, appeas in one equaion. Afe decoupling he elecic field and he magneic field we can fuhe he discussion b look ino he elaii of he maeial paamees fo he elecic and elecomagneic popeies. Wihou lose of geneali, we can assume he ime aiaion of he elecic field is in a simple hamonic fom, i.e., e iω iω, iωe iω, iω iω e ω iω eeinafe we le ha is a newl defined quani wihou ime aing componen and omied he subscip eo fo he eas wiing. Pu his definiion ino he second equaion of eqn. we hae σ ε iωσ ω ε σ ω ε i ωε k Follow a simila appoach we can ge he same esuls fo he magneic field and we hus can aie a a se of wo he so-called elmhol equaions. The elmhol equaions k k wih he definiion fo σ k ω ε i ωε as he squaed comple wae numbe. I is clea ha in he aboe equaion when he second em in he backe on he igh hand side has a alue much lage han one hen he k-squae has a significan imagina pa so ha he elmhol equaion is esseniall epesening a diffusion equaion. Vice esa, if he aio is much less han one, he k- squae has a significan eal pa so ha he elmhol equaion is esseniall epesening a wae equaion. Tha is o sa, in he complee equaions

12 if σ>>ωε, we hae σ ε σ ε σ σ ince mos eah maeials do no hae song magneic suscepibili, he elecic conducii is he conolling paamee in he pocess. Thus he aboe equaion epesens a conducie, o diffusion pocess, simila o he diffusion equaion used o descibe hea conducion, goundwae flow ec. Mahemaicall, his is a paabolic equaion. On he ohe hand, if σ<<ωε, we hae ε ε In his equaion he dielecic pemiii is he peailing paamee again, magneic pemeabili is elaie weak fo mos eah maeials. ielecic polaiaion is he conolling pocess ohe han conducion. The phsical feaue is he wae popagaion like pocess, simila o he mechanic waes. Mahemaicall, his is a hpebolic equaion. We will discuss in a moe deailed fashion on he diffusion equaion in elecomagneic inducion, and he wae equaion in gound peneaing ada. We also define and discuss in deails on he elecic conducion and dielecic polaiaion in maeial popeies in Chape 3..5 lecomagneic bounda condiions lecomagneic shows ha he nomal componen of cuen, elecic displacemen, and magneic inducion should be coninuous when coss a maeial ineface o bounda; while he angenial componen of he elecic field and he magneic field should be coninuous coss he maeial ineface. Le us ake he magneic bounda condiion as he eample o illusae he calculaion. Fom he Gaussian heoem we hae A d A ds

13 Replacing A b he magneic inducion eco and make a small disc wih he hickness of h and is cenal line is coinciden wih he bounda of wo media Figue.4 we hae d uppesuface ds ds lowesuface uppesuface ds ds lowesuface ds ds nds nds sidesuface Figue.4. Illusaion of he elecomagneic bounda condiions. In he aboe deiaion, we can see ha afe define he small olume enclosed b his small disc, and eenuall le he hickness of he disc h ends o be eo, he flu hough he aea of he side suface is also eo. The uppe and lowe sufaces hae he same aea bu opposie eenal nomal diecion, hus he do poducs of he magneic inducion and he suface nomal hae opposie polai. Ug subscip and o denoe he inducion in wo medium we ge and. Fom he Mawell equaion we knew ha he diegence of he magneic inducion is eo, his leads o he las sep in he aboe deiaion. Finall we ge n, o n n This means ha he nomal componen of he magneic inducion is coninuous cosg he medium bounda. Combining wih ohe phsical pinciples as shown in he Mawell s equaions, we can ge he following se of he so-called elecomagneic bounda condiions. In hese saemens we assume hee is no fee elecic chage eewhee. These elaionships of bounda condiions can be mahemaicall epessed as

14 n n n n n Afe we discussed he phsical equaions he Mawell s equaions, he consiuie elaionships, and he bounda condiions, we ae ead o discuss he elecomagneic phenomena in he eah maeials. Aenuaion: The uni of aenuaion coefficien is in Nepes fo science and decibels d fo engineeing. The definiions ae α ln Np, and α log d, so ha he conesion is log ln ln log e log e Np d d d log ed 8.686d ln ln ln The following equaliies ae used in deiing he aboe elaion. ln ln e e, and log b a, if a>, and b> log a b log e ln In summa, nepe Np is a uni used o epess aios, such as gain, loss, and elaie alues, hee is he aenuaion coefficien. The nepe is analogous o he decibel, ecep ha he Napeian base e is used in compuing he aio in nepes. One nepe Np d, whee loge/ln. The nepe is ofen used o epess he aio of ampliude, wheeas he decibel is usuall used o epess powe aios. Like he d, he Np is a dimensionless uni.

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