一种用保角变换求解带电Kelvin电容器边缘效应所产生静电力的解析模型∗

摘要

We present a new realization of weak electrostatic force based on Kelvin capacitor, which, compared with the conventional realization, simplifies the mechanical design without synchronously moving the central and guard-ring electrodes. However, a residual force due to the edge effect of the capacitor should be accurately known. In this paper, an analytic model of the electrostatic force generated from the edge effect is presented based on conformal transformations. The accuracy of the model is verified by the finite element calculation based on a micro-Newton weak force generation.%提出了一种用于弱力测量的带电Kelvin电容器产生静电力的新型工作方式。该工作方式的优点是在机械上可明显简化静电力的实现，避免了带电Kelvin电容器在传统工作方式下需要同步移动中心电极和保护电极的困难；缺点是因Kelvin 电容器的边缘效应，会不可避免地随之产生一个小的静电力分量，需要准确建立它的数学模型，方可清晰地评估它对测量结果的影响。为解决此问题，本文采用部分电容模拟和保角变换，将二维电场的计算转换为一维电场计算问题，并基于此，得到了一种合理表征Kelvin电容器边缘效应所产生静电力分量的解析模型。通过与一个具体计算实例的有限元计算求解结果相比较，分析了所建立的解析模型的准确性。