Question

In the infinite grid, the value of each resistance is $r=2(\sqrt{5}-1)\Omega$. Find the equivalent resistance between the points $a$ and $c$ (in $\Omega$)

Answer 1

Let the equivalent resistance between a and c be $r_{eq}$.

Also due to symmetry, the resistance connected to the line passing through b is useless.

From figure, effective resistance between P and M $R^{\prime }=2r\parallel r_{eq}=\frac{\left(2r\right)r_{eq}}{2r+r_{eq}}$

$\therefore$ Equivalent resistance between a and c $r_{eq}=2r+\frac{\left(2r\right)r_{eq}}{2r+r_{eq}}$

$⟹$ $r_{eq}^2-\left(2r\right)r_{eq}-4r^2=0$

$\therefore$ $r_{eq}=\frac{2r\pm \sqrt{4r^2-4\left(4r^2\right)}}{2}=(1\pm \sqrt{5})r=(1+\sqrt{5})r$

$⟹$ $r_{eq}=(1+\sqrt{5})\times 2(\sqrt{5}-1)=2\times 4=8\Omega$