0WAVE PROPAGATION IN MATERIAL SPACE

Transcription

1 0WAVE PROPAGATION IN MATERIAL SPACE All forms of EM nrgy shar thr fundamntal charactristics: 1) thy all tral at high locity 2) In traling, thy assum th proprtis of was 3) Thy radiat outward from a sourc without bnfit of any discrnibl physical hicls So far, th mdium of propagation is fr spac (th spac oid of xtrnal chargs and currnts). Th goal is to xtnd our prious rsults to othr typs of mdia of propagation (dilctric, magntic matrial, conductor, tc.) collctily known as matrial spac. As a rsult, Maxwll s quations must b modifid accordingly.

2 MATERIAL CLASSIFICATION All mattrs ssntially consist of chargd particls (protons, lctrons) Th prsnc of xtrnal E and H in matrials causs ractions in th form of intrnal flow of currnts and chargs. Ths inducd currnts and chargs will in turn crat intrnal E and H to countract th imposing E and H. Thr ar 3 basic typs of ractions: conduction, polariation, and magntiation. All thr ractions can occur simultanously dpnding on th charactristics of th matrial. Th prdominant raction will b usd to classify matrials into diffrnt typs.

3 CONDUCTORS Th prdominant raction is conduction which dpnds on th aailability of fr lctrons conduction currnt Conduction currnt inols th momnt of lctrons through conducti mdium in rspons to an applid lctric fild. It can b shown that conduction currnt dnsity is gin by J cd = σ E whr E is an applid lctric fild 2 n τ σ = is th conductiity of th conductor m n is th numbr of lctron pr olum is th charg of an lctron m is th mass of an lctron τ is th arag tim intral btwn collision of lctrons

4 DIELECTRIC MATERIALS OR INSULATORS Th prdominant raction is polariation which is basd on bound lctrons For a crtain class of dilctric matrials (linar, isotropic and homognous), polariation can b quantifid in trms of polariation ctor P, which has th sam dirction as E P = ε 0 χe whr χ = lctric suscptibility of th matrial (a masur of how snsiti a gin dilctric is to lctric filds) χ +1 = ε = rlati prmittiity or dilctric constant r This P will crat polariation currnt and polariation charg. P J p = and ρ = P t

5 MAGNETIC MATERIALS Th prdominant raction is magntiation dpnds on th ralignmnt of magntic dipols which can b quantiation in trms of magntiation ctor M which has th sam dirction as H M = χ H whr χ m = magntic suscptibility of th matrial (a masur how snsiti a gin dilctric is to lctric filds) χ +1 = μ = rlati prmability m r This M will crat magntiation currnt J m = M = magntiation olum currnt dnsity K m = M nˆ = magntiation surfac currnt dnsity If μ = 1, th matrial is said to b nonmagntic r m

6 SUMMARY Thr paramtrs ar ndd in th classification of matrials: σ, ε, μ according to th thr constituti rlations. D = ε E = ε rε 0 E = ( χ + 1) ε 0E B = μ H = μrμ0 H = ( χm + 1) μ0h = σ E J cd Rstrict to matrial mdia that ar simpl, i.., linar, isotropic, homognous and tim-inariant. Linar D aris linarly with fild intnsity E Homognous charactristics of matrial do not chang from point to point Isotropic sam proprtis in all dirctions

7 MODIFICATIONS OF MAXWELL S EQUATIONS Rcall Ampr s law for tim-harmonic filds H s = J s + jωds = E + jωε E + jωε χ E + J σ s 0 s 0 s othrs Assuming J othrs = 0, σ H s = σ Es + jωε Es = ( σ + jωε ) Es = jω ε j E ω σ H s = jωεe s whr ε = ε j (ε is complx) ω Likwis, E s = jω B s = jωμh s (μ is ral) Ds = ρs + ρ ps = 0 (assuming no chargs) B s = 0 s

8 From th nw st of Maxwll s qs, th wa quation bcoms 2 2 ( ω με ) E = 0 With th sam assumptions as in th cas of fr spac, th solutions rprsnting EM wa propagating insid a matrial in γ E0 γ + dirction ar Es = E0 xˆ and H s = yˆ. η whr 2 2 γ = ω με is th disprsion rlation σ γ = j ω με = j ω μ ε j = jωμ( σ + jωε ) ω γ is calld propagation constant (unit of pr mtr) μ η = = complx intrinsic impdanc of th mdium ε ε is calld complx prmittiity of th mdium s

9 Sinc γ is complx, w may lt γ = α + j β = jωμ( σ + jωε whr α = R{γ } is calld attnuation constant (Npr/m) β = Im{γ} is calld phas constant (rad/m) 2 μ ε σ 2 μ ε σ α = ω 1+ 1 and β = ω ω ε 2 ω ε Likwis, η is complx and can b xprssd as η = μ = ε jωμ = η φ η = η σ + jωε jφ η whr η = μ / ε σ 1 + ωε 2 1/ 4, and σ tan 2φ η =, ωε 0 φ 45 η o

10 Th solutions can b rwrittn as j j s x E x E E β α β α + = = 0 ) ( 0 ˆ ˆ φ η β α η j j s E y H + = ) ( 0 ˆ Th instantanous xprssions for filds in matrial spac bcom x t E t E ) ˆ cos( ), ( 0 β ω α = y t E t H ˆ ) cos( ), ( 0 η α φ β ω η =

11 Comparison with filds in fr spac rals that 1) Th filds E and H and th dirction of propagation ar mutually orthogonal as bfor. Howr, th phas locity u p is rducd du to a nw dfinition of th phas constant β. A wa propagat with a phas constant of 1 rad/m will xprinc a phas shift of 1 rad as it trals a distanc of 1 m 2) Th magnitud of E in matrial is rducd by a factor of and that of H by and η. α α Wa in th mdium is attnuatd xponntially with an attnuation constant or a spatial rat of dcay of α Npr/m (Np/m) An attnuation of 1 Np/m quals 20log10 = db/m For α = 1 Np/m, th wa of unit amplitud gts attnuatd to a magnitud of 1 = as it trals a distanc of 1 m

12 If α is nonro, th mdium is said to b a lossy mdium; othrwis, it is losslss. 3) Th filds E and H ar out of phas ( E lads H or H lags E ) by φ η rad at any instant of tim du to th complx intrinsic impdanc of th mdium. σ 4) Rcall th complx prmittiity: ε = ε j = ε θ ω σ whr θ = tan 1 = 2φη is th loss angl of th mdium. ωε σ Or, tan θ = is calld th loss tangnt of th mdium. ωε Th loss tangnt can b intrprtd as th ratio of th magnitud of conduction currnt dnsity J cd to that of th displacmnt currnt dnsity J Jcds σ Es σ d in lossy mdia = = J jωε E ωε ds s

13 Ex. Writ a spac-tim xprssion for a 1-V amplitud 100 MH x-polarid wa in a losslss mdium (ε r =2, μ r = 1) propagating in th + dirction.

14 Ex. Suppos a propagating lctric fild is gin by o E(, t) = 34 cos(2π 10 t 10π + 45 ) xˆ V/m. Find (a) th initial amplitud (b) th attnuation constant (c) th wa frquncy (d) th walngth () th phas shift in radians