On the force between a dielectric cylinder in a constant electric field and a conducting half space

Abstract

Abstract.The force between an infinitely long dielectric cylinder in a constant electric field and a conducting half space is determined using the separation of variables technique on the Laplace equation in bipolar coordinates.The force is obtained as a series containing the relative distance between the cylinder and the half space as a parameter. This series is not uniformly convergent for the cylinder approaching the half space and the corresponding force cannot be obtained by performing the limit term per term. A special asymptotic analysis is presented leading to an analytic expression for this limiting value of the force.

A note on a problem in asymptotics

“…Alternative expansions are also developed for Fc in terms of c, the distance of the cylinder axis from the face of the conducting half-space. The note concludes with a further comment on the work in [1] …”

“…In [1] Dhondt and Kohl have solved the two-dimensional problem of a dielectric cylinder placed in front of a conducting half-space, there being a constant electric field applied perpendicular to the half-space. Using a bipolar coordinate system, the potential field is determined, and the force on the cylinder is shown to be proportional (1 -k2){Sx-S2) (2) in an obvious notation.…”

“…Using a bipolar coordinate system, the potential field is determined, and the force on the cylinder is shown to be proportional (1 -k2){Sx-S2) (2) in an obvious notation. In (1), the radius of the cylinder is taken to be unity and c = cosh uq is the distance of the cylinder axis from the conducting half-space; further, k = e/eo is the ratio of the cylinder dielectric permittivity to that of free space. The value Fc for a conducting cylinder is obtained from (1) in the limit n -> oo as t?…”

“…f [ sinh nu0 smh(n + l)u0 sinh^ nu0 J The authors of [1] then derive the leading terms in the asymptotic developments of (1) and (3) as uq -> 0, i.e., the cylinder approaches the plane face of the conducting half-space. The mathematical arguments used are somewhat long and intricate, and a 786 R. SHAIL crucial step (the replacement of sinh tluq sinh(n + l)uo in (3) by sinh2 uuq) is justified at length in [2].…”

“…Although (1) and (3) can be expanded asymptotically as they stand, it turns out that a preliminary transformation of the infinite series produces much simpler and preferable representations of Fd and Fc. This transformation is described in Sec.…”