Professor: Chungpin Liao (Hovering) 廖重賓 （飛翔） cpliao@alum.mit cpliaq@nfu.tw

1. Electromagnetism - I （電磁學－I） Chapter 1 Maxwell’s Integral Laws in Free Space Professor: Chungpin Liao (Hovering) 廖重賓 （飛翔） cpliao@alum.mit.edu cpliaq@nfu.edu.tw EM (I) -- Hovering

2. 1.0 Introduction EM  Theory of Fields learning Maxwell’s equ’s（EM）develops math language ＆ methods for many other areas Ordinary circuit theory fails at high frequencies（ω’s ） proper modification ＆justification needs understanding EM field theory In circuit model, when ω↑↑ such that the modeling fails ω↑↑ inductor capacitor becomes ω↑↑ AC loss causes : resistor transistor transducer transformer “capacitive” effects “eddy” current Anyone concerned with developing circuit models for physical systems requires a field theory background to justify approximations and to derive the values of the circuit parameters EM (I) -- Hovering

3. Overview of subject See Fig.1.0.1 Maxwell’s equ’s EM in the language of circuit theory in the broad sense（i.e., EM field theory） (Ch. 4~7 ) EQS MQS ( Ch. 8~10 ) Chap.1Integral form In simple configurations relating fields to sources Helping inventions in a qualitative fashion Chap.2 Differential form using operators of differential geometry More general（r , t）variation Static fields as 1st topic. EM (I) -- Hovering

4. Fields are not directly measurable, let alone of practical interest, unless they are dynamic. In fact, fields are never static. EQS MQS ∴ Chap.3 quasistatics（QS） EQS, MQS justified if time rates of change are slow enough (i.e., ω↓↓ ) so that time delays due to the propagation of EM waves are unimportant Full appreciation of quasistatics won’t come until chap.11~15 where EQS ＆ MQS are drawn together. Although capacitors and inductors are examples in the EQS and MQS categories, respectively, it is NOT true that quasistatic systems can be generally modeled by frequency – independent circuit elements. even~ GHZ in electromics Dynamics are important in EQS e.g. change migration in a cap. or transistor & MQS e.g. magnetic field diffusion in time. EM (I) -- Hovering

5. 1.1 Lorentz Law in free space 2 pts of view for formulating EM theory： action at a distance：Coulomb’s law： the vacuum space between q1, q2 is filled with fields continuum theory ( E and H defined at r, even when no change present. ) Maxwell’s equ’s 1 2 Fields（E & H）are defined in terms of the force that would be exerted on a test change q if it were introduced at r, moving at a velocity υ, at the time of interest. Lorentz force on that test charge q： ﹝N﹞﹝C﹞ electric field intensity magnetic flux density right-hand rule EM (I) -- Hovering

6. ﹝N﹞﹝C﹞ electric field intensity magnetic flux density right-hand rule ∴ (υ×μ0H ) ⊥υ ⊥ μ0H Lorentz force law was experimentally found. E = E (r, t), H = H (r, t) In general, E and H are not uniform or static： EM (I) -- Hovering

7. x x=0 Ex.1.1.1 e－motion in vacuum in a uniform static E I.C. ξx= 0, and υx =υi at t = ti － 1 ∴ － － 2 3 3 into ： 1 1 EM (I) -- Hovering

8. EX.1.1.2 Electron motion in vacuum in a uniform static magnetic field. imposed. ＊ E= 0, μH0 = μH0y ∴ f = (-e)( υ×μ0 H0 ) ⊥υ and ⊥μ0H0 z ≡ B0 ∵υiz spiral motion ⊥υ and ⊥μ0H0 ＊i.e. Effect of particle charge density and current density is not taken into account. EM (I) -- Hovering

9. System equ’s： Newton’s law e- gyro-freq. ωce = Component equ’s I.C. － 1 － 2 － 3 － 4 － 5 － 6 EM (I) -- Hovering

10. Conservation of energy（e- gyro motion） ⊙ does NO work Parallel motion （υ//） ∴ － 5 － 6 EM (I) -- Hovering

11. perpendicular motion I.C. － 1 ∵ － 3 － a a ∵ I.C. 7 8 ∴ 7 8 Note： υx2+υz2 =υ⊥2 = const. for all t → circular motion？ EM (I) -- Hovering

12. Integrate ∵ ∴ EM (I) -- Hovering

13. z ⊙ z x x ⊙ Guiding center positions let ∴ ∴ gyro-radius ( Larmor radius) ∴ Circular motion indeed in the perpendicular direction If υ//≠ 0helical trajectory EM (I) -- Hovering

