Question

Write Kirchhoff's first rule. A battery of $10V$ and negligible internal resistance is connected to the diagonally opposite corners of a cubical network consisting of $12$ resistors each of resistance $2\Omega$. Determine the equivalent resistance of the network.

Answer 1

According to Kirchhoff's first Law, at any node in an electrical circuit, total currents entering the node equals the currents exiting the node.

Using symmetry, equal currents flows through resistors $a,b\&c$.

Hence, equal potential drops occur across them and they can be connected in parallel near node $A$.

Similarly, equal potential drops occur across resistors $j,k\&l$ them and they can be connected in parallel near node $B$.

After this, it can be observed that the ends of remaining resistors are at same value of potentials. Hence, the remaining six resistors can be connected in parallel as shown.

For the equivalent circuit, the equivalent resistance can be found as a simple series-parallel resistance circuit.

$R_{eq}=\left(2||2||2\right)+\left(2||2||2||2||2||2\right)+\left(2||2||2\right)$

$=\frac{2}{3}+\frac{2}{6}+\frac{2}{3}$

$=\frac{5}{3}\Omega$