One embodiment of a plasma fusion reactor, in vogue a number of years ago at the Lawrence Livermore National Laboratory, employed a cylindrical chamber having its ends capped by massive, Yin-Yang (Y-Y) magnetic coils serving as barriers against charged particle escape.' Such Y-Y coils, by their very geometry, require opposed current flow in close proximity, circumstance which summons forth a dilatational magnetic pressure raising device disintegration to the level of a calamitous possibility. And, while such Y-Y fragmentation is surely not a welcome design outcome, nevertheless it does invite a preliminary analysis as to its potential violence, analysis which enjoys besides a modicum of theoretical interest by virtue of making relevant a scenario of electrodynamics in an expanding cavity.With these dual aims in mind, we had many years ago undertaken the study of the very simplest of such expanding cavity situations, namely, the growing interstitial (vacuum) wafer separating two massive metallic plates undergoing a symmetric flight from one another. Quick penetration into the heart of this problem was provided by the observation that, on the one hand, a quasi-static (QS) field computation would surely suffice, while, on the other, that a moving boundary condition (MBC) could be fashioned in lowest relativistic order by combining laboratory-frame electric E and magnetic B fields, and the boundary velocity v, and thence requiring that the effective tangential electric field ~vX {E + vxB} vanish upon both plate boundaries. In this process, a secondary computation of E was bootstrapped upon a primary, QS one for B via Faraday's law, whereby the obligatory time derivative of the latter was implicitly tethered to the dynamic evolution of its underlying separation parameter r](t).2 Under this viewpoint there easily emerged the invariance against time of the product of B by r (or else of current / by r/) leading to a simple differential equation for the dynamical evolution of the net separation r](t), and, in particular, to the identification of a characteristic time scale t suggesting a most vigorous magnet disintegration. This aspect of the work has been previously reported in summary form,' and is set out anew here for the purpose of building an intuitive, heuristic base concerning field evolution within the primitive, expanding wafer cavity now at hand.A heuristic base of this sort is far too coarse to account for field retardation effects due to signal transit at finite light speed c. We remove this defect by returning to the Faraday/Ampere equations in their primitive form and subjecting them first to Fourier transformation in coordinate z along the direction of cavity expansion perpendicular to magnet walls. Such transformation embraces the entire interval —oo < z < oo and, as such, submits to a null-field attitude which regards the field, in both its electric E and magnetic B manifestations, as being zero exterior to the expanding wafer, i.e., V|z| > r/(t)/2. Due deference must of course be paid, in the form of Dirac delta function sources placed at z = ±r](t)/2, to the radiation emanating from surface current density ±/(i) flowing on cavity walls. Elimination of either field transform leads then to a simple harmonic differential equation in time t having a source gauged by I(t). Its solution is readily gotten in a form that allows inverse Fourier transformation to proceed smoothly and, in particular, to identify a retarded signal emission time t, < i as gauged from either plate which obeys the intuitively pleasing condition c(t — tt) = {rj(t) + r)(t,)} /2. All in all one confronts at this point a relatively simple pattern of connections between the field and its source / as reckoned at retarded times t* suitably structured so as to track upper/lower plate emissions, connections which succumb at length to an a posteriori enforcement of the non-relativistic limit r)(t) 展开▼
