Short- and long-range penetration of fields and potentials through meshes, grids or gauzes

doi:10.1016/S0168-9002(98)01564-2

Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment

Volume 427, Issues 1–2, May 1999, Pages 363-367

https://doi.org/10.1016/S0168-9002(98)01564-2 Get rights and content

Abstract

When a conducting partially-transparent mesh is used to separate two regions of different electrostatic field strength there are short-range changes to the field and potential distributions in the vicinity of the mesh and also long-range changes far from the mesh. The forms and magnitudes of these changes are investigated computationally for meshes that consist of parallel or crossed round wires or crossed flat strips.

Introduction

Partially transparent meshes (sometimes referred to as grids or gauzes) are often used to separate or screen two regions of static or quasi-static electric field of different strengths. Short-range non-uniform fields that have the periodicity of the mesh are then created in the vicinity of the mesh, and can affect the trajectories of charged particles that pass through the holes in the mesh. The resulting `micro-lensing’ effects are well-known and have been investigated in detail (see Williams et al. [1] and references therein).

What appears to be less well appreciated is that because of the penetration of the potential through the holes in the mesh the effective average potential at the plane of the mesh becomes different from the potential applied to the mesh itself. The change in the effective potential has the effect of changing the long-range potentials and fields on both sides of the mesh. This effect was recognized by Verster [2], but with the advent of powerful computational techniques it is now possible to provide accurate values of the changes in effective potential for a range of mesh transparencies, as reported in Ref. [3]. Here we extend the previous study by considering the form and magnitude of the short-range components of the penetrating field and potential.

For the computational simulations we have used the CPO-2D and CPO-3D programs [4], which are based on the Surface Charge Method [5] (also known as the Boundary Element or Charge Density Method). The present study has been made possible by the ability of these programs to accurately simulate electrostatic fields near wires that are much thinner than the spacings between them. This type of problem cannot usually be well simulated by programs based on the Finite Difference and Finite Element Methods.

The present results apply to static and quasi-static electric fields – i.e. fields for which the wavelength is much larger than the mesh spacing.

Section snippets

Meshes of parallel round wires

We start by considering a mesh composed of parallel round wires that lie along the y direction in the z=0 plane. The wires have a diameter d and a spacing s in the x direction between their axes. Flat plates are situated at distances l on both sides of the mesh, to provide the electric fields in which the mesh is immersed. Fig. 1 shows an xz section of a mesh that has the parameters d=0.025 mm, s=1 mm, and l=10 mm. The contours in this figure are those that result when the potential of the mesh is

Meshes of parallel round wires

To investigate the short-range changes to the field and potential distributions we express the general solution of Laplace's equation for |z|>d/2 in the form $φ(x,z)=a_{0} +bz+ ∑ m=1l>∞ a_{m} cos 2 π mx s exp − 2 π m z s$ where the origin of coordinates is mid-way between two wires and where $E_{1} =0, E_{2} =−1$ V/s. Here the first two terms depend on the equivalent potential of the mesh and on the potentials that are applied to neighbouring electrodes, while the summation contains the short-range terms of present interest. We

Acknowledgements

Two of the authors (PDB and RRAW) are grateful to the Department of Physics and Astronomy for the facilities to carry out a preliminary study of the potential distribution in the vicinity of a flat mesh of crossed wires as part of their design projects in the final year of their undergraduate M. Phys. course.

References (5)

D.J Williams et al.
Nucl. Instr. and Meth. A
(1995)
J.L Verster
Philips Res. Rep.
(1963)

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Short- and long-range penetration of fields and potentials through meshes, grids or gauzes

Abstract

Introduction

Section snippets

Meshes of parallel round wires

Meshes of parallel round wires

Acknowledgements

Nucl. Instr. and Meth. A

Philips Res. Rep.