Abstract The axisymmetric steady-state electron flow emitted from an electrified semi-infinite perfectly-conducting filament in vacuum is theoretically analyzed using an Eulerian model. The high symmetry allows for a solution in terms of self-similar functions, with radial dependence described by power-laws, leading to an eigenvalue problem for the angular dependence, which is numerically solved. Our results show a unique eigenvalue compatible with the boundary conditions, leading to a r − 1.6308 dependence for the electron number density, r 0.3692 for the electric potential, r 0.1846 for the velocity field. Hence, for an infinitely thin filament, space charge limited emission is not compatible with our boundary conditions.

中文翻译：

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从无限尖锐的带电锥体发射的电子流的欧拉分析

摘要 利用欧拉模型从理论上分析了真空中带电的半无限完美导电灯丝发出的轴对称稳态电子流。高度对称性允许在自相似函数方面的解决方案，具有由幂律描述的径向相关性，导致角度相关性的特征值问题，该问题通过数值求解。我们的结果显示了与边界条件兼容的独特特征值，导致电子数密度的 ar - 1.6308 依赖性，电势的 r 0.3692，速度场的 r 0.1846。因此，对于无限细的灯丝，空间电荷有限的发射与我们的边界条件不兼容。