question_answer

Thirteen resistances each of resistance R ohm are connected in the circuit as shown in the figure. The effective resistance between A and B is:

A)  \[\frac{4R}{3}\Omega \]                             

B)  \[2R\Omega \]

C)  \[R\Omega \]                                  

D)  \[\frac{2R}{3}\Omega \]

Correct Answer: D

Solution :

Resistance R bisecting the circuit can be neglected due to the symmetry of the circuit. Now there are four triangles Effective resistance of each triangle                 \[\frac{1}{R}=\frac{1}{R}+\frac{1}{2R}\]                 \[=\frac{2+1}{2R}=\frac{3}{2R}\] \[\therefore \]  \[R=\frac{2}{3}R\] Now the given circuit reduced to Therefore, effective resistance between A and  B,                 \[\frac{1}{{{R}_{AB}}}=\frac{1}{2R}+\frac{1}{2R}=\frac{1}{R}\] \[\Rightarrow \]               \[{{R}_{AB}}=R=\frac{2R}{3}\]