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

1. Electromagnetism - I （電磁學－I） Chapter 2 Maxwell’s Differential Laws in Free Space Professor: Chungpin Liao (Hovering) 廖重賓 （飛翔） cpliao@alum.mit.edu cpliaq@nfu.edu.tw EM -- Hovering

2. 2.0 Introduction Maxwell’s integral laws encompass the laws of electrical circuits. The transition from fields to circuits is made by associating the relevant volumes, surfaces, and contours with electrodes, wires, and terminal pairs. Indeed, many of the empirical origins of the integral laws are in experiments involving electrodes, wires and the like. The remarkable fact is that the integral laws apply to any combination of volume (V) and enclosing surface (S) , or surface (S) and enclosing contour (C), whether associated with a circuit or not. Even though the integral laws can be used to determine the fields in highly symmetric configurations, they are not generally applicable to the analysis of realistic problems. EM -- Hovering

3. Reasons： Ⅰ) geometric complexity ★ Ⅱ) ρ, J are general unclear ∵self-consistent interplay between sources and fields Applying to arbitrary volumes, surfaces, and contours, the integral laws ales contain the differential laws that apply at each point in space. The differential laws provide a more broadly applicable basis for predicting fields. The point relations must involve information about the shape of the fields in the neighborhood of the point. Integral laws are converted to point relations by introducing partialderivatives of fields curt spatial coordinates. operators EM -- Hovering

4. The game is first to write each of the integral laws in terms of one type of integral. E.g. : Gauss’E law //← to be derived ― (*) (*) results only when V is arbitrary Q & A Q：Is it true that if two integrals are equal, their integrands are as well？ Ans.：In general , NO！ EM -- Hovering

5. 2.1. The divergence operator (div) If Gauss’ E law : Gaussion surface. is to be written with the surface integral replaced by a volume integral, it is necessary that an operator called：〝divergence operator 〞(div) be found such that： ― @ for general A Q & A div A = ? Apply @ to △V (since V is arbitrary) EM -- Hovering

6. Z (x, y, z) y (x, y, 0) X We need: def. a function of r With this condition satisfied, the actual shape of △V is arbitrary. A convenient choice of △V a rectangular parallelepiped △X△Y△Z →0, centered at (X,Y,Z) in Cartesian coordinate. EM -- Hovering

7. Z (x, y, z) y (x, y, 0) X = = △X△Y△Z = EM -- Hovering

8. need ∴ in Cartesian coord. del： div suggests that this combination of derivatives describe the outflow of A from the neighborhood of r. EM -- Hovering

9. 2.2. Gauss’ integral theorem (math) The operator “div” (div A) that is required to validate has been identified by considering △V→0 Q & A Q：Does the identified div A hold for △V → 0 ？ (or V) × Ans. ：subdivision of V into Vi’s EM -- Hovering

10. = ∴ ≡ divA when △V → 0 ∴ Gauss’ integral theorem (math) EM -- Hovering

11. 2.3 Gauss’ law, magnetic flux continuity, and charge conservation ∴ Gauss’ E law : = V arbitrary magnetic flux continuity = charge conservation ∴ = = differential charge conservation law EM -- Hovering

12. 2.4. Curl operator Ampere’s : Faraday’s : If ＆ Open surfaces are to be written in one type of integral, it is necessary to have a new operator (called curl ) such that： for general A － b open to be determined def : da // n Again, the curl A form is to be identified by making the surface an incremental one, △a EM -- Hovering

13. z C y 0 x At r, pick a direction n and construct a plane ⊥n through the point r ∴ b /// －(*) Then, in this plane choose a contour C around r circircling △a The shape of C is arbitrary except that all its points are assumed to approach r under study when △a→0. (*) is indep. of coordinates type EM -- Hovering

14. △a z ⊙ △Z center (X,Y,Z) △Y y x To express (*) in Cartesian coord.： Let n = x (i.e., △a on yz-plane) (*) EM -- Hovering

15. Similar procedure on other faces (ie：△Z△X, △X△Y faces) ∵del operator ▽ : ∴ EM -- Hovering

16. 2.5. Stoke’s integral theorem (math) The operator 〝curl 〞(giving curl A ) that is required to validate has been identified by considering △a →0 open Q & A Q：Does this relation hold for S and C of finite size and arbitrary shape？ Ans.： positive sense of Ci EM -- Hovering

17. positive sense of Ci ≈ curlA at r when △ai → 0 ∴ Stoke’s integral theorem EM -- Hovering

