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

1. Electromagnetism - I （電磁學－I） Chapter 4. EQS Fields: The Superposition Integral Point of View – A Convolution Approach Professor: Chungpin Liao (Hovering) 廖重賓 （飛翔） cpliao@alum.mit.edu cpliaq@nfu.edu.tw EM -- Hovering

2. EQS: ( E be irrotational ) Most of this chapter Find E(r) , given ρ(r) Later, also：Find E(r) given conductor B.C. In this 〝limited region bounded by conductors〞 case, the distribution of charge on the boundary surfaces is not known until after the fields have been determined. General boundary value problems in chap.5 EM -- Hovering

3. We’ll establish electric scalar potential ψ(r)that uniquely represents an irrotational E(r). Poisson’s equ.Linear in ψ The field (ψ) due to a superposition of charges is the superposition of the fields associated with the individual charge components. The art of arranging the charge so that, in a restricted region, the resulting fields satisfy boundary conditions (BC’s) will be shown. EM -- Hovering

4. Solving Poisson’s equ. ： Superposition approach －The impulse response of Poisson’s equ. is the field (ψ) of a point charge. 1 －Evaluating total ψ as the result of all charges needs the convolution (Superposition) integral. 2 Boundary value approach particular solu. →caused by the drive Homogeneous solu → to satisfy BC’s －Solution ψ －We’ll find that the superposition integral approach 1 is one way of finding the particular solution. EM -- Hovering

5. 4.1 Irrotational field represented by scalar potential： The gradient operator (▽) and gradient integral theorem Parth－independent integral The integral of an irrotational electric field from some reference point rref to the position r is independent of the integration path. (EQS) proof: = rref Stoke’s law path Ι pathⅡ Path Ι PathⅡ EM -- Hovering ＃

6. A field that assigns a unique value of the line integral between two points independent of path of integration is said to be conservative. Definition ：ψ(r) －the electric potential of the point r wrt. the reference point rref . With the endpoints consisting of 〝nodes〞 where wires could be attached, the potential difference would be the voltage difference. usually set to be the 〝ground〞 potential EM -- Hovering

7. Thus, for an irrotational field, the EMF defined becomes the voltage at the point of a relative to point b. ρR=0 E = 0 a+ - b (p.32) We shall show that specification of ψ(r)(scalar) contains the same information as specification of E(r) (vector). remarkable！ Surfaces of constant potential（ψ(r)=const.）are called equipotentials e.g. spherical equi. pot. x2+y2+z2 = R2 EM -- Hovering

8. Gradient： 梯度. 2 equipotentials passing r and r+Δr respectively △n = shortest distance between ψ=const. and ψ+ △ψ=const. △r⊥≡ △n ∵differential geometry →△r⊥⊥both equipoten. to show Def. ∴ △n≡ △r⊥= △r． cosθ (▽ψ)≡gradψ if ∴ ∴ △ψ≡ gradψ•Δr Def. EM -- Hovering

9. gradψ≡▽ψ Proof: in Cartesian Coord. = ∵ grad ψ•Δr Recall: del operator EM -- Hovering

10. To show that the potential function ψ(r)uniquely defines. Recognizing Δr = ds & Δr → 0 ∴ Δψ= -E•Δr but Δψ= ▽ψ•Δr E = -▽ψ Further, ∵ ▽×E = ▽×(-▽ψ) = 0 ∴ E = -▽ψ fulfills EQS : ▽×E= 0 &ψ→ψ+c →E = -▽(ψ+c)=-▽ψsame automatically ∴ Whatever ψ(r) contains full info about E(r) EM -- Hovering

11. EM -- Hovering

12. An useful grad. Integral theorem. About rref: －We have not made any specific assignment. －In general , it is arbitrary. －Provided that the potential behaves properly at infinity, it is often convenient to let the reference point be at infinity. －there are some exceptional cases for which such a choice is not possible. EM -- Hovering

13. －All such cases involve problems with infinite amount of charges. －e.g. : ∞＝z - ∞＝z Recapitulation： For ▽×E = 0 (EQS), E can only be in the form E = -▽ψ, not others. So, ψ(or ψ+c) uniquely determines E. EM -- Hovering

