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

Electromagnetism - II （電磁學－II） Chapter 8. MQS Fields: Superposition Integral & Boundary Value Point of View Professor: Chungpin Liao (Hovering) 廖重賓（飛翔） cpliao@alum.mit.edu cpliaq@nfu.edu.tw EM -- Hovering

8.0 Introduction We now follow the study of EQS with that of MQS. MQS primary (recall) (see Table 3.6.1 p.83) ( Ampere’s law ) displacement current density or, magnetic flux continuity law ( Gauss’ H law ) ＆ B.C.’s at interfaces : surface current density [A/m] In the absence of magnetizable materials, these laws determine the magnetic field intensity H given its source, the current density J. (∵ H ≈ J in MQS ) EM -- Hovering

By contrast with the EQS field intensity E (satisfying E≈ 0 ), H is not everywhere irrotational (i.e., H= J≠0 in general ). However, it is solenoidal everywhere. ( 0H = 0 ) Cf. EQS MQS E = 0 E = - ψ H= J ε0E =ρ 0H= 0 Recall EQS starts from : Chap.4, 5 : E = 0 E = - ψ Poisson’s then Chap.6 : P = ε0χ0E BC D = εE Chap.7 : “ RC ” dynamics IC EM -- Hovering

Cf. MQS approach : ∵ H= J≠0 in general unless J= 0 ∴ Chap.8 :0H = 0 (  = r0 ) 0H =A 2A = -0J vector Poisson’s vector potential Chap.9 : m BC Chap.10 : “ L/R ” dynamics IC Chap.4Chap.8 Cf. ψ interpreted with insights A not interpreted until §8.6 where J≠0 in general For 2 reasons EM -- Hovering

For 2 reasons : directly  Often, J’sH without the intermediary superposition of a potential ( A) • In many situations of interest involving current carrying coils, use of surface current is adequate Approx. & ∵ H= 0 except where there is such surface current scalar potential  H = -  ∵ 0H= 0 when with no magnetizable materials ∴ 2=0 Laplace’s equ. EQS methods can be readily used. In Chap.9, we’ll extend the approach to problems involving piece-wise uniform and linear magnetizable materials. EM -- Hovering

Vector field uniquely specified A vector field ( F ) is uniquely specified by its cure ( F ) and divergence ( F ) see 〝proof〞 below : F = C(r)vector－ F = D(r)scalar－ If we let and assume that FaandFb are two different solutions of  &  , then the difference solution Fd≡Fa - Fb is both irrotational ( Fd = 0 ) and solenoidal ( Fd = 0 ). ∵ Fd= 0 Fd = -ψd ∵ Fd= 0 2ψd = 0 and ψd = 0 on bounding surfaces same equ’s as in § 5.2 uniqueness of EQS The only way ψd can both satisfy Laplace’s equ. and be zero on all of the bounding surfaces is for it to be zero. see proof in § 5.2 notes EM -- Hovering

F = C(r) F = D(r) uniquely determines F Cf. Chap.4. (EQS) Chap.8. (MQS) F → E C → 0 D → F → H C → J D → 0 The strategy in Chap.8 parallels that for Chap. 4 & 5 ∴H= Hp + Hh again to satisfy B.C. (together with Hp) due to current density Hp = J 0Hp = 0 Hh = 0 0Hh = 0 ∴ & EM -- Hovering

8.1 The vector potential ( A ) and the vector Poisson equ. ∵ 0H = 0 ∴ a general solution H is : － (+) 0H = A － (++) (++) as “ integral ” of (+) Cf. EQS : E = -ψ as “ integral ” of E = 0 In EQS, we can let ψ´→ψ+ c without affecting E (∵E = -ψ´= -ψ -c = -ψ) no need to uniquely specify  =0 In MQS, we can let A→A´=A+ without affecting H ( ∵ 0H = A´ = A+  = A) =0 We can view 0H = A are the specification of A in terms of the physical H field. (i.e. : A= 0H ) EM -- Hovering

However, to specify A uniquely we need to further specify A for ease of math →(*) In MQS, it is traditionally convenient to make the vector potential ( A ) solenoidal : called 〝setting the gauge〞 to the Coulomb gauge. 標準量度 A = 0 － ( & ) ∴ ∵ H = J (MQS) ∴ 0H = 0J = － (++) A =  (A) = (A) - 2A － (&) Coulomb gauge = 0 ∴ We arrive at the vector Poisson’s equ. : 2A = -0J － (*) EM -- Hovering

