The equivalent resistance between the points a and c of the network s…

The correct answer is Option (2) → \( 2r/3 \)

Use symmetry. From a to c there are three identical paths, a-d-c, a-o-c and a-b-c, each made of two resistors \( r \) in series. With a potential difference \( V \) applied between a and c, the midpoint of each identical path is at the same potential.

    \( \displaystyle V_d = V_o = V_b = \frac{V}{2} \)

So there is no potential difference across the two vertical resistors (d-o and o-b); they carry no current and can be removed. The network is then three paths of \( 2r \) in parallel.

    \( \displaystyle \frac{1}{R_{ac}} = \frac{1}{2r} + \frac{1}{2r} + \frac{1}{2r} = \frac{3}{2r} \)

    \( \displaystyle R_{ac} = \frac{2r}{3} \)