We present a simple approach, different from approaches in the literature, for calculating the magnetic field of a finite solenoid with a circular cross section, carrying an ideal surface current with an ordinary azimuthal component and a small axial component due to nonideal winding. The results are given as an infinite series whose terms are elementary functions or simple polynomials rather than other infinite series. It is easy to deal with the results because the polynomials involved are partial binomial expansions. These series converge rapidly everywhere, except near the edges of the solenoid. From these results, it is very easy to derive some general properties of the magnetic field, to reduce the field to greatly simplified forms in various special regions, and to address the special cases of infinite and semi-infinite solenoids. The azimuthal field due to nonideal winding is shown to be small everywhere. A solenoid with a small but finite thickness in the radial direction is investigated in detail, and we present simple analytic approximate results for its magnetic field. A comparison with an exact numerical evaluation shows that these are very good approximations.

我们提供一种不同于文献中方法的简单方法，用于计算具有圆形横截面的有限螺线管的磁场，该有限螺线管承载理想表面电流，该表面电流具有普通的方位角分量，并且由于非理想绕组而具有较小的轴向分量。结果以无穷级数给出，其项是基本函数或简单多项式，而不是其他无穷级数。由于涉及的多项式是部分二项式展开式，因此处理结果很容易。除了螺线管的边缘以外，这些系列均在所有位置快速收敛。从这些结果中，很容易得出磁场的某些一般属性，将磁场简化为各种特殊区域中的简化形式，并解决无限和半无限螺线管的特殊情况。由于不理想的缠绕而导致的方位角磁场在任何地方都很小。详细研究了螺线管的径向厚度较小但有限的问题，我们对其磁场给出了简单的解析近似结果。与精确数值评估的比较表明，这是非常好的近似值。