A wire frame forms a square ABCD with a center point O . Eight resistors, each of resistance R , are connected: four along the perimeter sides ( AB , BC , CD , DA ) and four along the radial segments from the vertices to the center ( OA , OB , OC , OD ). The equivalent resistance between the opposite vertices A and C is

Step-by-step solution

Let a voltage be applied across the opposite vertices A and C . The diagonal BD forms an axis of symmetry for the circuit. Due to this symmetry, the nodes B , O , and D are at the same equipotential state ( V_B = V_O = V_D ). Because there is no potential difference between O and B , and between O and D , no current flows through the radial resistors OB and OD . These two resistors can be removed from the circuit without affecting the equivalent resistance. After removing OB and OD , the circuit simplifies to three