perturbation attempt REVIEWED

Perturbation Attempt PhL 5.2.14 This was an early precursor to Section P.1's perturbation theory approach. Here I have a formal dimensionless α, but there I just fudge the smallness parameter and say it is ω because I don't want to have to worry about some ω1 though I could no doubt do it using skin effect δ to set the ω scale. We now imagine a scenario where some "external" B field Bext is created by some external current source Jext. A system of interest (call it DUT for Device Under Test), is immersed in this external field Bext and this time-changing field "induces" currents Jeddy in the DUT. With this partitioning, we write the previous two equations as curl (Bext + Beddy) = μ(Jext + Jeddy) (P.1.3) curl (Jext + Jeddy) = - jωσ(Bext + Beddy) . (P.1.4) We assume next that Jeddy in the DUT exists in a region of space which is disjoint from the region where Jext flows in the external generating apparatus. In the DUT region, Jext = 0, and in the generating region we have Jeddy = 0, following our definitions of Jext and Jeddy. Any eddy currents which the generating system induces into itself we assume are included in Jext. Then we can write (P.1.3) and (P.1.4) as curl (Bext + Beddy) = μJeddy // within the DUT (P.1.3)' curl Jeddy = - jωσ(Bext + Beddy) . // within the DUT (P.1.4)' In principle, Beddy could alter Jext in the external apparatus, but we shall assume that Jext is regulated by some kind of current source and so Jext does not change when the DUT is put in place. Correspondingly, we assume that Bext does not change either. Assuming also that the DUT and the region of space into which it is inserted both have permeability μ, we have the following equation applying both before and after the DUT is put in place: curl Bext = μJext = 0 // within the DUT Thus we once more rewrite our pair of equations using this fact, curl Beddy = μJeddy // within the DUT (P.1.3)" curl Jeddy = - jωσ(Bext + Beddy) . // within the DUT (P.1.4)" We now define ω0 to be some frequency, and α ≡ ω/ω0 is a dimensionless parameter which is small at low frequency. The above equations are then curl Beddy = μJeddy // within the DUT curl Jeddy = - jω0σ α (Bext + Beddy). // within the DUT We next expand Beddy and Jeddy in this smallness parameter α, Beddy = α Beddy(1) + α2 Beddy(2) + ..... = Σn=1∞ αn Beddy(n) Jeddy = α Jeddy(1) + α2 Jeddy(2) + ..... = Σn=1∞ αn Jeddy(n) Inserting these expansions into (P.1.3)" and (P.1.4)" then gives curl (α Beddy(1) + α2 Beddy(2) + .....) = μ(α Jeddy(1) + α2 Jeddy(2) + .....) curl (α Jeddy(1) + α2 Jeddy(2) + .....) = -jω0σ α (Bext + α Beddy(1) + α2 Beddy(2) + .....) If we now equate the coefficients of α in the two equations, and alternate back and forth between the equations, we obtain this sequence: curl Jeddy(1) = -jω0σ Bext curl Beddy(1) = μ Jeddy(1) curl Jeddy(2) = -jω0σ Beddy(1) curl Beddy(2) = μ Jeddy(2) etc. curl (Beddy(0) + α Beddy(1) + α2 Beddy(2) + ....) = μ(Jeddy(0) + α Jeddy(1) + α2 Jeddy(2) + .....) curl (Jeddy(0) + α Jeddy(1) + α2 Jeddy(2) + .....) = - jωσ(Bext + Beddy(0) + α Beddy(1) + α2 Beddy(2) + .....) We can then regard each of these equations as a set of equations in each order of α : 0 = 0 order α0 curl Beddy(1) = μ Jeddy(1) order α1 curl Beddy(2) = μ Jeddy(2) order α2 .... 0 = - jωσBext order α0 curl Jeddy(1) = - jωσ Beddy(1) order α1 curl Jeddy(2) = - jωσ Beddy(2) order α2 ....