PART 1 GENERAL INFORMATION. ELECTROMAGNETIC FIELDS AND WAVES. LAWS OF ELECTROMAGNETICS

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1 PART 1 GENERAL INFORMATION. ELECTROMAGNETIC FIELDS AND WAES. LAWS OF ELECTROMAGNETICS The skill to evlute books without eding cn be ttibuted, to my mind, without doubts to the numbe of getest discoveies, to which the humn intellect ecently comes. G.K. Lichtenbeg CHAPTER 1 GENERAL DEFINITIONS AND RELATIONS OF ELECTRODYNAMICS 1.1. GENERAL DEFINITIONS An electomgnetic field (EMF) is unified pocess in spce nd time, i.e. it exists in ny spce point ( ), in ny medium nd in ech time moment (t ). Its electicl pt - n electic field is chcteized by electic vectos E = E(, t) nd D D( =, t), ccodingly, which e clled s n electosttic field intensity nd its induction. In the SI- units system the electosttic field intensity is mesued in olts / m ) (in hono of Alessndo olt: ) *), n induction in pe mete ([ ] coulomb pe sque mete ([ / m ]) Cl (Chles-Augustin de Coulomb: ). Similly, its mgnetic pt mgnetic field is chcteized by mgnetic vectos H H, t B = B, t, ccodingly, which e clled s the mgnetic field = ( ) nd ( ) intensity nd its induction. The mgnetic field intensity H is mesued in Ampee pe mete nd hs dimension ([ A/ m] ) (Andé-Mie Ampèe: ), nd its induction in Tesl with dimension ( Tl ) (Nicol Tesl: ). Chges distibuted in the spce nd time with the density ρ = ρ (, t) nd electic cuents (conduction cuents) distibuted with the density j = j(, t) e the souces of n electomgnetic field. Hee we must define exctly densities of cuents nd chges becuse they cn be ttibuted to ech spce point in the given time moment. A chge is mesued in coulombs ([Cl]), nd cuent in mpees ([А]). If we e speking bout the chge density ρ nd the cuent density j, then they hve dimensions 3 [ / m ] Cl nd [A/m ], ccodingly. The cuent density j nd the chge density ρ e mutully elted by the continuity eqution, which is the mthemticl fomultion of the lw of electic chge consevtion: ρ t + div j =. (1.1) *) Detiled infomtion bout scientists, which contibution into physics nd electodynmics s well s into the othe knowledge es cn be found in Appendix. 1

2 In othe wods, the electic chge cnnot be nnihilted nd cnnot be isen fom nothing, it cn be edistibuted only mong the bodies t thei contcts. The fundmentl physicl sense of this eqution (1.1) is quite simple: the ny chge vition (moe exct the chge density) ρ in the time moment t ( ρ t ) coesponds to the cuent density vition t the sme spce point nd in the sme time moment, which the second tem in the lst pt of (1.1) descibes. This is divegence, which cn be descibed, sy, in the simplest cse of ectngul coodintes ( x, y, z) s: div j = djx dx + djy dy + djz dz. Hee j x, jy, jz e components of the cuent vecto j (sometimes, when it does not led to misundestnding, we shll use insted of the cuent density the simply cuent). Fom the continuity eqution (1.1) it diectly follows tht the conduction cuent j = j(, t) is cused by the fee chge motion ( ρ = ρ, t) obeyed to the electicity consevtion lw, which cn be elted to Russin scientist M. Lomonosov ( ). An electomgnetic field ffects to point (indefinitely smll, concentted in point) chge q, moving in the spce with velocity v unde foce F = F(, t) - the Loentz foce (Hendik Antoon Loentz: ) equled to ( E + [ v, B] ) F = q. (1.) The foce in the SI system is mesued in newtons ([ ]) N (Isc Newton: ). In the lst eqution (1.) sque bckets designte the coss poduct of vectos v nd B. Thus, t bsence of mgnetic field ( B = ; electosttics cse) the foce F cting on the point chge q will be equl to the scl poduct F = qe, i.e. vectos F nd E will be pllel nd the chge motion will coincide with the diection of the electic field vecto E. In nothe cse, when n electic field is bsent ( E = ; F = q v, B is defined by the coss poduct of vectos v mgnetosttics cse) the foce [ ] nd B. Hence, the foce F in this cse will be diected t ight ngle (nomlly) to the plne, in which vectos v nd B e situted. At tht, the tiplet of vectos F, v, B will be ight-hnd (if to see fom the end of the vecto F, the ottion fom v to B should occu counte-clockwise). An electomgnetic field induces (excites) in the medium o in the body with conductivity σ the electic cuent, which density j is elted to the field electic component E s: j = σ E. (1.3) This is so-clled diffeentil fom (i.e. elted to the one point of spce o sufce) of the Ohm lw (Geog Simon Ohm: ). The conductivity σ is included in the eqution (1.3), which is mesued in siemens pe mete (Sim/m) (Cl Heinich von Siemens: ). Depending on its vlue, medi cn be divided into conductos, when σ 1 Sim/m, semiconductos if 1 < σ <1 Sim/m, nd t lst dielectics, fo which σ <1 1 Sim/m. The diffeentil fom of the Ohm lw is necessy when the cuent is nonunifomly distibuted long the conducto section o in some volume s it cn be seen in Figue 1.1,а. We cn see tht the cuent fom two electodes evidently non-unifomly