18. 2.6 Differential laws of Ampere ＆ Faraday Ampere’s open = = ∵ arbitrary S : ∴ differential form of Ampere’s law displacement current density ∂/∂t is used to make it clear that the location (x,y, z) at which the expression is evaluated is held fixed as the time derivative is taken. EM -- Hovering

19. Faraday’s = = ∵arbitrary S ∴ differential form of Faraday’s law EM -- Hovering

20. Charge conservation ∵Ampere’s： = = ρ 0 Verify it your-self ∴ differential form of Charge conservation Or, from the integral form： = = ∵ arbitrary V： EM -- Hovering

21. 2.7 Visualization of fields and the divergence and curl General vector A(r) : r - dependent 3 - componects Hard to draw 3D scalar, let alone 3D vector. Convention： Field lines to represent field Number of field lines to represent field strength Particular fields： No diveragence ( A = 0 ) A solenoidal No curl ( A = 0 ) A irrotational EM -- Hovering

22. solenoidal A： field lines of A endfaces Solenoidal field lines form hoses within which the lines never begin nor end. Asolenoidal field is represented by lines that are continuous： they do not appear or disappear within the region where they are solenoidal. EM -- Hovering

23. + + + r + R Ex.2.7.1 Fields with divergence but no curl ( irrotational but not solenoidal ) A = 0 A ≠ 0 ∵ Ex.1.3.1 (P.12) Spherical coord. EM -- Hovering

24. ＋＋ ＋ ＋ ＋ ＋ ＋ ＋ ＋ ＋ ＋ ＋ ∴ E ≠ 0 not solenoidal on the whole (ε0E= ρ of course agrees with original. ρ(r) ) (Just to emphasize that the differential laws apply point by point throughout the negion.) The E field has divergence only where there is a charge density. ( ε0E=ρ) picture Area  r2 ∴E1A1 = E2A2 ∴solenoidal outside ∴E1A1≠ E2A2 ∴E1 < E2 A1 < A2 ∴ not solenoidal inside EM -- Hovering

25. curl：Spherical coord. ∵E = Err & indep. of θ,ψ∴ E = 0 irrotational e.g. n into paper C= C1+C2+C3+C4 picture ∵⊥E cancel ncan be arbitrary same result e.g. n into paper ∵Eindep. of θ,ψ Recall : EM -- Hovering

26. i R Ex. 2.7.2 Fields with curl but no divergence (Solenoidal but rotational) A = 0 A ≠ 0 ∵ Ex.1.4.1 (P.20) Q & A Q：Where does this field ( H ) have curl？ Ans.： Ampere’s ∴ Curl H is the current density ∴ H ≠0 within (r < R) EM -- Hovering

27. Math：cylindrical coordinate γ→ψ→Z indeed Picture flux tube inner flux tube outer Q：Can the azimuthally directed field vary with r (⊥ψ) and ( ) have no curl in the outer region？ Ans. ：yes outer EM -- Hovering

28. outer C = C1+C2+C3+C4 & C1,C3 r H 1/r ∴C1 & C3 cancel. ∵⊥H Cf. interior : C1, C3 r H r2 H≠0 Hψψ & indep. of ψ,z. ∴ solenoidal See, flux tube inner ＆ outer EM -- Hovering

29. ＋ ＋ ＋ ＋＋ ＋ ＋ ＋ ＋ ＋ ＋ ＋ ＋ ＋ ＋ ＋ ＋ ＋ R S Some point to be clarified Q：Is it possible for a field that has no divergence at each point on a closed surface S to have a net flux thru that surface？ Ans. ：yes Ex. 2.7.1 on S. (r >R) but only tells local information (not global) EM -- Hovering

30. C ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ r R ⊙ ⊙ ⊙ S Q：Is it possible for a field to have a circulation on some contour C and yet be irrotational at each point on C？ Ans. ：yes Ex. 2.7.2 but ( r > R ) It is known that (a b) ⊥a & ⊥b. So, would A⊥A = ? Ans.：with Ex.2.7.2. for r < R, it would seem so (H ↔ A) →J // Z, H //ψ But, it is not. = H ∵ by definition, (H+H1) = H if H1 = 0. So, if H1 has a component // ( H ), then (H+H1) ≡ H’ ⊥ H’ EM -- Hovering

31. 反證法 That is, if (false), then: (false) If then: EM -- Hovering

32. 2.8 Summary of Maxwell’s differential laws in free space Gauss’E law Ampere’s law Faraday’s law Magnetic flux continuity (or Gauss’H law) Charge conservation EM -- Hovering