question_answer

12 resistors each of 10 \[\Omega \] are connected as shown in figure. The effective resistance between A and B is:

A)  120 \[\Omega \]                             

B)  8\[\Omega \]

C)  12\[\Omega \]                                

D)  10\[\Omega \]

Correct Answer: B

Solution :

 \[r=10\Omega \] By symmetry The current entering at A= current leaving at B Hence, the equivalent circuit is shown as figure. It is clearly shown that equivalent resistance of upper or lawer branch is given as \[=r+\frac{2r}{3}+r=\frac{8r}{3}\] The resistance in the middle branch is \[r+r=2r\] So, resistances \[\frac{8r}{3},2r,\frac{8r}{3}\] are in parallel  combination. So,          \[\frac{1}{{{R}_{AB}}}=\frac{3}{8r}+\frac{1}{2r}+\frac{3}{8r}\] \[\Rightarrow \]               \[\frac{1}{{{R}_{AB}}}=\frac{10}{8r}\]                 \[\frac{1}{{{R}_{AB}}}=\frac{8}{10}r\] (since \[r=10\Omega \]given) \[=\frac{8\times 10}{10}=8\Omega \]