complex E and B issues REVIEWED

Confusion with complex E and B PhL 9.24.13 I was truly confused, and the notes below led me to get things straightened out. The subject is now fully documented in Section 1.7 of lines.doc. This little working doc turned out to be very useful. There is no need to read it ever again, but I will keep it for the record. Big Bug just found. In my Fourier Transform stuff in general one must have B(x,ω) = !Syntax Error, Idt e-jωt B(x,t) and although B(x,t) is real, we know that B(x,ω) is in general complex, so my little arrows in my pictures showing real quantities don't mean much! On the other hand, one could say B(x,t) = eiωt B(x) but then again B(x,t) is complex. I have not clarified this always-murky idea. So today will be lost getting this mess cleaned up. But now it is time for caulk and paint! // Am back. We like the eiωt idea because it makes for simple rules on ∂t and so on. The theory of course is that B(x,t) = Re[ eiωt B(x) ] = cos(ωt) B(x) = real times real. But how does this pass through differential equations like those of Maxwell. Consider: curl E(x,t) = - ∂B(x,t)/∂t ok In full FT this does say curl E(x,ω) = iωB(x,ω) ok Ouch. This has the feel of a whole new Appendix once I get this cleaned up which will take a while. As I look through lines before section 2.1, I am fine with equations having E(x,ω) which is a complex entity. Where I have trouble is in those Stoke's loops. Look at (2.1.1) and (2.1.2). I think as equations these are all just fine, and express the relations between complex vectors. curl B(x,ω) = μ (jωε E(x,ω) + J(x,ω) ) B(x,ω)•ds = μ ∫(jωεE(x,ω) + J(x,ω) )•dA (2.1.1) curl E(x,ω) = jωB(x,ω) E(x,ω)•ds = -jω∫B(x,ω)•dA . all ok (2.1.2) Good Question: But how do you draw E(x,ω) in a diagram if it is a complex vector? I don't think I have done this before. It has a direction like , that part is OK, but the component in the z direction is complex. So perhaps my drawings really show Re(B) and not B [no], say. But in (2.1.2) things are mixed. Go back to E(x,t)•ds = -∂t∫B(x,t)•dA Now the vectors are real, but time derivative sits there. This is such an elementary level subject, I wonder if I have dealt with it somewhere before. If not, time to produce some notes. Note: I think I resolved this issue of what the little arrows mean, added it to lines doc. Perhaps it will come up below in this doc. A complex vector can still have a direction!!! Like E(x,t) = Ez(x,t) . Suppose we assume this for FT electric field vector: E(x,ω) = E(x)2πδ(ω-ω1) [ but this form is not correct for the round wire which is more E(x,ω) = E1(x,ω)2πδ(ω-ω1) ] Then: E(x,t) = ejωt E(x) = (cosωt + jsinωt)(Re E(x)+ j Im E(x)) [cosωt Re E(x) - sinωt Im E(x)] + j [cosωt Im E(x) + sinωt Re E(x) ] If E(x,t) is real, this implies a relationship between Re and Im parts of E(x). [ but don't want this E(x,t) to be real! ] Maybe I have started at the wrong point. In time domain write E(x,t) = ejωt E(x) (1) ok, a special assumed form Note that E(x) might depend parametrically on ω1 Think now of E(x) as being real [ok], and then E(x,t) is the thing that is complex [ok] , and then actual field = Re E(x,t) = cosωt E(x) by fiat ok Now do the FT on both sides of (1) to get E(x,ω) = 2πδ(ω-ω1) E(x) (2) ok Now you cannot give B the same phase because of Maxwell: curl E(x,t) = - ∂B(x,t)/∂t claim stated is true, you cannot So start over once again and postulate that