Let the vertices of the cube be A, B, C, D, E, F, G, H. Let the resistance of each edge be R = 3Ω. The potential difference across the diagonally opposite corners is 10V.
Due to symmetry, the current distribution is as follows:
Consider the path from one corner (say A) to the opposite corner (say G). The equivalent resistance can be found using several methods, including superposition or symmetry arguments. One common approach is to use symmetry to simplify the circuit. Due to the symmetry of the cube and the equal resistances, the potential at points B, D, and F will be equal. Similarly, the potential at points C, E, and H will be equal.
We can simplify the circuit by considering the parallel combination of resistances. Let's analyze the current distribution from A to G:
Applying symmetry and Kirchhoff's laws becomes complex for direct calculation. Let's instead use the method of equivalent resistance.
Imagine injecting a current I at A and extracting it at G. The equivalent resistance between A and G is given by R_eq = V/I, where V = 10V.
By applying the principles of network reduction and symmetry, the equivalent resistance between opposite corners of a cube with edge resistance R is 5R/6.
Therefore, the equivalent resistance is:
R_eq = (5/6) * R = (5/6) * 3Ω = 2.5Ω
The total current flowing through the circuit is I = V/R_eq = 10V / 2.5Ω = 4A.
The current along each edge of the cube is I_edge = I/6 = 4A/6 = 2/3 A
Therefore: