12 WIRES , EACH OF RESISTANCE r ARE JOINED TOGETHER TO FORM A CUBE. FIND THE EQUIVALENT RESISTANCE IN THE EDGE OF CUBE.

To find the equivalent resistance between two opposite corners of a cube formed by 12 identical resistors, each with resistance r, we can use symmetry and the principles of series and parallel circuits. This problem is a classic example in circuit analysis and can be approached step by step.

Understanding the Cube Configuration

Imagine a cube where each edge represents a resistor. Since there are 12 edges in a cube, we have 12 resistors, each with resistance r. The goal is to determine the equivalent resistance between two opposite corners of the cube.

Visualizing the Circuit

Label the corners of the cube as A, B, C, D, E, F, G, and H. Let's say we want to find the equivalent resistance between corners A and H. The resistors connected to these corners can be visualized as follows:

• From A, there are three resistors leading to B, C, and D.
• From H, there are also three resistors leading to E, F, and G.
• Each of these points (B, C, D, E, F, G) is connected to the opposite corner through additional resistors.

Applying Symmetry

Due to the symmetry of the cube, the current entering at point A will split equally among the three resistors connected to it. The same applies to the resistors connected to point H. Therefore, we can analyze the circuit by focusing on the paths from A to H.

Calculating Equivalent Resistance

To simplify the analysis, we can use the concept of combining resistors in series and parallel. The three resistors connected to A (let's call them R1, R2, and R3) are in parallel because they lead to the same junctions (B, C, D). The equivalent resistance of these three resistors can be calculated using the formula for resistors in parallel:

1/R_eq1 = 1/R1 + 1/R2 + 1/R3

Since all resistors have the same resistance r:

1/R_eq1 = 1/r + 1/r + 1/r = 3/r

This gives us:

R_eq1 = r/3

Continuing the Analysis

Next, we need to consider the resistors that connect the junctions from B, C, and D to E, F, and G. Each of these junctions will also have resistors leading to H. The resistors from B, C, and D to E, F, and G can be analyzed similarly, leading to another equivalent resistance of:

R_eq2 = r/3

Final Steps to Find Total Resistance

Now, we have two sets of equivalent resistances, R_eq1 and R_eq2, which are in series with the resistors connecting the junctions to H. The total equivalent resistance R_eq between A and H can be calculated as follows:

R_total = R_eq1 + R_eq2 + R_eq3

Where R_eq3 is the resistance of the resistors connecting the junctions to H, which is also r/3. Therefore:

R_total = r/3 + r/3 + r/3 = r

Conclusion

The equivalent resistance between two opposite corners of the cube formed by 12 resistors, each with resistance r, is:

R_eq = (5/6) r

This result showcases how symmetry and the principles of series and parallel resistances can simplify complex circuit problems. Understanding these concepts is crucial for analyzing more intricate electrical networks.