4. Electrodynamic fields

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1 4. Electodynamic fields D. Rakhesh Singh Kshetimayum 1

2 4.1 Intoduction Electodynamics Faaday s law Maxwell s equations Wave equations Lenz s law Integal fom Diffeential fom Phaso fom Bounday conditions Fig. 4.1 Electodynamics: Faaday s law and Maxwell s equations 2

3 4.1 Intoduction We will study Faaday s law of electomagnetic induction (the 4 th Maxwell s equations) Faaday showed that a changing magnetic field ceates an electic field He was the fist peson who connected these two foces (magnetic and electic foces) Maxwell, in 19 th centuy, combined the woks of his pedecessos (the fou Maxwell s equations) 3

4 4.1 Intoduction Since eveything in natue is symmetic, he postulated that a changing electic field should also poduce magnetic field Wheneve the dynamic fields inteact with a media inteface, thei behavio is govened by electomagnetic bounday conditions We will discuss how to obtain Helmholtz wave equations fom the two Maxwell s cul equations and geneate electomagnetic (EM) waves 4

5 4.2 Faaday s law of electomagnetic induction B t 2 B s B t I dl 1 v I v (a) (b) (c) Fig. 4.2 (a) Case I (b) Case II and (c) Case III 5

6 4.2 Faaday s law of electomagnetic induction A majo advance in electomagnetic theoy was made by Michael Faaday in 1831 He discoveed expeimentally that a cuent was induced in a conducting loop when the magnetic flux linking that loop changes The possible thee cases of changing magnetic flux linkage is depicted in Fig

7 4.2 Faaday s law of electomagnetic induction Case I: A stationay cicuit in a time vaying magnetic field It has been obseved expeimentally that when both the loop and magnet is at est But the magnetic field is time vaying (denoted by B t in Fig. 4.2(a)), a cuent flows in the loop Mathematically: ξ E dl = dφ = dt B t ds E ds = = B t ds 7

8 4.2 Faaday s law of electomagnetic induction It is fo any abitay suface Hence E B = t The emf induced in a stationay closed cicuit is equal to the negative ate of incease of magnetic flux linking the cicuit The negative sign is due to Lenz s law 8

9 4.2 Faaday s law of electomagnetic induction Case II: A moving conducto in a static magnetic field When a conducto moves with a velocity in a static magnetic field (denoted by B s in Fig. 4.2(b)), a foce F m = qv B will cause the feely movable electon in the conducto to dift towad one end of the conducto and leave the othe end positively chaged 9

10 4.2 Faaday s law of electomagnetic induction The magnetic foce pe unit chage u F m v B q = u can be intepeted as an induced electic field acting along the conducto and poducing a voltage V V = E dl = ( v B) dl

11 4.2 Faaday s law of electomagnetic induction Case III: A moving cicuit in a time vaying magnetic field When a chage moves with a velocity in a egion whee both an electic field and a magnetic field exists The electomagnetic foce on q is given by Loentz s foce field equation, F = q( E + v B) 11

12 4.2 Faaday s law of electomagnetic induction Hence, when a conducting cicuit with contou C and suface S moves with a velocity in pesence of magnetic and electic field, we obtain, B E dl = ds + v B dl t ( ) C S The line integal in the LHS is the emf induced in the moving fame of efeence C 12

13 4.2 Faaday s law of electomagnetic induction The fist tem in the ight side epesents tansfome emf due to the vaiation of magnetic field The second tem epesents the motional emf due the motion of the cicuit in magnetic field Faaday s law of electomagnetic induction shows that thee is a close connection between electic and magnetic fields In othe wods, a time changing magnetic field poduces an electic field 13

14 4.2 Faaday s law of electomagnetic induction Lenz s law Fig. 4.3 Lenz s law 14

15 4.2 Faaday s law of electomagnetic induction The induced cuent ceates a magnetic flux which pevents the vaiation of the magnetic flux geneating the induced emf Fo example, the loop L is dawn to the coil c, the magnetic flux though the loop inceases The cuent induced in the loop in this case is in the counteclockwise diection 15

16 The theoy of EM field was mathematically completed by Maxwell It was James Clek Maxwell who has combined the pevious woks of Cal Fedeick Gauss, Ande Maie Ampee and Michael Faaday into fou laws 16