B(x,t) = ejωt B(x) B(x) = real ok, but a completely new form Then curl E(x,t) = -iω1 B(x,t) = -iω1 ejωt B(x) = ω1 eiωt-π/2 B(x) ok I will continue this a bit. Suppose it were this simpler 1D equation ∂xE(x,t) = -jω eiωt B(x) What conclusions can one reach about E(x,t) just from this equation? After a short digression into PDE theory (see new folder in math on same), the general solution to the above is E(x,t) = !Syntax Error, I dx [-jω eiωt B(x)] + g(t) = -jω eiωt !Syntax Error, I dx B(x) + g(t) . One conclusion is that one possible solution of the equation is this E(x,t) = -jω eiωt !Syntax Error, Idx B(x) So I crudely claim that, apart from a possible arbitrary time function G(t), for which curl G = 0, the curl E equation above does imply that E is monochrome. ******************* = jωB(r) . (2.1.9) This does NOT imply that = ω|B(r)| correct, ∂r and | | do not commute Example: ∂r |ejr| = ∂r 1 = 0, whereas | ∂rejr| = | j ejr| = 1. so I really cannot make any clean statement regarding skin effect. yes Question: In this picture, one might say " the vector J points in the direction " . What do those words mean when J has complex components? Even with real components, you don't really know whether Jz is positive or negative when you say this. It has to do with my right hand rules I think. J = J(r) same issue raised earlier and resolved in lines doc. ***************** OK, I have resolved this issue and the resolution is explained in a simple comment in transmission lines doc. The phrase" B points in the z direction" is meaningful even if B is complex. See that comment. The FT has survived, the world is back in joint. yes! *********************** ok to here Let's try again, here is an imagined new section to stuff somewhere. One quickly runs out of symbols to represent different quantities, I will give it a try though. Appendix I: Dealing with Fourier Transforms of electric and magnetic fields. 1. Electric Field Let the real, physical electric field be e(x,t). We can write e(x,ω) = !Syntax Error, Idt e-jωt e(x,t) e(x,ω) = volt-sec/m (I.1.1) ok e(x,t) = (1/2π)!Syntax Error, Idt ejωt e(x,ω) e(x,t) = volts/m (I.1.2) ok In this transform, e(x,ω) is complex as a general rule (ie, for some general real e(x,t)). As a special case, one can consider a special e(x,t) that is sinusoidal in time (monochromatic) with a phase that may depend on spatial location (as I think in the round wire situation) e(x,t) = cos[ωet + φe(x)] e(x) e(x) = volts/m (I.1.3) ok where all functions are real. Now suppose we define complex E(x,t) as, E(x,t) ≡ ej(ωt+φ(x)) e(x) . E(x,t) = volts/m (I.1.4) ok If one is given E(x,t), one can always recover the physical field using e(x,t) = Re{ E(x,t)}. (I.1.5) ok Now we can consider this Fourier Transform, E(x,ω) = !Syntax Error, Idt e-jωt E(x,t) E(x,ω) = volt-sec/m (I.1.6) ok E(x,t) = (1/2π)!Syntax Error, Idt ejωt E(x,ω) E(x,t) = volts/m (I.1.7) ok where now both E(x,t) and E(x,ω) are (in general) complex. Were we to somehow compute E(x,ω), we could use (I.1.7) to get E(x,t) and then (I.1.5) to get the physical e(x,t). In the special monochromatic case (I.1.4), E(x,t) ≡ ej(ωt+φ(x)) e(x) . E(x,t) = volts/m (I.1.4) ok we find that E(x,ω) = !Syntax Error, Idt e-jωt E(x,t) = ejφ(x) e(x) 2πδ(ω-ω1) (I.1.9) ok volt-sec/m volt/m sec complex complex real real The above line now appears in lines doc as (1.7.10) but with ω1 dependence of φ and e shown. We shall now repeat this