3 speds in the volume, nd theefoe, the cuent J in the mutul cicuit cn be detemined by integtion of its distibution by section. Figue 1.1. Pttens of cuent flowing in the conducting semi-spce (а) nd though the conducto segment (b) The integted fom of the Ohm lw is well known fom the secondy school s physics, nmely: if the voltge U is pplied to the esistive cicuit piece with esistnce R (Figue 1.1,b), the cuent J will flow in this cicuit: J = U R. (1.4) The lst eqution is the Ohm lw in the moe usul fom. In numbe of cses, one hs to use the eqution (1.3), nd we shll be sue moe thn once in this lectue couse. Fo instnce, if the cuent with the density j flows though the section S of some conducto with conductivity σ nd this density is bitily distibuted ove section S, the net cuent though this section S cn be detemined s J = S j d s. (1.5) Hee the elementy e in the section S is designted s d s. 1.. GENERAL LAWS OF CLASSICAL ELECTRODYNAMICS. SOURCES OF AN ELECTROMAGNETIC FIELD Now, fte eltions mentioned in the pevious section, we e edy to spek bout the genel lws of electomgnetics. This point is not vey simple in its fom but it is deep in its sense. The ede must simply get ccustomed to the somewht unusul (t the fist glnce) fom of electomgnetics lws. An electomgnetic field (EMF) is descibed by its intensities: the electic field intensity E nd the mgnetic field density H. ectos of the electomgnetic field E, D, H, B nd its souces j, ρ e elted 3

4 between ech othe by system of the Mxwell equtions (Jmes Clek Mxwell: ) (in the diffeentil fom, i.e. concened to the single point of spce in the given time moment), which cn be witten s: D ot H = j, (1.6) t B ot E + =, (1.7) t divd = ρ, (1.8) divb =. (1.9) When we spoke bout the continuity eqution (1.1), we hve got the cquintnce with the opetion div. The opetion ot ( oto, litelly, whilwind, ottion) is included into the eqution system (1.6)-(1.9). In the coodinte nottion, fo instnce, the eqution (1.6) will be witten s: H z y Dy z = jx, Hx y Dz z = jy etc. We hope tht the ede is well cquinted with the concept of the odiny deivtive du dx ; in equtions (1.6)-(1.9) thee e the ptil deivtives of the field components by coodintes. Since the field vlues e functions of sevel vibles ( x, yzt,, t, ), the deivtive on the one fom them is ptil deivtive. Undestnding will come to the ede: one should hve will. We diectly see fom Mxwell equtions (1.6)-(1.9) tht outside cuents jout nd outside chges ρ out e the souces, which excite the electomgnetic field E, H. At the sme time, the field E, H induces cuents nd chges j, ρ on the conducting bodies o in dielectic o othe medi. Fo instnce, the electic field E stimultes the conduction cuent jcon = σe, ccoding to the Ohm lw (1.3), in ech point of the conducting body, nd the totl cuent J will be defined ccoding to the integted Ohm lw (1.4). In ccodnce with the fmous Mxwell hypothesis, the conduction cuent in the cicuit bek (fo exmple, between cpcito pltes; Figue 1.) should be supplemented by the displcement cuent jdis, so tht the net cuent included in the eqution (1.6) is sum of the conduction cuent nd the bis cuent: j = σ E + jdis. The sttement tht field H, E vitions in time (the opetion / t in time) coespond to ny field E, H vitions in spce (the opetion ot in spce coodintes) is nothe impotnt consequence of Mxwell equtions. The eqution system does not include the vlues elted to the medi chcteistics, in which the EMF E, H exists nd electomgnetic wves popgte. Tht is why, the min eqution system of electodynmics the Mxwell eqution system (1.6)- (1.9) should be supplemented by some conditions. Fist of ll, these e constitutive equtions, i.e. equtions connecting the field intensities E, H nd thei induction pmetes D, B though the medi pmetes ε, µ, which e the dielectic nd mgnetic pemebility Figue 1.. To explntion of of the medi. In the genel cse, ε, µ e the complex the bis cuent effect in vibles: ε = ε ' + iε", µ = µ' + iµ ". The vecto D cpcito. is 4