14. EX.4.1.1. Equipotential surfaces Saddle point EM -- Hovering

15. EX.4.1.2 Evaluation of gradient ＆ line integral cf. fig above. －To check the fact ψ2-ψ1=2V0using line integral along C1＆ C2 c1 : y = a, ds = dx x = ∵C1 (-a → a) EM -- Hovering

16. c2: ( parabola 拋物線 ) = ※ indeed. EM -- Hovering

17. EX.4.1.3. Potential of spherical cloud of charge ( Piece-wise continuous ψ(r)) ρ0 = const. balancing charge at r →∞ r < R r > R － 1 － 2 EM -- Hovering

18. ∴ from outer to inner : let rref be at r → ∞ ∵ →E(r → ∞)=0 2 r > R = let 0 & ds= dr 2 = = r > R q where r = R EM -- Hovering

19. ∴ for r < R = = & ds= dr 1 = = ※ forr < R Like coulomb potential of a point change q. EM -- Hovering

20. 4.2. Poisson’s equ. Given that E is irrotational ( i.e. ×E = 0 ) (EQS) and ε0E=ρ,what is E(r) ? = ▽2ψ≡ Laplacian of ψ cf. circuit theory ODE (ordinary differential equ.) RHS= -ρ/ε0 is a 〝driving function〞 to push the LHS circuit. Linear ( in ψ), 2nd-order (in ) In Cartesian Coord. cylindrical spherical coord’s cf. ＆ Constant coefficient: 1 , though time does not appear explicitly, describes the instantaneous variation of ψ(t)due toρ(t) . EM -- Hovering

21. 4.3 Superposition Principle An important consequence of the linearity of Poisson’s equ. is thatψ(r) obeys the superposition principle. Cf. Linear ODE for circuit theory Consider 2 different spatial distributions of charge density, ρa(r) andρb(r), which are relegated to different regions or occupy the same region. Suppose 2 potentials are found to satisfy Poisson’s equ. with ρa ,ρb respective charge distrib. ∵ def. Adding  = ∵same result (∵ linearity) Therefore, ifρa →ρathenρa +ρb→ψa+ψb ρb →ρb EM -- Hovering

22. 4.4 fields associated with charge singularities The potential proves invaluable in picturing fields not highly symmetric. Some often used potential functions will be derived. Solutions to Poisson’s equ. ( ) filling all of space will turn out to be used in solving Laplace’s equ. (2ψ=0) in subregions that are devoid of charge ( ie: to fit BC’s ) Point charge This 〝impulse response〞 for the 3D Poisson’s equ. is the starting point in derivations and problem solutions. → worth remembering！ see blow EM -- Hovering

23. Consider ∵ Superposition principle － 1 in Cartesian coord. ： distance － － 2 3 － 1 4 EM -- Hovering

24. shows that in the immediate vicinity of one or the other of the charger, the respective charge dominates the potential. 1 Thus, close to the point charges, the equipotentials are spheres enclosing the charge. Further, ψ= 0 at z=0 (see ) 4 d →0 (to be shown) Z=0, ψ=0 ↑Z E E E half-wave dipole field d → ∞ (to be shown) E yz plane (x=0) E EM -- Hovering

25. Dipole at the origin (d→0). ~θ θ ~θ For an observer (P) for from either of the charges, d→0. (or r >> d). Charge pairs of opposite sign are the model for polarized atoms or molecules. (＊) (＊) EM -- Hovering

26. (＊) － (in spherical coord.) see previous page for figure 5 The dipole model is made mathematically exact by defining ： such that － 6 d → 0 q →∞ Another more general representation: － 7 EM -- Hovering

27. Pair of charges at infinity having equal magnitude and opposite sign (d → ∞, or r << d) Here we look at the appearance of the field for an observer located between the charges of + q and – q, in the neighborhood of the origin. in spherical coord. EM -- Hovering

28. r << EM -- Hovering

29. Similarly, = z ∴uniform on each xy plane, but ψ z Note : q P (see fig. on previous-page) same -q EM -- Hovering

30. 4.5 Solution of Poisson’s equ. for specified charge distributions Using the superposition principle Representing any arbitrary charge density distribution ρ(r) as a sum of 〝elementary〞 charge distribution： An elementary volume of charge at r' gives rice to a potential at the observer P at r . Note that each of these elementary charge distrib. Has zero charge density at all points outside of the volume element dV' situated at r' . EM -- Hovering