17 which completely explains the entie electomagnetic phenomenon in natue except quantum mechanics Electodynamics befoe Maxwell Table 4.1 Electodynamics befoe Maxwell (next page) 17

18 Equation Equations Laws No. u 1. E ρ v = Gauss law fo electic field ε 0 2. B = 0 Gauss law fo magnetic field u 3. B = µ J Ampee s law 0 u u B E = t 4. Faaday s law 18

19 Thee is fatal inconsistency in one of these fomulae It has to do with the ule that the divegence of cul is always zeo Apply this ule to 4, we get, This is consistent LHS is zeo ( ) B 0 ( E) = = t since it is the divegence of cul of electic field vecto 19

20 RHS is also zeo fom Gauss s law fo magnetic field which states that divegence of a magnetic flux density vecto is zeo Let us apply this ule to 3, we get, ( B) = 0 µ 0 ( µ J ) = ( J ) 20

21 The RHS is zeo fo steady state cuents only wheeas the LHS is zeo fo all cases since it is the divegence of cul of magnetic flux density vecto One fundamental question is that what is happening in between the paallel plates of the capacito while chaging o dischaging 21

22 Fom ou cicuit analysis, no cuent flows between the two plates since they ae disconnected In accodance with Gauss theoem, fo time vaying electic flux density and fixed o static closed suface D ds = q t ( D ds ) = q t 22

23 So the tem on the LHS of the above equation is ate of change of electic flux o it can be also temed as displacement cuent we can add this exta tem in the Ampee s law and we get, H = J + D D t The 2 nd tem on the RHS can also be temed as displacement cuent density Both tems on the RHS togethe is known as the total cuent. 23

24 4.3.2 Maxwell s equations in integal fom Maxwell equations ae the elegant mathematical expession of decades of expeimental obsevations of the electic and magnetic effects of chages and cuents by many scientists viz., Gauss, Ampee and Faaday 24

25 Maxwell s own contibution indeed is the last tem of the thid equation (efe to Table 4.2) But this tem had pofound impact on the electomagnetic theoy It made evident fo the fist time that vaying electic and magnetic fields could poduce o geneate each othe and these fields could popagate indefinitely though fee space, fa fom the vaying chages and cuents whee they ae oiginated 25

26 Peviously the fields had been envisioned bound to the chages and cuents giving ise to them Maxwell s new tem displacement cuent feed them to move though space in a self-sustaining fashion, and even pedicted thei velocity of motion was the speed of light Electodynamics afte Maxwell can be epesented by the fou Maxwell s equations in integal fom (listed in Table 4.2) Table 4.2 Electodynamics afte Maxwell (next page) 26

27 Equation numbes Equations Laws D ds = ρ dv v B d s =0 H dl = J + D ds t B E dl = ds t Gauss law fo electic field Gauss law fo magnetic field Ampee s law Faaday s law 27

28 The elations fo linea, isotopic and non-dispesive mateials can be witten as: D = ε E, B = µ H whee ε is the pemittivity and µ is the pemeability of the mateial Suppose, we estict ouselves to time-independent situations That means nothing is vaying with time and all fields ae stationay 28

29 If the fields ae stationay (electic and magnetic field ae constant with time), Maxwell s equations educes to fou goups of independent equations D ds 1. = q 2. B ds = 0 H dl 3. = I 4. E dl = 0 29

30 In this case, the electic and magnetic fields ae independent of one anothe These ae laws fo electostatics and magnetostatics Diffeential fom of Maxwell s equations We can obtain the 4 laws of Maxwell s equations in diffeential fom listed below fom the integal fom of Maxwell s equations by applying Divegence and Stokes theoems 30

31 1. E = ρ v ε 2. B = D H = J + t u u B E = t 31

32 Equation 1 means that static o dynamic chages in a given volume ae esponsible fo a diveging electic field Equation 2 means that thee is no physical medium which makes a magnetic field divege Equation 3 means eithe a cuent flow though a medium o a time-vaying electic field poduces a spatially culing magnetic field Equations 4 means that a spatially vaying (culing) electic field will cause o poduce a time-vaying magnetic field 32

33 Maxwell equation include the continuity equation J + D ds = 0 t Time hamonic fields and Maxwell s equations in phaso fom Fo time hamonic fields F( x, y, z, t) = F( x, y, z) e jωt whee ω is the angula fequency of the time vaying field 33