section for the magnetic field, putting in appropriate units. The monochromatic frequency will be taken ad ωb and phase φb. 2. Magnetic Field Let the real, physical electric field be b(x,t). We can write b(x,ω) = !Syntax Error, Idt e-jωt b(x,t) b(x,ω) = tesla-sec (I.2.1) ok b(x,t) = (1/2π)!Syntax Error, Idt ejωt b(x,ω) b(x,t) = tesla (I.2.2) ok In this transform, b(x,ω) is complex as a general rule (ie, for some general real b(x,t)). As a special case, one can consider a special b(x,t) that is sinusoidal in time (monochromatic) with a phase that may depend on spatial location (as I think in the round wire situation) b(x,t) = cos(ωbt + φb(x)) b(x) b(x) = tesla (I.2.3) ok where here all three functions are real. Now suppose we define complex B(x,t) as, B(x,t) ≡ ej(ωt+φ(x)) b(x) . B(x,t) = tesla (I.2.4) ok If one is given B(x,t), one can always recover the physical field using b(x,t) = Re{ B(x,t) }. (I.2.5) ok Now we can consider this Fourier Transform, B(x,ω) = !Syntax Error, Idt e-jωt B(x,t) B(x,ω) = tesla-sec (I.2.6) ok B(x,t) = (1/2π)!Syntax Error, Idt ejωt B(x,ω) B(x,t) = tesla (I.2.7) ok where now both B(x,t) and B(x,ω) are (in general) complex. Were we to somehow compute B(x,ω), we could use (I.2.7) to get B(x,t) and then (I.2.5) to get the physical b(x,t). In the special monochromatic case (I.2.4), B(x,t) ≡ ej(ωt+φ(x)) b(x) . B(x,t) = tesla (I.2.4 ok we find that B(x,ω) = !Syntax Error, Idt e-jωt B(x,t) = ejφ(x) b(x) 2πδ(ω-ω1) (I.2.9) ok tesla-sec tesla sec complex complex real real Comment: In the above two sections I have just given names to things so we can tell what is what, and know what is real and what is not real. I am at this point unsure whether you can insist that both these complex field relations be true at the same time when σ is present [ you can! ] B(x,t) ≡ ej(ωt+φ(x)) b(x) b real E(x,t) ≡ ej(ωt+φ(x)) e(x) e real (I.2.10) But we could assume both these forms are valid and see if we can find a solution. If a solution exists, then these claims are both justified at the same time. Notice that the above forms allow Bθ/Ez to have a phase which varies with space x (with radius r), something we observe in the round wire solution as now presented in lines doc. 3. Implications of the electric field Maxwell curl equation Now consider the physical Maxwell curl equation: curl e(x,t) = - ∂tb(x,t) (I.3.1) ok If we apply !Syntax Error, Idt e-jωt to both sides, we obtain [ might want to show his this works with parts ] curl e(x,ω) = -jω b(x,ω) (I.3.2) ok Now we wonder if there is meaning to the curl equation (I.3.1) for our "complex fields": curl E(x,t) = - ∂tB(x,t) ?? (I.3.3) If we decompose E and B into real and imaginary part we have curl { Re[E(x,t)] + j Re[E(x,t)] } = - ∂t {Re[E(x,t)] + j Re[E(x,t)] } Since multiplication by j commutes with curl and ∂t, we can equate real and imaginary parts to get curl { Re[E(x,t)] } = - ∂t {Re[B(x,t)] } curl { Im[E(x,t)] } = - ∂t {Im[B(x,t)] } (I.3.4) These are both "physical" curl equations based on our choice of the physical fields. Perhaps write these as curl e(x,t) = - ∂t b(x,t) e(x,t) = Re[E(x,t)] b(x,t) = Re[B(x,t)] curl e'(x,t) = - ∂t b'(x,t) e'(x,t) = Im[E(x,t)] b'(x,t) = Im[B(x,t)] (I.3.5) Thus our "complex field curl equation" (I.3.3) can be regarded as a combination of "physical field curl equations" for two different sets of physical fields [ yes ]. Thus, (I.33) is completely justified in terms of the complex