On the current induced in a conducting ribbon by a current filament parallel to it

Abstract

It has been shown that, if a plane wave is incident normally on a flat conducting ribbon, both components of induced current density differ little from what would obtain if the width of the ribbon was infinite, save very near the edges. The density very near the edges was evaluated for ribbons whose total width was 2?/? or 4?/? it emerged that the distribution of density was very nearly the same in each case and did not differ significantly from the distribution at the edge of a very wide ribbon, which is known from another solution. In other words, the distribution very near an edge is sensibly independent of the width of the ribbon. An example of great practical interest is the current induced in a flat reflecting sheet by a current filament parallel to it, or?speaking generally?when the incident wave has a cylindrical, in contrast to a plane, front. The exact solution for this case is known in principle but is impracticable to evaluate numerically. However, the solution can be evaluated when the ribbon is very wide indeed and the cylindrical wave starts from a line in the vicinity of the single bounding edge. Experience indicates that it seems reasonable to suppose that the ?edge effect? for this limiting case will not differ significantly from the effect at both edges when the cylindrical source is situated on the normal to a ribbon of finite width. The distribution of induced current has been evaluated when the cylindrical source is (a) distant ?? from a half-plane and 0.592? from its bounding edge, (b) distant 0.96? from a half-plane and 0.555? from its bounding edge, (c) distant ?? from a half-plane and 0.99? from its bounding edge. In each case the disturbance of density is found to be concentrated in a width of about 0.1? from the bounding edge. Then follows a Section which shows that the net electric force at the foot of the perpendicular from a filament distant ?? from a plane is almost zero when the induced currents which are more distant than 0.99? from this point are ignored. And that the small residue of force is sensibly equilibrated by the force which arises from the effect of two bounding edges, as estimated from the edge effect which obtains in a corresponding half-plane. The Section is intended to show that the calculable edge effects appropriate to a half-plane provide a very close solution for a ribbon of finite width. Then it is shown that the diffraction pattern for a filament in the presence of an infinite conducting plane depends dominantly on the currents which are induced in the width, say, ?? from the foot of the perpendicular. Curves from which the diffraction pattern can readily be deduced are given for a filament distant from a plane and due to the currents induced in the central strips of various widths; they are intended to be of practical use as an aid to design. The modifications to these patterns which would result from the estimated ?edge effects? which must occur when the reflector has a finite width, in contrast to the pattern accruing from the current induced in a finite width of an infinite plane, are considered and found to be insignificant. Accordingly, it is suggested that the diffraction pattern for a filament and a reflector of finite width can be estimated very closely by crediting the reflector with the currents which would be induced in the same width of an infinite plane. It is held that this has established a principle of considerable use in practice. Although an aerial and a flat reflector is not a case of much practical concern, at least it is one which can now be worked out simply and with considerable accuracy; it appears that there is little to be gained from making the reflector wider than 3/2?. The principle developed in the paper must be capable of wide extension, and it should be possible to extend it to apply to cylindrical and to parabolic reflectors. It is hoped to do this in due course.