Question

The network shown consists of infinitely large number of identical sections. Resistance between any two nearest joints is R. Then resultant resistance between $A$ and $B$ is $\frac{R}{\sqrt{x}}$. The value of x is.(Answer upto two digits after the decimal point)

Answer 1

No current flows in lines PQ and RS

So, by eliminating the lines PQ and RS, the circuit turns out to be

Let equivalent resistance on either sides be $R_o$

$2R+\frac{{RR}_0}{R+R_0}=R_0$

$2R^2+2R{R}_0+R{R}_0=R{R}_0+R_0^2$

$R_0^2-2{RR}_0-2R^2=0$

$R_0=\frac{2R+\sqrt{4R^2+8R^2}}{2}$

$R_0=\frac{2R+2\sqrt{3}R}{2}=(\sqrt{3}+1)R$

$\therefore R_{AB}=\frac{R\times \frac{(\sqrt{3}+1)R}{2}}{(1+\frac{\sqrt{3}+1}{2})R}=\frac{(\sqrt{3}+1)R}{(\sqrt{3}+3)}=\frac{R}{\sqrt{3}}$