Question

Find the equivalent resistance between two opposite corners of a cube made of wires, each having resistance \(R\).

• A. \( \dfrac{R}{6} \)
• B. \( \dfrac{5R}{6} \)
• C. \( \dfrac{R}{3} \)
• D. \( \dfrac{2R}{3} \)

Hint:

In symmetric resistor networks like cubes or tetrahedrons, nodes at equal potential can be grouped together. This reduces the circuit into simple series–parallel combinations.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We have a cube where each of the 12 edges is a resistor of resistance R. We need to find the equivalent resistance across a body diagonal (between two opposite corners).
Step 2: Key Formula or Approach:
We can solve this problem using symmetry. Let's name the corners. Let the current enter at corner A and leave at the diagonally opposite corner G.
1. Identify points of equal potential.
2. Re-draw the circuit as a simpler combination of series and parallel resistors.
Step 3: Detailed Explanation:
Let's apply a voltage V between A and G.

Symmetry at the input: From corner A, there are three identical paths to its nearest neighbors (let's call them B, D, E). Due to symmetry, the current must split equally into these three paths. Therefore, points B, D, and E are at the same potential.
Symmetry at the output: Current leaves from corner G. It arrives from its three nearest neighbors (C, F, H). Due to symmetry, the potential at these three points must be the same.
So, we can group the vertices by their potential:
Level 1: Point A (Highest potential)
Level 2: Points (B, D, E)
Level 3: Points (C, F, H)
Level 4: Point G (Lowest potential)
Now let's count the resistors between these levels:
Between Level 1 and 2: Three resistors (A-B, A-D, A-E) are in parallel. Their equivalent resistance is \( R_1 = R/3 \).
Between Level 2 and 3: Six resistors (B-C, B-F, D-C, D-H, E-F, E-H) connect the two groups of equipotential points. These six resistors are in parallel. Their equivalent resistance is \( R_2 = R/6 \).
Between Level 3 and 4: Three resistors (C-G, F-G, H-G) are in parallel. Their equivalent resistance is \( R_3 = R/3 \).
The entire circuit is now a series combination of these three equivalent resistances. \[ R_{eq} = R_1 + R_2 + R_3 = \frac{R}{3} + \frac{R}{6} + \frac{R}{3} \] \[ R_{eq} = \frac{2R}{6} + \frac{R}{6} + \frac{2R}{6} = \frac{5R}{6} \] Step 4: Final Answer:
The equivalent resistance between the opposite corners of the cube is \( \frac{5R}{6} \).