Therefore, as the solution to the vector Poisson’s equ. : 2A = -0J － (*) A = ? View (*) in the Cartesian Coord. then (*) is equivalent to 3 scalar Poisson’s equ.’s, e.g., z-component 2Az = -0Jz Recall that for EQS : Now, for MQS, with the identification of ψ→Az , 0Jz→ρ/ε0 , we realize : Combining with x and ycomponents, we obtain the superposition integral for the vector potential : Unique solution specified by A = 0H & A= 0 (coulomb gauge) EM -- Hovering

Note that, However, in order that A = 0Hbe a physical flux density, J(r) cannot be an arbitrary vector field. ∵ div (curl) of any vector = 0 ∴  (H) ∴J = 0 for MQS = (MQS) J I.e., the current distributions of MQS must be solenoidel. Cf. J = 0 also happens in EQS when ( e.s.) As a reminder, even under dynamic conditions, J = 0 remains valid for MQS systems. 1. A uniquely specified now, so must J too 2. using steady state J (J = 0) of Chap 7 (EQS), where J =  (-), so  J =   (-) = 0 3. steady J = 0 can be extended to dynamic situation for MQS where J = 0 always EM -- Hovering

Q & A Q : To uniquely specify J , we know that only J = 0 is not enough (we also need J ), so what so we normally do for MQS systems actually ? A : Low-freq. High-freq. (τ> τe ) (τ< τe ) If Ohmic, the stationery (i.e. : J(r)=σ(r) E(r)) (e.s.) J(r) found in § 7.1-7.5 ( satisfying J= 0 already) can be used in (Faraday’s) has to do applied to determine J for use in A integral (§ 8.4-8.6 & Chap.10 ) EM -- Hovering

2D current and vector potential distributions If J = Jz (x, y) z exists through the space of interest, then i.e. : － (i) MQS Note that this Azexpression is formally the same as － (ii) produced by a charge distrib. ρ(x´, y´) EQS Recall that it was inconvenient to integrate (ii) directly, and it was indirectly approached by first evaluating ψℓ due to a line charge (λℓ ) to be : using symmetry & Gauss’ E EM -- Hovering

distance from the line charge r0 = reference radius Then, using λℓ→ρ(r) da´, r → |r - r´|to integrate over all 〝line charger〞 to get : － (iii) = da´ In dealing with charge distributions that extends to infinity in the z direction, the potential at infinity cannot 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

Cf. MQS EQS － (i) － (ii) and the integrated result of (ii) : － (iii) we have : making － (iv) EM -- Hovering

3 important consequences emerge from this derivation : • Every 2D EQS potential ψ(x, y) produced by a given charge distrib. ρ(x’,y’), has an MQS analog vector potential Az (x, y) caused by a current density Jz (x’, y’) with the same spatial distrib. as ρ( x’, y’) . (B) Lines of constant Az are lines of magnetic flux ( 0H) ( Proof ) : Az: gradient of Az largest variation 0H⊥Az EM -- Hovering

EQSMQS -Az ψ= const. Az = const. E = -ψ -Az 0H⊥Az ∴ 0His along Az = const. Note that, however, 0H is not constant along a line of constant Az a Az = const. line 0H = -zAz varies along a Az = const. line since along which Az ≠const. | 0H | = | Az | steeper EM -- Hovering

(C). The vector potential of a line current of magnitude i along the z direction obtained by analogy with i.e.: － (**) is consistent with the previously obtained ( from ) 2πrHψ= i ( Proof ) : AZ symmetric wrtψ ( see(**) ) Aψ=0 Aψ=0 Ar=0 Ar=0 EM -- Hovering

(**) A special case of: EM -- Hovering

EX. 8.1.1 Field associated with a current sheet J = Jzz from x1 to x2 uniformly ( →Jz = const. ) const. Az surface remember, these are also the ines of0H along Jz K0≡JzΔ u≡x-x´ EM -- Hovering

∵ Gradshteyn's table p.205 x1-x2 EM -- Hovering

Note that, as we’ll see, ∵ n·0H = 0on the surface of a perfect conductor (§8.4) & ∵0H = -zAz ie : 0H resides on the constant Az surface ( or, no 0H⊥Az surface) ∴ Constant Az surface ≡ perfect conducting surface (2D) Cf. Constant ψ surface ( equipotential surface) ≡ perfect conducting surface (2D, 3D) Az=const. //0H For 2D in 3D Note that, in general, 0H= A⊥Az (x, y, z) Ax (x, y, z) Ay (x, y, z) EM -- Hovering