5 sometimes clled s vecto of electic displcement. The pmete ε s eltive dielectic pemebility is connected with ε = ε ε - the bsolute pemebility mesued in fd pe mete ([ F / m] ) (Michel Fdy: ); ε is the pemebility of the vcuum. In pctice one cn fequently use the eltive vlues of ε, µ. The concept of n index of efction of the medium n = ε µ is used s well. The sense of signs in font of imginy pts of the pemebility nd the index of efction we shll discuss lte. Constitutive equtions in the simplest cse ( ε, µ, σ e scl pmetes) will tke the following view: D = ε E, B = µ H, (1.1) In eqution (1.1) the quntity of σ mens the specific (diffeentil) conductnce in the given spce point. The coefficient of popotionlity µ between vectos of the mgnetic field B nd H is clled the bsolute mgnetic pemebility of the medium nd is mesued in Heny pe mete ([ Hn / m] ). Constitutive equtions (1.1) in vcuum will be witten s: D E = ε, B H 9 = µ, whee ε = ( 1 36π ) 1 (F/m), 7 µ = 4π 1 (Hn/m). The eqution (1.6) is usully clled s the fist Mxwell eqution, nd the eqution (1.7) the second one. The eqution (1.7) is the diffeentil fom of the Fdey lw of mgnetic induction. Eqution (1.6) is the diffeentil fom of the genelized totl cuent lw (Ampee lw). We shll be cquinted with the moe genel integted fomultion of Mxwell equtions little lte (see Section 1.4), nd now we shll wite, fo instnce, the totl cuent lw in the integted fom: H dl = j ds = J. The totl cuent lw connects the mgnetic field cicultion long L S the contou L nd the totl cuent j = σ E + jdis, which is coveed by the contou. In the concept of the totl cuent we include, s we ledy noted little elie, the outside cuent j out, which is the field souce nd consideed to be given, howeve, it itself is not esults of the EMF unde considetion. In tun, the outside cuent jout is the consequence of the electomotive foces E out (, t) ction, i.e. foces of the nonelectomgnetic oigin. Such foces e pocesses of chemicl, bio-electomgnetic, spce, diffusion nd etc. chctes. Thus, fo the outside cuent the diffeentil fom of the Ohm lw is coect: j out = σ E out. The tem j dis = D / t, which is clled s the displcement cuent epesenting of the fmous Mxwell hypothesis bout this cuent existence supplementing in the spce the conduction cuent j, is included lso into the totl cuent j. Thus, the lines of the totl cuent e closed in the spce eithe to themselves o to souces (electic chges), s it is shown in Figue 1. on n exmple of cpcito. The eqution fo the conduction cuent (1.3) is the diffeentil fom of the Ohm lw. It is impotnt tht in the Mxwell eqution system the medium pmetes ( ε, µ, σ ) e not included nd since these equtions e suitble fo ny medi, then they must be complemented by the constitutive equtions (1.1) (they e sometimes clled s equtions of sttes becuse they chcteize the medium). A little ltely we shll give the summy of the min clsses of medi, with which in pctice of ntenn systems nd dio wve popgtion we need to del. 5