fields shown, and the same is then true for the other Maxwell equations and in fact any other equations involving E and B fields such as J = σE. [ yes ] So we know exactly how to get from the complex fields E and B to either set of physical fields. Now what happens when we do a Fourier transform to the complex fields? Consider !Syntax Error, Idt e-jωt E(x,t) = !Syntax Error, Idt e-jωt [e(x,t) + j e'(x,t)] which tells us that E(x,ω) = e(x,ω) + j e'(x,ω) B(x,ω) = b(x,ω) + j b'(x,ω) These equations have the same form as E(x,t) = e(x,t) + j e'(x,t) B(x,t) = b(x,t) + j b'(x,t) but there is a big difference. In the second set of equations, we can identify the pieces with the real and imaginary part of the F(x,t) fields, whereas in the first set of equations since all functions are in general complex, we cannot say for example that e(x,ω) is the real part of E(x,ω). Just a small warning to be careful. [ correct ] Now we can compare different versions of the FT of a Maxwell equation. For example, apply !Syntax Error, Idt e-jωt to both sides of (I.3.3) to get curl E(x,ω) = - jωB(x,ω) (I.3.6) which we can write as curl { e(x,ω) + j e'(x,ω)} = - jω{ b(x,ω) + j b'(x,ω)} or curl e(x,ω) + j curl e'(x,ω) = -jω b(x,ω) + ω b'(x,ω) We cannot at this point claim that curl e(x,ω) = -jω b(x,ω) for example! So just be careful! I think there will never be any need to use the functions like e(x,ω). We will use only E(x,ω). [ correct ] Now let's assume the monochromatic situation for both electric and magnetic fields, E(x,t) ≡ ej(ωt+φ) e(x) . E(x,t) = volts/m (I.1.4) B(x,t) ≡ ej(ωt+φ) b(x) . B(x,t) = tesla (I.2.4) Inserting these into (I.3.3) gives ej(ωt+φ) curl e(x) = - ∂t[ej(ωt+φ) b(x) ] = -jωb ej(ωt+φ) b(x) [ wrong! In general φe = φe(x) so does not pass through the curl ! ] = ωb ej(ωt+φ-π/2) b(x) or curl e(x) = ωb ej([ω-ω]t+[φ-φ]-π/2) b(x) // wrong Since curl e(x), ωb and b(x) are all real, the phasor must be unity. For this to be true for all time, we clearly need to have ωb = ωe and φb-φe = π/2. Without loss of generality, we shall set φe = 0 and let the frequencies both be ω1 and φb = π/2. We then have this monochromatic set of complex fields E(x,t) ≡ ejωt e(x) E(x,t) = volts/m B(x,t) ≡ ej(ωt+π/2) b(x) . B(x,t) = tesla (I.3.5) and the curl equation may be written curl e(x) = ω1 b(x) (I.3.6) where e(x) and b(x) are real fields. The physical fields are these e(x,t) = cos(ω1t) e(x) e(x) = volts/m (I.1.3) b(x,t) = cos(ω1t + π/2) b(x) b(x) = tesla (I.2.3) The magnetic field leads the electric field by phase of π/2. [ wrong ] Fact: In any monochromatic electromagnetic field situation such as free space waves, reflection, boundary conditions, transmission lines, etc. the time-domain electric and magnetic fields at any point always have this rigid phase relationship due to the electric curl equation. [ wrong!] (I.3.7) What about ω domain stuff? Question: How does all this affect my Section 2.1 and 2.2? Here are Maxwell's equations in terms of the complex fields. curl b(x,t) = μ j(x,t) // these are for the actual physical fields curl e(x,t) = - ∂b(x,t)/∂t j(x,t) = σe(x,t) curl B(x,t) = μ J(x,t) // these are for the complex fields curl E(x,t) = - ∂B(x,t)/∂t J(x,t) = σE(x,t) where we have [ these are not general since possible phases φi(x,ω1) are not included. ] but ok, suppose you DID have these limited forms, what conclusions would you reach? E(x,t) ≡ ejωt e(x) e(x,t) = Re{E(x,t)} = cos(ω1t) e(x) J(x,t) ≡ ejωt j(x) J(x,t) = Re{J(x,t)} = cos(ω1t) j(x) B(x,t) ≡ ej(ωt+π/2) b(x) B(x,t) = Re{B(x,t)} = cos(ω1t +π/2) j(x) Also we know that E(x,ω) = e(x) 2πδ(ω-ω1) (I.1.9) B(x,ω) = ejπ/2 b(x) 2πδ(ω-ω1) = j b(x) 2πδ(ω-ω1) (I.2.9) Now try to translate this result from the round wire analysis, = jωB(r) . (2.1.9) ∂r E(x,ω) = jω B(x,ω) where E(x,ω) = E(r) and B(x,ω) = B(r) Now put in the results above to get ∂r [e(x) 2πδ(ω-ω1)] = jω [j b(x) 2πδ(ω-ω1) ] or ∂r e(x) = -ω b(x) or ∂r e(r) = -ω b(r) Now everything is real and maybe I can really say that ∂re(r) < 0 which seems the wrong conclusion! Given our limited assumed forms, yes, ∂re(r) < 0 if b(r) > 0, but those limited forms simply are not compatible with the round wire problem! This is leading to many new questions and problems. Consider my solution E(r) = f(x) = C J0(βr) β = ej3π/4 (2.1.18) I now think this has a constant phase! [ wrong ] Look at these results: E(r) = E(a) [J0(βr) / J0(βa)] (2.1.22) B(r) = - (β/jω) E(a) [J1(βr) / J0(βa)] B(r)/E(r) = - (β/jω) [J1(βr) / J0(βa)] / [J0(βr) / J0(βa)] = - (β/jω) [J1(βr)/ J0(βr) ] I CLAIM, but want to verify it, that the ratio [J1(βr)/ J0(βr) ] must have a constant phase as a function of ω. [ wrong ] It just seems unlikely [yup!], but Maple can tell me. Let's write β = ej3π/4 b where b is real. What does Maple have to say here? Aside: Let β = ej3π/4 b Jν(βr) = Jν(ej3π/4 br) The Iν(z) = (-i)ν Jν(iz) so this is NOT the right phase! So we don't just have I or K functions. Well, here is what Maple says about my so-called constant phase: So now I have a serious contradiction. E and B in ω space must have a constant phase shift of 90 degrees [ wrong! ] E/B in the above specific case shows this is not true. [ right! ] Is there something wrong with my set of assumptions for the round wire? Consider for example the divergence equation inside the conductor (recall no charge there). div E = 0 ok Recall E = E(x,ω) = E(r). Looking at the cyl formula for div we really do get div E = 0 with our assumptions because only Ez exists, but we have no variation in z. correct What about the other divergence equation? div B = 0 This is also perfectly respected in our assumption set. So I think our wire solution inside the wire solves all four Maxwell's, so why do I get this discrepancy? At ω = 0 for the wire, it seems the phase between E and B would be 0? But pretty vague. Go back to these monochrome equations E(x,ω) = e(x) 2πδ(ω-ω1) (I.1.9) B(x,ω) = ejπ/2 b(x) 2πδ(ω-ω1) = j b(x) 2πδ(ω-ω1) (I.2.9) I can translate these to the round wire to read E(r) = e(r) 2πδ(ω-ω1) E(r) = e(r) 2πδ(ω-ω1) B(r) = ejπ/2 b(r) 2πδ(ω-ω1) B(r) = ejπ/2b(r) 2πδ(ω-ω1) and therefore B(r)/E(r) = ejπ/2 argument(B(r)/E(r)) = π/2 and this is what Maple is showing but not for small ω. [ I am persisting in this π/2 error ] Maple says that for small ω, the phase goes to 3π/4 instead of π/2. I can surely show this as follows: B(r)/E(r) = (β/jω) [J1(βr)/ J0(βr) ] β = ej3π/4 = ej3π/4 b J0(βr) ≈ 1 J1(βr) ≈ βr/2 B(r)/E(r) ≈ (β/jω) [βr/2] = β2r/(2jω) = [-(2j/δ2)]r/ (2jω) = - (r/[ωδ2]) But here I get a phase if -π, which disagrees with everything else! OK, after writing this I realized the business of phase φ(x) being spatially dependent in the round wire problem and that removed all the above paradoxes. Today 9/30/13 I read through this entire doc and I think all mysteries are resolved, and Section 1.7 I hope clarifies everything.