12 wires of equals resistance x Upon y are connected to form a cube . the effective resistance between the two diagonal ends will be
Answers
Explanation:
Method 1:
This is the cube structure consisting of 12 resistors electrically connected between the 8 vertices. Each resistor is 1 Ω, but any value can be used so long as they are all the same.
Here is where the intuition comes into play. Color coding is used to help keep track of the resistors and associated nodes (below). Due to symmetry, the potential (voltage) at the three nodes labeled "α" are equal. Since no current flows between nodes with a potential difference of 0 V, they can be shorted together without affecting the circuit's integrity. The same can be done for the nodes labeled "β."
Once you short those nodes, you obtain the equivalent circuit shown below. As you can see, there are two sets of three resistors in parallel, in series with one set of six resistors in parallel. So, you have 1/3 Ω in series with 1/6 Ω in series with 1/3 Ω, which equals 5/6 Ω.
Now I will present my method of solving the resistor cube problem. The structure is repeated again here.
Method 2:
Kirchhoff's current law, which states that the sum of the currents entering and exiting a node is zero, is essential in the analysis.
The first step is to recognize that at a node where equal resistances exist, current entering the node will be distributed equally between the number of output branches - in this case three. For convenience sake, I assigned an input current of 3 amperes at the corner labeled "A," so that 1 amp will flow through each output branch. Note that 1 A flows through each branch.
On the far side of each of those branches is another node with two output branches. Again, due to symmetry, the input current will divide evenly so that ½ A flow into each branch. Looking at the cube's output node labeled "B," it is apparent that the same situation exists as with "A."
Take a moment to sum the currents into and out of each node to verify that they all add up as required.
Now that you know the current through each branch, and you know that each branch has a single 1 Ω resistor in it, Ohms law allows you to calculate the voltage across each resistor.
The next step is to sum the voltage from input node "A" to output node "B." Any path you take travels along three edges, and all total to 2½ volts.
Finally, apply Ohms law, which says that the resistance is equal to the voltage divided by the current. As with the other analysis method, the resulting equivalent resistance is 5/6 Ω.