14. A A 1.2 Charge ＆ current densities In Maxwell’s day, it was not known that charges are not infinitely divisible but occur in elementary units of 1.6×10-19C, the charge of an electron. Hence, Maxwell’s macroscopic theory deals with continuous charge distrib. Def. ﹝c/m3﹞as the volumetric charge density Current（I）is charge transport and is effectively time rate of change of charge（Q） ﹝C/s﹞=﹝A﹞ Def. current density ﹝A/m2﹞ ﹝＃/m3﹞ total ＃ of e- trapassing A EM (I) -- Hovering

15. X EX.1.2.1. Charge and current densities in a vacuum diode （low e- flow） → i.e. e--gun Ex=const.（uniform） ~ Caused by charges much more intense than the space charge e-/s e-s continuously injected at x = 0 with υi ∴ At each x = const plane ρ(x)·υx(x) = J0 = const. wrt. x & t （proved later in P.26） p.7 Ex1.1.1 take〝－〞to satisfy t = tiξx = 0 EM (I) -- Hovering

16. x x δ ∵ p.7 Ex1.1.1 = x EM (I) -- Hovering

17. 1.3 Gauss1 integral low of electric field intensity ( E ) closed surface (Gaussian surface;GS) always imaginary permittivity of free space ≡ε0 = 8.854×10-12 F/m defined directed outward V enclosed charge ie：G.S. field source ε0E ≡ D = electric displacement flux density Out of any region containing net charge, there must be a net D. EM (I) -- Hovering

18. Ex. 1.3.1 E due to spherically symmetric charge distrib. r < R Given : r > R EM (I) -- Hovering

19. = r < R = r > R r < R r > R EM (I) -- Hovering

20. Singular charge distrib. Point charge An infinite charge density occupying zero volume From Ex. 1.3.1：for r > R 〔C〕 Line charge density λl 〔C/m〕 EM (I) -- Hovering

21. r q Surface charge density σs 〔C/m2〕 Illustration : field of a point charge EM (I) -- Hovering

22. z ＋ ＋ ＋ ＋ ＋ ＋ ＋ ＋ ＋ ＋ G.S. r Illustration ：field associated with straight uniform line charge observer at any z sees same field ∴ E＝E(r) To see：if EZ ≠ 0 ＝C1 then rotation of an r axis by 180o must lead to Ez＝ -C1 However, rotation leaves the charge distrib. intact. ∴ Ez＝0 EM (I) -- Hovering

23. ＋ ＋ ＋ ＋ G.S. － － － － ＋ ＋ ＋ ＋ － － － － Ex. 1.3.2 Field of a pair of equal ＆ opposite infinite planar charge densities → a [[ E ]] jump across σs surface A Ez = ? Eoutside bottom = E0 Eoutside upper = ? ∵ Source distrib. indep. of x, y E = Ezz also indep. of x, y choose G.S. : If E0 = 0 EM (I) -- Hovering

24. q ○ Oq2 Illustration：Coulombs force law for pt’ charge （a check of Lorentz law） For a charge q at rest = 0 If E is caused by q2: Coulomb’s law ∴ Lorentz law is consistent with Coulomb’s law. EM (I) -- Hovering

25. Gauss’ continuity condition The general rule about the E jump across charged surface. d Qenclosed by G.S. EM (I) -- Hovering

26. 1.4 Ampere’s Integral law The law relating the magnetic field intensity H to its source：the current density J D = ε0E = displacement flux density displacement current thru S density Open surface S (can be curved) Contour integral along C：edge of S (right-hand rule sense) EM (I) -- Hovering

27. S = S1 S = S2 J ≠0 H≠0 J = 0 H = 0 Cf. Lorentz law f = q (E+υ×μ0H)there is no μ0 in Ampere’s law in front of H. H : [ C/m-s ] Permeability of free space ≡μ0 = 4π×10-7 henry/m （henry =voltsec/amp） Permittivity of free space ≡ε0 = 8.854×10-12farad/m Originally was known. However, when charging a capacitor OK. ∴inconsistent ∴The 〝 displacement current〞was added. EM (I) -- Hovering

28. Ex.1.4.1 Magnetic field due to axisymmetric current →ψ-indep. Given a constant－ z-directed (from -∞to +∞）current distrib. r < R ψ-symmetric H = Hψψonly azimuthal field intensity exists r > R Why？ EM (I) -- Hovering

29. z C’ Hr = 0 ∵rotation of r-axis reverses the source ( ie：J // -z now ), and hence must reverse the field. But Hr does not reverse under such an r-axis rotation apparently ∴Hr= 0 Hz, Hψ left Hz is at most uniform. = = 0 0 0 = = EM (I) -- Hovering

30. Hψ= ? = = EM (I) -- Hovering

31. Singular current distrib. line current i [A] Surface current density [A/m] ∴ K is tangential to the surface, in amp / meter along the surface meter K EM (I) -- Hovering

32. Illustration : H field produced by a uniform line current Hψ Illustration : Uniform axial surface current → H = ? Given uniform, z-directed, from -∞ to +∞, radius-R shell of surface current density K0( = const. ) = = Hψ ∴ surface current H jump EM (I) -- Hovering

33. Ampere’s continuity condition A surface current density in a surface S cause a discontinuity of the magnetic field intensity ( H ) To obtain a general relation concerning this H jump : of s’ S span >>( l, w ) chosen the surface of a sheet of current & w<< l ∵though w → 0 we have | J | →0 def. of K < ∞ 0 EM (I) -- Hovering

34. = ∵ inis arbitrary so long as it lies in S (position-dependent) EM (I) -- Hovering

35. 1.5 Charge conservation in integral form Embedded in the laws of Gauss and Ampere is a relationship that must exist between the charge and current densities. Gauss : E’s source : ρ Ampere : H’s source : J Cf. ? Apply Ampere’s law to a closed surface : S : 〝bag〞 C : 〝drawstring 〞 Drawing C tight C shrink to zero S becomes closed EM (I) -- Hovering

36. 0 closed closed = Gauss’ law =0 Law of conservation of charge If there is net current out of V , then the net charge enclosed by S must decrease in time. EM (I) -- Hovering

37. A compelling reason for Maxwell to add the electric displacement term to Ampere’s law was that without it. Ampere’s law would be inconsistent with the charge conservation. Specifically, if is missing in Ampere’s law dropped always The net current cannot enter or leave a volume, always There are some special cases for which are true. (cf. ρ=const. incompressible flow) or at steady state (穩態) EM (I) -- Hovering

38. EX. 1.5.1 Continuity of convection current (cf. EX 1.2.1 vacuum diode) ∵ steady state EM (I) -- Hovering

39. EX. 1.5.2 Current density and time-varying charge known With chosen S &V = ∵axisymmetric ρ(r,t) = EM (I) -- Hovering

40. EM (I) -- Hovering

41. Charge conservation continuity condition You may obtain the result by comparing with the continuity condition of Gauss’ law, as the text did. But we’ll do it directly here, instead. ∵ working on t & indep. of r ∴ used d/dt out side V pillbox-shaped surface S, with h→0 closed Assuming | J | <∞ ∴Knot included EM (I) -- Hovering

42. 1.6 Faradays integral law Cf. Gauss’ : E------ρ Ampere’s : H------J Charge Conservation : ρ-----J Faraday’s and Gauss’H laws involve no ρ or J Faraday’s induction law The circulation of E around a contour C is determined by the time rate of change of the magnetic flux linking the surface enclosed by that contour. EM (I) -- Hovering

43. Def. electromotive force (EMF) (電動勢) between 2 pt.s along a path P : 2 major circumstances for Faraday’s law : (EQS) & (MQS) A. Free of circulation B. with circulation EM (I) -- Hovering

44. A. Electric field intensity with no circulation (EQS) negligible For whatever contour chosen always in fact EQS, as we’ll see Illustration : A field having no circulation A static field between plane parallel sheets of uniform charge density has no circulation. C1 ∴ indeed EM (I) -- Hovering

45. EX. 1.6.1 Contour integration too = = EM (I) -- Hovering

46. (1) y r (2) x  (1)  (3) (4)  (2)  dr = dr &  sin cos   EM (I) -- Hovering

47. Electric field intensity with circulation (MQS) E is generated by the time-varying magnetic flux density μ0H (≡B) Faraday’s law holds for any contour, whether in free space or in a material. Often, however, the contour of interest coincides with a conducting wire, which comprises a coil that links a magnetic flux density. wire, C. EM (I) -- Hovering

48. Illustration : Terminal EMF of a coil = = = ∵∞ conducting ∴E = 0 ∞ conducting wire = Flux linkage (≡Φ) i.e. : the total flux of magnetic field linking the coil ∴Faraday’s law makes it possible to measure μ0H electrically. EM (I) -- Hovering

49. Faraday’s continuity condition The tangential electric field is continuous across a surface of discontinuity, provided that the magnetic field intensity is finite in the neighborhood of the surface of discontinuity. You may secure this result by comparing and inferring the Ampere’s continuity condition. But here we will do it directly from Faraday’s law. Consider a surface of a sheet of magnetic flux (μ0H ) S: EM (I) -- Hovering

50. = 0 as w →0 ≈0 ∵| H | < ∞ = Since in is arbitary, (position-dependent) EM (I) -- Hovering