34 t F( x, y, z, t) F( x, y, z, t) = jωf ( x, y, z, t); F( x, y, z, t) dt = + c jω 1 whee c 1 is a constant of integation Since we ae inteested only in time vaying quantities, we can take, c 1 =0 t 2 jω; = jω jω = ω 2 t 2 34

35 Maxwell s equations in phaso fom If we take the time vaiation explicitly out, then we can wite 1. E% = jωb% ~ ~ 2. H = j ωd + ~ J 3. D ~ = ρ υ ~ 4. B = 0 35

36 Some points to be noted on pefect conductos and magnetic fields: A pefect conducto o metal can t have time-vaying magnetic fields inside it At the suface of a pefect metal thee can be no component of a time-vaying magnetic field that is nomal to the suface Time-vaying cuents can only flow at the suface of a pefect metal but not inside it Electomagnetic bounday conditions 36

37 Table 4.3 Electomagnetic bounday conditions fo time vaying fields Sl. No. Scala fom Vecto fom 1. E t1 = E t2 2. H t1 -H t2 =J s ~ ~ nˆ ( E1 E2 ) = 0 ~ ~ n ( H H ) = ˆ 1 2 J s 3. B n1 =B n2 ~ ~ n ˆ ( B 1 B 2 ) = 0 4. ~ ~ n ( D D ) ˆ 1 2 = ρ s D n1 D n 2 = ρ s 5. J n1 =J n2 ~ ~ nˆ ( J1 J2) = 0 6. J J σ σ t = nˆ = t 2 2 J% J% σ σ

38 The tangential component of the electic field is continuous acoss the bounday between two dielectics The tangential component of the magnetic field is discontinuous acoss the bounday between two magnetic mateials by the suface cuent density flowing along the bounday The nomal component of the magnetic flux density is continuous acoss the bounday between two magnetic mateials 38

39 The nomal component of the electic flux density is discontinuous acoss the bounday between two dielectics by the suface chage density at the bounday It states that the nomal component of electic cuent density is continuous acoss the bounday The atio of the tangential components of the cuent densities at the inteface is equal to the atio of the conductivities 39

40 4.4 Wave equations fom Maxwell s equations Helmholtz wave equations In linea isotopic medium, the two Maxwell cul equations in phaso fom ae E = jωµ H H = jωε E + J = jωε E + σ E = jωε + σ E ( ) 40

41 To solve fo electic and magnetic fields, we can take cul on the fist equation and eliminate the cul of magnetic field vecto in the RHS of the fist equation by using the second equation given above 2 E = E = ( E) E 41

42 Fo a chage fee egion, E = 0 = = ωµ = ωµ ωε + σ = γ Hence, Similaly, ( ) 2 2 E E j ( H ) j j E E 2 2 E γ E = γ H H = 0 42

43 These two equations ae known as Helmholtz equations o wave equations γ=α+jβ is a complex numbe and it is known as popagation constant The eal pat α is attenuation constant, it will show how fast the wave will attenuate wheeas the imaginay pat β is the phase constant Popagation in ectangula coodinates 43

44 The Helmholtz vecto wave equation can be solved in any othogonal coodinate system by substituting the appopiate Laplacian opeato Let us fist solve it in Catesian coodinate systems Let us assume that the electic field is polaized along the x- axis and it is popagating along the z-diection ) E x y z t E z t x (,,, ) = (, ) x 44

45 Theefoe, the wave equation now educes to E 2 E γ E = 0 γ E = 0 2 z Hence, the solution of the wave equation is of the fom E x y z E z x ) E e E e x ) + γ z + γ z (,, ) = x ( ) = ( + ) 0 0 whee supescipt + and fo E 0 the abitay constants denote fo wave popagating along + and z-axis espectively 45

46 Assuming that the electic field popagates only in the positive z-diection and of finite value at infinity then the abitay constant fo wave taveling along z axis must be equal to zeo, which means ) E x y z E z x E e x ) + γ z (,, ) = x ( ) = 0 Dopping the supescipt + in the constant E 0 of the above expession, we can wite ) (,, ) = 0 γ z E x y z E e x 46

