On the current induced in a conducting ribbon by the incidence of a plane electromagnetic wave

Morse and Rubenstein did not investigate the induced currents when they published a solution of the problem of the diffraction of waves by ribbons and slits. The authors of the present paper regretted this omission because the diffraction pattern of the scattered wave is directly dependent upon these induced currents and it is very easy to predict the pattern from the distribution of induced current, but not vice versa. Moreover, the distribution of induced current near a bounding edge is a matter of great interest in itself and is likely to serve as a guide in other problems which have not yet been solved. The first two Sections of the paper are devoted to extending the analysis developed by Morse and Rubenstein in terms of Mathieu functions, so as to make possible the calculation of the induced currents. Curves are given which show the distribution of these currents across ribbons whose width is 2λ/π and 4λ/π. The necessary analytical development was very difficult and cumbersome and the evaluation of the results was very troublesome and exacting: the senior author wishes to disclaim any share in this portion of the work, beyond bringing that part of the problem to the attention of the junior author.The senior author had previously obtained an expression for the currents induced in a half-plane and had derived the curve of this distribution. It differed little from the curves derived from the Mathieu function solution for a plane of finite width, thus showing that the disturbance at one edge is very little dependent on the existence of a second parallel edge, provided it is more distant than about ½λ It became clear that the disturbance due to an edge may be considered to be substantially contained within a width of, say, ⅛λ.If this conclusion is accepted as general there is no further necessity to use Mathieu functions in this problem: they are extremely involved and poorly tabulated, and it is advantageous to dismiss them after they have served a useful turn.The third Section of the paper is devoted to predicting the diffraction pattern from the currents induced in the ribbon and to discussing the effect which their distribution will have on the shape of the pattern. It is found that the patterns calculated by Morse and Rubenstein are not compatible with the current distributions derived from the extension of the Morse and Rubenstein analysis. The discrepancy must be due to the inclusion of insufficient terms of an infinite series in the process of numerical evaluation. In evaluating the curves of current distribution given in the paper, the junior author has included as many terms as the published tables of Mathieu functions will permit and is confident that they are adequate. But since there was no direct method of deciding whether the higher degree of accuracy of evaluation was to be credited to the Morse-Rubenstein patterns or to the current distributions obtained by the junior author, the two authors were led to consider, very closely and generally, the form of current distribution which must obtain to satisfy the essential condition of the problem. This condition is that the current distribution must be such as to produce an electric force which is constant in magnitude and phase all across the ribbon, thereby equilibrating the field of the wave incident normally on it.In Section 7 it is shown, by very simple methods, which are in no way dependent upon either the Mathieu solution or the half-plane solution, that the surface field can be constant over the major portion of the width of a wide ribbon only if the distribution consists of a uniform co-phased density together with a second and quadrature distribution which is virtually concentrated in a filament located at each edge, the strength of each filament being equal to the current in a strip of width 0.16λ of the uniform density, and thus independent of the width of the ribbon. It is shown, moreover, that if the uniform co-phased density in fact becomes not uniform very near the edges there must be regions, near the edges, over which this density is greater in the one region and less in the other region than the mean density over the width, in such a way that the mean density is unchanged by the lack of uniformity near the edges. But all these properties had already been inferred and derived from the distribution evaluated from the Mathieu functions and for the half-plane. These evaluated distri butions have thus been found to possess the characteristics which are now found to be inherently necessary. In fact this apparently very difficult problem turns out to be a simple one if only it is attacked by examining that distribution which must obtain to produce a uniform force across the ribbon at its surface, in contrast to deriving the currents which must be induced by a given incident field.Once this fact had been recognized, it became apparent that the problem of a very narrow ribbon could now be evaluated, though this is not practicable by the Mathieu process because the limiting values of these functions have not yet been tabulated for small arguments. Section 8 gives the solution for a narrow ribbon, valid within the range such that we may write Y0(z)≃2/π(log z+y).When the ribbon is wide it is obvious that the echoed field must be proportional to width. A curve is given in which the echoed field per unit width is plotted as a function of width, expressed as a multiple of a wavelength: it is substantially constant provided the width of the ribbon exceeds about 0.2λ It is of great interest to find that a result which is obvious for a wide ribbon is still substantially correct even when the width is only ⅕λ; this result is believed to be novel.It is hoped that the paper throws considerable light on the mechanism of the induction of current in a ribbon; it certainly shows the reason why the phase of this current must necessarily change across the width: an effect which was to be expected from general experience of comparable problems, where it had but emerged in the final result of the analysis without disclosing the physical necessity which causes it to exist.Section 10 discusses the case when the direction of propagation of the wave is parallel to the plane of the ribbon. It includes curves showing the distribution of induced currents.An Appendix (Section 12) contains details of the solution for the half-plane, and several formulae are worked out which are required for various computations used in the main part of the paper.