6 1.3. ENERGY OF AN ELECTROMAGNETIC FIELD. THE UMO-POYNTING ECTOR. THE EQUATION OF THE ENERGY BALANCE (THE UMO THEOREM) The system of min equtions of electodynmics (1.6)-(1.9) is the totl nd sufficient system to povide n nlysis nd clcultions of the specific poblems in the tuly boundless e of humn ctivity including the poblems of ntenn-feede devices nd dio wve popgtion. Howeve, this system is the system of equtions in ptil deivtives, nd theefoe, it llows the boundless viety of solutions. Tht is why, t exmintion of pcticl o simultion poblems, the min eqution system should be supplemented by numbe of conditions, mong which thee e the necessy considetion of the field behvio chcte ne boundies of medi with diffeent pmetes (boundy conditions), the field behvio on the infinite seption fom the souce (conditions of dition, the pinciple of extinguishing, the pinciple of the limited mplitude etc.), nd, t lst, the field behvio in the e of the knee of the fomtive sufce (the non-nlyticl boundy epesenttion) of the object unde considetion, ne shp bends etc. (conditions on edges). At solution of non-sttiony poblems, when the field quntity dependence in time is mnifested (insted of hmonic one), thee e necessy the initil conditions s well, i.e. field vlues in the some initil time moment (fo exmple, t t = ). Mentioned conditions e necessy fo the coect poblem sttement stisfying the equiements of the unicity theoem of the Mxwell eqution solution. The ction of the electomgnetic field becomes ppent, s we ledy mentioned, on the bsis of its influence upon moving chge (1.1) s well s ccoding to its impct on ny body, which is situted in the e occupied by the field. This ction is mnifested owing to the electomgnetic enegy W distibuted in some e, limited by the sufce S (Figue 1.3; the sufce S cn be both el (elizble) nd imginy (vitul): W = ( 1 ) ( ε E + µ H ) d. (1.11) Thus, the electomgnetic enegy W is distibuted in spce with the bulk density ( 1 ) ε E + ( 1 ) H. w = µ (1.1) а Fom (1.11) it diectly follows tht EMF enegy W epesents sum of the el m electic W nd mgnetic W field enegy, so el W = ε E D d, (1.13) ( 1 ) E d = ( 1 ) ( 1 ) H d = ( 1 ) m W = µ H B d. (1.14) The EMF enegy W (1.11) in the e cn chnge in time owing to, t lest, two pocesses. Fist of ll, it cn be tnsfomed into nothe enegy type of the nonelectomgnetic chcte, fo instnce, into theml enegy, fo heting the body with conductivity σ, chemicl, bio-physicl enegy nd mny othes. The vition of enegy mount in the volume cn be lso t the expense of its going wy fom the volume though some holes S o though the sufce S itself by vitue of complete o 6

7 ptil tnspency (fo exmple, owing to mentioned imgintion) o due to some fetues of the field itself (see below). It is evident tht enegy my not only go wy fom the volume but to ente it fom outside though the sme sufce S o though some of its pt (fo instnce, S Σ ). The fist of the mentioned pocesses of EMF enegy vition in the volume W (tnsfomtion to nothe enegy types) is descibed s powe deliveed (deliveble) by the field in the time unit s P = j E d, (1.15) The integtion element in (1.15) defines the bulk powe density p, which is p = j E. Enegy going wy (emitted) fom the volume in time unit (o coming into it fom outside) is defined s follows: P = S ds, S (1.16) whee S is the Umov-Poynting vecto (N.A.Umov: ; Poynting John Heny: ) epesenting the enegy flow density. Figue 1.3. The scheme of the bity volume, closed by some combined sufce S = S + S + S. Sufces 1 S 1, S diffe by thei popeties, nd the sufce S epesents hole though which enegy is going wy fom the volume o entes into it. The Umov-Poynting vecto U is defined s the coss poduct of vectos of electic E nd mgnetic H field intensities in ech point of the sufce S : U = E, H. (1.17) Enegy quntities W, P, PΣ e elted ech othe by the integted eltion the eqution of powe blnce: d W d t P + P =. (1.18) + Σ In essence, this is lw of enegy consevtion fo EMF: ech vition of enegy W in some volume is coesponded by the vition owing to losses P o by 7