12 wires each of resistance 'r' ohm are connected to form a cube. Find the resistance if current enters through one edge and leaves through other corner of the same edge of the cube

I like this question. Draw the figure.

Trail the currents. Let the entering current be 12I, so each branch current is 4I. EAch of those branch currents splits into two currents of 2I each. Now look draw the currents on the diagrams. You see coming into the final node, three currents of 4I each.

Now consider the voltage drop.
first path, voltage drop of 4I*r=4rI.
second path, voltage drop of 2I*r=2rI
final path, voltage drop of 4I*r=4rI

sum of voltage drops=10I

total current=12I

network resistance=V/I=5r/6 ohms

check my thinking.

http://www.rfcafe.com/miscellany/factoids/kirts-cogitations-256.htm

To find the equivalent resistance of the cube, we need to consider how the individual resistors are connected.

Given that the current enters through one edge and leaves through the other corner of the same edge, the cube can be visualized as a series-parallel combination of resistors.

To start, we know that there are 12 wires of resistance 'r' ohm each. Since each wire is connected to its adjacent wires, they are in parallel to one another.

So, let's consider a single edge of the cube. If we imagine it as a line with 12 wires in parallel, the equivalent resistance of these 12 parallel resistors can be calculated using the formula:

1/Req = 1/r + 1/r + 1/r + ... (12 times)

Since all the resistors are identical (r ohm), the equation becomes:

1/Req = 12/r

By taking the reciprocal on both sides, we get:

Req = r/12

Thus, the equivalent resistance of a single edge of the cube is r/12 ohm.

Now, since the current enters through one edge and leaves through the other corner of the same edge, the cube can be split into two halves, each having an equivalent resistance of r/12.

These two halves are connected in series, so to find the total resistance, we simply sum up these individual resistances:

Total Resistance = 2 * (r/12) = 2r/12 = r/6

Therefore, the resistance when current enters through one edge and leaves through the other corner of the same edge of the cube is r/6 ohm.