EX. 8.1.2 2D magnetic dipole field surface of const. Az and also lines of0H d oppositely directed currents extending -∞ → + ∞ along z2D right, out →∵ 2 currents left, in polar coord. (approx.) EM -- Hovering

∴ see (*) d << r ≈ Sequence r→ ψ → z And (*) in the above EM -- Hovering

8.2. The Biot-Savart Superposition integral Once the vector potential A has been determined from the superposition integral : the magnetic flux density 0Hfollows directly from A = 0H However, in certain field evaluations, it is best to have a superposition integral for the field itself. For example in numerical calculations, numerical derivatives should be avoided. The field superposition integral : wrt r ( not r ) =Ψ·J(r’ ) Ψ≡ | r – r’| -1 EM -- Hovering

r r’ ≡ source – observer unit vector ( ΨJ ) =  ΨJ + Ψ  J = J indep. of r 0 Biot-Savart law for H This integrand represents the contribution of the current density at r’ to the field (H) at r . EM -- Hovering

Cf. EQS a contributing charge element cf. q ‧ EM -- Hovering

EX. 8.2.1 On-axis field of a circular cylindrical solenoid ∴r // z z N-turn in d Hz = ? r z In cylindrical coordinate : sequence r → φ→ z& dv´= (rdφ´)dr´·dz´ EM -- Hovering

dV’ Δ<< r’=a EM -- Hovering

z" ≡ z "-z ∴dz"= dz´ factor out a ≈ EM -- Hovering

Stick model for computing fields of electromagnet The Biot-Savart superposition integral ( ) can be completed analytically for relatively few configurations. However, it’s no more than a summation of the field contributions from each of the current elements. →∴straight forward on the computer. Many practical current distrib’s are, or can be approximated by, connected 〝current sticks 〞. ( i ) On a current stick, represented by a vector a , the current is uniformly distributed between the base of a at (r+b) and the tip of a at (r+c) an i-stick I.e., system of currents P (observer) EM -- Hovering

Note that because the i-stick does not represent a solenoidal current density ( i.e., satisfying J = 0 ) at its ends, the field derived is of physical significance only if used in conjunction with other i-stick that 〝together〞 represent a continuous current distrib. Detailed view of a current stick : P Distance of ξ=0 to P ≡r0 = closest distance of a to P EM -- Hovering

now due to one i-stick = = = dξ 1 = = = EM -- Hovering

due to one i-stick This expression can be incorporated into a subroutine of a computer code evaluating H(r) and used repetitively. P (observer) evaluating H due to the i-stick a1 EM -- Hovering

8.3 The scalar magnetic potential (Ψ) MQS : displacement current density In the space free of current : H = 0(irrotational H ) For J= 0, H = 0 leads to H = -Ψ Further, ∵  0H = 0→ 2Ψ= 0 Cf. EQS : • Charge free or • Solving 2ψh = 0 & B.C.’s Good that EQS results can be used for MQS EM -- Hovering

EX. 8.3.1 The scalar potential of a line current (i) A line current is a source singularity. ∵ H≈ J (MQS) = -Ψ have Really not equal! = don’t even need to utilize EQS analogy Note that Ψ ( scalar potential ) is multiple valued as the origin is encircled more than once. → let 0 ≤φ<2π artificially This property reflects that strictly H is not curl free in all of space. ( i.e., i : source singularity ). EM -- Hovering

The scalar potential (Ψ ) of a current loop A current loop carrying a current i has a magnetic field that is curl free ( H = 0→Q : Ψ = ? Current loop) everywhere except at the location of the wire. ( i is a singular source. ) Remember that the scalar potential Ψ came from H= -Ψ owing to H = 0 Since the line integral enclosing the current does not give zero, a definite (well-defined) Ψ does not come out. Q & A Q : So, what do we do to get Ψ ? A : Path’s (C’s) that enclose the current i in the loop are not allowed, if the desired scalar potential Ψ is to be single-valued. EM -- Hovering

Spanning S Cl i C S Solution: Mounting over the current loop Cl a surface S ( spanning the loop ) which is not crossed by any path of integration C. i.e., ∵ H ≈J (MQS) In reality Cf. H = 0 (approx.) So, to secure a scalar potential, some artificial means must be implemented. Hpointing inward In fact, H =J // J EM -- Hovering

Spanning S Cl i C S Treat the inward point on screen as a portion of the aforementioned line current and let C be broken! Since there are infinite such points along Cl, we’d better have a spanning surface S as stated. EM -- Hovering

With the above arrangement, the scalar potential Ψ is then made single-valued. The discontinuity ( jump ) of potential across the surface S follows from Ampere’s law : ∫ c note the sign of a broken circle ∫ ∫ i.e.: = i ∵ H = J So, Ψ (r) =? (for a current loop ) Cf. EQS uniform dipole layer We have found in EQS that a uniform dipole layer of magnitude πs on a surface S produces a potential that experiences a constant potential jump πs / ε0 across the surface S. Its potential : (p.111) πs=σs·d Ω= solid angle subtended by the rim of S as seen by on observer at r EM -- Hovering

Cf. Where Ω is the solid angle subtended by the contour along the wire as seen by an observer atr EM -- Hovering

In the EQS dipole layer case, the surface specified the physical distribution of the dipole layer. In the present MQS current loop case, S is arbitrary as long as it spans the contour Cℓ of the wire. This is consistent with the fact that the solid angle Ω is invariant wrt the changes of the surface S and depends only on the geometry of the rim. EM -- Hovering

EX. 8.3.2 The H field of a small loop r´// z & at origin Ψ = ?, H = ? ( This is the potential of a magnetic dipole, as we’ll see ) =0 Dipole field EM -- Hovering

Cf. Physical insight : In fact, as far as its field around and far from the loop is concerned, the current loop can be viewed as if it were a 〝magnetic〞 dipole, consisting of two equal and opposite magnetic charges ±qm spaced a distance d apart. Due to the dipole field shape of H(r) To see if we’ll get the same results on Ψ and H The magnetic charges ( ±qm ) (monopoles ) are sources of the divergence of the magnetic flux density 0H, analogous to electric charges ( q ) as sources of divergence of the displacement flux density ε0E . EM -- Hovering

Thus, if Maxwell’s equ’s are modified to include the action of a magnetic charge density in units of V-s/m4, then the new magnetic Gauss’ law must be : ·0H=ρm－ (*) ( cf. ε0E = ρ) The magnetic monopoles have been postulated by Dirac and have been located though very rarely. Here the introduction of magnetic charge (monopole) is a matter of convenience. ∵The potential of an electric dipole is : EM -- Hovering

Recall : EQS ( E = -ψ) MQS ( H= -Ψ) J = 0 － (†) where where p ≡ qd monopole dipole (moment) electric dipole moment magnetic dipole (moment) = dipole moment of a current loop. EM -- Hovering

Note that : magnetic dipole moment in literature Pm or m (†) ∴ of a magnetic dipole at origin ( r´= 0 ) same as that from using the notion of a current loop Of course, the details of the field produced by the current loop ( using m = ia ) and the magnetic charge dipole ( using Pm=qmd) differ in the near field. qm : 0H≠0 (=ρm) Loop : 0H = 0 EM -- Hovering

8.4 MQS fields in the presence of perfect conductors ( 3D in general ) There are physical situations where the current distrib. J(r) is not pre-specified, but is given by some equivalent information. E.g., a perfectly conducting body in a time-varying magnetic field H(t) supports surface currents K(t) that shield the H field from the interior body. The effect of the conductor on the magnetic field H(t) is reminiscent of the EQS situations where charges ρ(r) distributed themselves on the surface of a perfect conductor in such a way as to shield the electric field E(t) out of the conducting body. We found ( in Chap. 7 ) that the EQS model of a perfect conductor described the high-frequency response of system in the sinusoidal steady state, or the short-time response to a step function drive, of a metal with σ<∞ EM -- Hovering

We’ll find ( in Chap.10 ) that the MQS model of a perfect conductor represents the low-frequency sinusoidal steady state response, or the long-time response to a step drive, of a metal with σ<∞. In MQS, i.e. the value of conductivity σ which justifies use of the perfect conductor model depends on the frequency (or, time scale in the case of a transient ) as well as the geometry and size, as will be seen in Chap.10 Cf. EQS : freq. or τ but not geometry and size EM -- Hovering

Superconductors (possible realization of perfect conductors ) : Type Ⅰ －metal coding －e.g. : Pb －low temp －room temp ↓ cooled Type Ⅱ －composite －e.g. : Y1Ba2Cu3O7 －〝high〞 temp. ≥77 K H H H H EM -- Hovering

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