Question

In the above arrangement, each side of the cube have the same resistance, and it is known that the effective resistance between $A$ and $B$ is $\frac{5}{9}\Omega$ . Now, if the resistor between $A$ and $B$ is removed, the new effective resistance (in $\Omega$ ) between the same two points is $\frac{k}{4}$ , find $k$ .

Answer 1

If $AB$ is removed, assuming the equivalent resistance between remaining resistors is $x,$
$\frac{1}{R_{e q}}=\frac{1}{R}+\frac{1}{x}$
$R_{e q}=\frac{R x}{R + x}$
$\therefore \frac{5}{9}=\frac{1 \left(\right. x \left.\right)}{1 + x}=\frac{x}{1 + x}....\left(\because R = 1 \Omega\right)$
$\therefore 5\left(\right.1+x\left.\right)=9x$
$\therefore 5+5x=9x$
$\therefore 5=4x$
$\therefore x=\frac{5}{4}\Omega$