47 Let us put the time dependence now by multiplying the above expession by e jωt, which gives ) ) ) (,,, ) = x (, ) = ( ) = ( ) z j t z j z j t E x y z t E z t x E e γ e ω x E e α e β e ω x 0 0 The eal pat of this electic field becomes Re E x, y, z, t = Re E z, t x ) = E e cos t z x ) { ( )} x ( ) α z { } ( ω β ) 0 47

48 Some points to be noted: Fist point is that the solution of the wave equation is a vecto Secondly, Fig. 4.5 shows a typical plot of the nomalized popagating electic field at a fixed point in time Thidly, its phase φ=ωt-βz is a function of both time, fequency and popagation distance 48

49 So, if we fix the phase and let the wave tavel a distance of z ove a peiod of time t Mathematically, this can be expessed as φ= ω t-β z = 0, theefoe, v p = z/ t = ω / β, which is the phase velocity of the wave Hence, v p = z/ t = ω / {ω (µε) } = 1/ (µε) 49

50 Fo fee space this tuns out to be v p = 1/ (µ 0 ε 0 ) m/s Note that the phase velocity can be geate than the speed of light Fo instance, hollow metallic pipe in the fom of ectangula waveguide has phase velocity v p ω = = β c fc 1 f 2 50

51 Fo guided-wave popagation inside ectangula waveguide f c < f and coespondingly such guided waves popagate with phase velocity geate than the speed of light this doesn t contadict the Eiensten s theoy of elativity because thee is no enegy o infomation tansfe associated with this velocity 51

52 Note that beside phase velocity, we have anothe tem called goup velocity which is associated with the enegy o infomation tansfe Inside ectangula waveguide, in fact, the goup velocity is much lowe than the speed of light This can be seen fom the anothe elation between the phase and goup velocity inside ectangula waveguide v p v g =c 2 52

53 phase velocity v p = ω / β goup velocity v g = dω / dβ Fouthly, phase constant is also geneally known as wave numbe β β = ω / v p = 2 ϖ f / v p = 2 ϖ / λ Fifth point is that atio of the amplitude of the electic field to the magnetic field is known as intinsic o wave impedance of the medium 53

54 η = E / H = ωµ / β = ωµ / {ω (µε) = µ / { (µε) = (µ/ε) Fo fee space η = (µ 0 /ε 0 ) 377 Ω Popagation in spheical coodinates We can also calculate the popagation of electic fields fom an isotopic point souce in spheical coodinates We will assume that the souce is isotopic and the electic field solution is independent of (θ,φ) Assuming that the electic field is polaized along θ-diection and it is only a function of 54

55 ) E t E t θ (, θ, ϕ, ) = (, ) we can wite the vecto wave equation as follows: θ d de γ 0 γ 0 d d E E = E = Fo finite fields, the solution is of the fom ) E0 E (, θ, ϕ ) = Eθ ( ) θ = e γ 55

56 Like befoe, we can add the time dependence to the phaso and get the eal pat of the electic field as follows E0 Re { E α (, θ, ϕ, t) } = Re { E (, t) ) θ θ } = e cos( ωt β) ) θ Note that thee is an additional 1/ dependence tem in the expession fo electic field hee this facto is temed as spheical speading fo the isotopic point souce 56

57 4.5 Summay Faaday s law B E dl = ds + v B dl t ( ) C S C Electodynamics Maxwell s equations γ 2 2 E E = 0 Wave equations γ 2 2 H H = Bounday conditions 0 Lenz s law 57 Integal fom D ds = ρ dv d B s =0 D H dl = J + ds t B E dl = ds t v Diffeential fom ρ E = v ε B = 0 D H = J + t B E = t Phaso fom ~ ρ E = v ε ~ B = 0 ~ ~ H = J + jωd ~ ~ E = jωb ~ ~ nˆ ( E 1 E 2 ) = 0 ~ ~ n ( H H ) = ˆ 1 2 ~ ~ nˆ ( B1 B2) = 0 Dn1 D n 2 = ρs ~ ~ nˆ ( J1 J2) = 0 J% 1 σ1 nˆ = J% 2 σ 2 Fig. 4.6 Electodynamics in a nutshell J s

3. Magnetostatic fields

3. Magnetostatic fields D. Rakhesh Singh Kshetimayum 1 Electomagnetic Field Theoy by R. S. Kshetimayum 3.1 Intoduction to electic cuents Electic cuents Ohm s law Kichoff s law Joule s law Bounday conditions