31. Thus, they represent point charges of magnitude dq , and the potential associated with this incremental charge is： where EQS. Calculation of －Triple integral in general. －Relying on numerical solution on Computers. (usually hard to obtain analytic solutions) EM -- Hovering

32. －There are special representation of ψ(r) when ρ(r') is confined to surfaces, lines or where the distribution is 2D. Triple integration reduced to 2 or even 1 integration. Difficulty in obtaining analytic solutions reduced. －3D charge distrib’s can be represented as the superposition of lines and sheets of charge and by exploiting the potentials found analytically for these distributions, the original numerical integration (a triple- integral ) can be reduced to two or even one numerical integration. EM -- Hovering

33. Superposition integral for surface charge density. If the charge density ρ(r') is confined to regions that can be described by surfaces having a very small thickness Δ, then ρ(r')dV' = da'Δρ(r') ≡ EM -- Hovering

34. EX.4.5.1 Potential of a uniformly charged disk = = EM -- Hovering

35. 2 limiting cases： |z| >> R |z| << R Like a charge sheet of ∞ extent. & (Gauss’) Check : EM -- Hovering

36. Superposition integral for line charge density A'→0 ≡ ρ(r') → ∞ [ C/m ] EM -- Hovering

37. EX.4.5.2 Field of collinear line charges of opposite polarity ∵axial symmetry → use cylindrical coord. elemental source： ±λ0dz' at (0, z') Observer P at (r, z) ∵ charge - (*) a = z2+r2 b=-2z c=1 (**) EM -- Hovering

38. I.S. Gradshteyn & I.M. Rithik Tables of Integrals, Series and Product. (*) For c > 0 where R = a+bx+cx2 (**) EM -- Hovering

39. different from that in the text. ∵axial symmetry, rotate around z to get the full picture by evaluating ψ at each (r, z) EM -- Hovering

40. Two dimensional charge and field distributions -∞ a fundamental element: line +∞ A 2D configuration of uniform or nonuniform charge distribution extending from Z=-∞ to Z=+∞ One of the 3 integrations of the general superposition integral is carried out by representing the charge by a superposition of line charges (-∞ < z < ∞) → fundamental element. EM -- Hovering

41. Recall： P -∞ Line change Z +∞ (Gauss’s E law ) Cylindrical coord. = = EM -- Hovering

42. r0 = a reference radius as an integration constant generalize： For － (*) Since the charge distrib. extends to infinity in the Z-direction, the potential at infinity can not be taken as a reference. The potential at an arbitrary finite position can be defined as zero by adding an integration constant to (*). EM -- Hovering

43. EX.4.5.3. 2D potential of an uniformly charged sheet (*) = σs(x') in general r= (x, y), r'= (x',0), and σs = σ0 In this case, at any z: EM -- Hovering

44. Gradshteyn p.205 － (*) Since 2D distributions of surface charge can be piece-wise approximated by uniformly charged planar segments, the associated potentials are then represented by superpositions of the potential given by the previous ψ2D . EM -- Hovering

45. σs S + + + + + + + d - - - - - - - - - - -σs Potential of uniform dipole layer Recall：a dipole of charges ±q spaced a vector distance d apart produces a potential: A dipole layer : An area element da of such a layer, with dapointing from -to +, can be regarded a differential dipole producing a differential potential dψ： EM -- Hovering

46. Def: surface dipole density： For dipole layer － (**) interpretation If πs = const. , then: 立體 單元角 dΩ:the extended angle of da•ir'rview from P solid angle EM -- Hovering

47. Discontinuity of potential in passing thru the surface S. Containing the dipole layer： Ω0 as the solid angle from + side :Ω0 from – side : -4π+Ω0 － (&) ∴ a discontinuity of potential across S ∵Δψ in d →0 E → ∞ in between . approaching From (**), contributions to (**) (ie:ψ) are dominated by πs are r →r ', the discontinuity of potential is till given by (&), even if πs =πs (r') EM -- Hovering

48. When πs =πs(x), ∵ Eint → ∞, we can show that (Prob.4.5.12) Exa +++++++ πs =σsd - - - - - - - ie: [[E//]]≠0 now Exb EM -- Hovering

49. P    EM -- Hovering

50. - Cut here P - + - complement: for - side EM -- Hovering