Twelve resistors, each of 1 Ω, are connected to form the sides of a cube. Find the effective resistance across any one side.

Steps to Solve

1. Identify the configuration of resistors The resistors are arranged to form the edges of a cube. Each edge of the cube has a resistor of 1 Ω, and there are 12 edges in total.

2. Understand the paths for effective resistance To find the effective resistance across one side of the cube (let's say between points A and B), we need to see how the resistors are connected. When we connect points A and B, there are multiple paths through which the current can flow.

3. Divide the resistances in parallel For the effective resistance across the side AB, there are three specific paths corresponding to the resistors on the cube:

   • Path 1: A → C → D → B (3 resistors in series)
   • Path 2: A → D → E → B (3 resistors in series)
   • Path 3: A → C → E → B (3 resistors in series)

Since all paths are identical, we find the resistance of one path only.

4. Calculate resistance along one path Each path has 3 resistors in series, hence the resistance for one path will be: $$ R_{\text{path}} = R_1 + R_2 + R_3 = 1Ω + 1Ω + 1Ω = 3Ω $$

5. Calculate total resistance of these paths in parallel Three identical paths (each with resistance of 3 Ω) are in parallel. The formula for total resistance $R_{total}$ in parallel is given by: $$ \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} $$

Substituting the values: $$ \frac{1}{R_{total}} = \frac{1}{3Ω} + \frac{1}{3Ω} + \frac{1}{3Ω} = \frac{3}{3Ω} = 1Ω^{-1} $$

6. Calculate effective resistance Now, taking the reciprocal gives the total effective resistance: $$ R_{total} = 1Ω $$