Resistor networks, used to model new types of natural or artificial matter, also provide generic examples for practising the methods of physics for obtaining estimates, revealing the main properties of a system and deriving exact expressions. Symmetric bracelet resistor networks are constructed by connecting n identical resistors in a circle, and then connecting two such circles by another set of n identical resistors. First, using van Steenwijk’s method, we establish that the equivalent resistance or two-point resistance (TPR) between any two nodes is derived when the layer-to-layer resistance R _0n is known. We then determine R _0n by an elementary recurrence relation which converges rapidly to its large n limit. Using this reference value of R _0n , accurate estimates of other TPRs follow for all values of n, characterised by a leading 1/n variation. In addition, exact explicit expressions of the TPRs can be calculated for any value of n. These networks, prototypes of three-dimensional networks considered in research, can be used to illustrate the diversity of the physical approach, the power of elementary methods, and to learn to be comfortable with approximations. Easy to make and use for experimental tests, they can support hands-on activities and conceptual changes.

中文翻译：

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对称手环电阻网络中的两点电阻：准确估计和精确表达式

电阻网络用于模拟新型自然或人造物质，还提供了实践物理方法的通用示例，以获取估计、揭示系统的主要属性并推导精确的表达式。对称手环电阻网络是通过连接构成的n在一个圆圈中放置相同的电阻，然后用另一组电阻连接两个这样的圆圈n相同的电阻器。首先，使用 van Steenwijk 的方法，我们确定当层间电阻为任意两个节点之间的等效电阻或两点电阻 (TPR)右 _0n 是已知的。然后我们确定右 _0n 通过基本递推关系，该关系快速收敛到其大值n限制。使用这个参考值右 _0n ，对于所有值，其他 TPR 的准确估计如下n，其特征在于领先 1/n变化。此外，可以针对任意值计算 TPR 的精确显式表达式n。这些网络是研究中考虑的三维网络的原型，可用于说明物理方法的多样性、基本方法的力量，并学会适应近似。它们易于制作和用于实验测试，可以支持实践活动和概念改变。