What's the Equivalent Resistance b/w A & B---Textual Information:
Above the diagram: "All resistors are 1 Ω"
Chart/Diagram Description:
Type: Electrical circuit diagram.
Main Elements:
- The circuit represents a 3D structure resembling a cube or a cubical lattice.
- There are 8 nodes represented by black dots, positioned at the vertices of the implied cube.
- There are 12 resistors, each represented by a blue zigzag line, positioned along the edges of the implied cube, connecting adjacent nodes.
- All resistors have a specified resistance value of 1 Ω.
- Two terminals, labeled A and B, are shown as lines extending from two specific nodes.
- Terminal A is connected to the top-left-front node of the cube structure.
- Terminal B is connected to the bottom-right-back node of the cube structure, which is diagonally opposite to the node connected to terminal A.
- Black lines represent the connections between the nodes and resistors.
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答案文本
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We have a cube resistor network problem. The circuit consists of eight vertices connected by twelve resistors, each with resistance one ohm. We need to find the equivalent resistance between terminals A and B, which are located at diagonally opposite corners of the cube.
To solve this problem, we use symmetry analysis. Due to the cube's symmetry, we can group vertices by their distance from terminal A. Group one contains only vertex A. Group two has three vertices adjacent to A. Group three has three vertices adjacent to B. Group four contains only vertex B. This symmetry allows us to simplify the analysis significantly.
Using symmetry, we can simplify the cube network into an equivalent linear circuit. Vertices V1, V2, and V3 are at the same potential due to symmetry, as are vertices V4, V5, and V6. This gives us three parallel one-ohm resistors from A to the V123 node, six one-ohm resistors in parallel between V123 and V456 nodes, and three parallel one-ohm resistors from V456 to B.
Now we calculate the equivalent resistance. The first section has three one-ohm resistors in parallel, giving one-third ohm. The middle section has six one-ohm resistors in parallel, giving one-sixth ohm. The last section again has three one-ohm resistors in parallel, giving one-third ohm. Adding these in series: one-third plus one-sixth plus one-third equals five-sixths ohm.
To summarize our solution: We analyzed a cube resistor network with twelve one-ohm resistors using symmetry principles. By grouping vertices at equivalent potentials, we simplified the complex three-dimensional network into a linear circuit with three sections in series. The final equivalent resistance between diagonally opposite vertices A and B is five-sixths ohm, or approximately zero point eight three three ohms. This symmetry-based approach is a powerful technique for solving complex resistor networks.