机械设计的基本原理7.5

Electricity & Magnetism: FUNdaMENTAL Principles

电磁学：基本原理

Electric and magnetic fields are analyzed by defining a boundary and then applying one of the fundamental principles (e.g., Faraday’s, Ampere’s, or Gauss’s laws) to that boundary. Given an electric or magnetic field that crosses a closed boundary (or surface), Faraday’s, Ampere’s, & Gauss’s laws essentially say that the sum of all of the products of the infinitesimal components of a field with all of the infinitesimal lengths (or areas) of a closed boundary are equal to some scaler value. Independent of the complexity of the field or closed boundary, they are expressed in a most general form as surface integrals of the dot products of field and boundary vectors. 

电磁场的分析是通过定义边界条件，然后使用基本定理（例如，法拉第、安培或者高斯定理）得到的。对于穿过封闭边界（或者曲面）的电磁场，法拉第、安培和高斯定理说明：每个场的微分单元与该场的长度（或者面积）微分单元的乘积的和都会等于一个常量。跟场或者封闭边界的复杂形式无关，其公式表达为最简单的场和边界向量的点积的面积分。

Real devices can often be modelled with a two-dimensional boundary and the multi-variable calculus problem becomes a simple summation of scaler quantities. An example is the application of Ampere’s Law to the force produced by an electromagnet, where a complete magnetic circuit can be analyzed as the sum of the products of the magnetic field intensities H aligned with the path and perpendicular to cross sectional areas A. 

实际应用情况中的建模通常可以采用二维边界条件，使用多重积分把复杂的问题变为常量数值的和。例如用安培定理来计算电磁铁产生的力，其中整个磁路可以被看作是磁场强度H沿着积分路径，垂直于截面A的计算公式。

When applying the fundamental laws to magnetic (or electric) circuits, the magnetic flux (or current) through each components in series is equal (they form a voltage divider), and the magnetomotive force (or voltage) through components in parallel is equal (they form a current divider). Gustav Robert Kirchoff (1824-1887) wrote these laws of closed electric circuits in 1845, which are now known as Kirchoff's Current and Voltage Laws: The sum of voltage drops around a circuit will be equal to the voltage drop for the entire circuit. In fact, the laws for magnetic and electric circuits (and also for fluid circuits!) are similar as shown in Table 1, and by Ohm’s law: 

在磁（电）路上应用基本原理时，磁通量（电流）通过每个零件的值是相同的（形成电压分配），而且平行的回路中电动势（电压）是相等（形成电流分配）。基尔霍夫（1824-1887）在1845年写出了封闭电路的定律，也就是为人熟知的基尔霍夫的电流和电压定律：沿着闭合回路的所有电动势的代数和等于所有电压降的代数和。事实上，电路和磁路(流体回路也是一样)的定理是一样等价的，如表格1所示，由欧姆定律得出：

Consider the force of attraction between an electromagnet and an object¹. For the figure shown, it can be assumed that the reluctance in the magnetic circuit is dominated by the air (permeability μo) in the region with gap δ and area A. Applying Ampere’s law yields: 

考虑物体和电磁铁之间的吸力。如图所示，可以假设磁路中的磁阻主要是空气（磁导率μo)）引起的，其间隙大小为δ、面积为A。根据安排定律可以得出：

The magnetic flux Φ also passes through the center of each turn of the coil. Faraday’s law says that this will induce a voltage in each turn of the coil. Since each turn is linked in series, the total magnetic flux linked together by the coils is λ = NΦ, where λ is called the flux linkage. The definition of inductance L is λ = Li, and since inductance in an electrical system is like mass in a mechanical system, and current is like velocity, the energy (work) U and hence the attraction force are: 

磁通量会通过每圈线圈的中心。法拉第定律表明，每圈线圈会产生电压。由于每圈线圈都是串联的，所以总的磁通量和为λ = NΦ，其中λ叫做磁通匝数。磁感应的定义为λ = Li，由于电路中的磁感应相当于机械系统中的质量，电流可以看作是速度，能量为U的话，可以得出吸力为：

Now would be a good time to review your electricity and magnetism notes and text from your freshman physics course! 

现在是时候看下你一年级物理课程关于电学和磁学的笔记了。

1. Many thanks to Prof. Jeff Lang for providing this clear explanation of a simplified system. For an in-depth discussion of this and other related topics, see Electromechanical Dynamics, HH Woodson and JR Melcher, John Wiley & Sons, 1968, Volumes I & II