Question

The effective resistance between $A$ and $B$ in the figure is $\frac{5}{9} \Omega$, if each side of the cube has $1\, \Omega$ resistance. The effective resistance between the same two points, when the link $A B$ is removed, is ______$\Omega$

Answer 1

i. For a cube of resistors each of resistance $R$, the equivalent resistance between two edges on the same face is
$R _{ eq }=\frac{5 R }{9}$
ii. 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(x)}{1+x}=\frac{x}{1+x} ....(\because R = 1\,\Omega)$
$ \therefore 5(1+x)=9 x $
$ \therefore 5+5 x=9 x $
$\therefore 5=4 x $
$ \therefore x=\frac{5}{4} \Omega $
$\therefore x=1.25\, \Omega $