Question

The equivalent resistance of the circuit between the points $A$ and $B$ as shown in figure is
(Given shape contains segments of equal length and resistance $1$ $\Omega$ each)

• A. $\frac{11}{25}\Omega$
• B. $\frac{13}{23}\Omega$
• C. $\frac{22}{35}\Omega$
• D. $\frac{11}{23}\Omega$

Answer 1

The correct option is C $\frac{22}{35}\Omega$

The given circuit posses mirror symmetry about a perpendicular axis joining $AB$, as shown below


The current distribution will be such that the mirror image will have same current as that of object element.

From the distribution of current, we infer that connection at junction ${}^{\prime }{O}^{\prime }$ is useless, hence it can be removed.

After junction removal :


$\Rightarrow$ For the two set of rectangular arrangement of resistors, two $r$ (in series), one $r$ and again two $r$ (in series) are in parallel combination.

$\frac{1}{r_{eq}}=\frac{1}{2r}+\frac{1}{r}+\frac{1}{2r}=\frac{1+2+1}{2r}$

$r_{eq}=\frac{2r}{4}+\frac{r}{2}$

Simillarly for the triangular arrangements, two ${}^{\prime }{r}^{\prime }$ (being in series) are in parallel with third resistance ${}^{\prime }{r}^{\prime }$.

$\Rightarrow \frac{1}{r_{eq}}=\frac{1}{2r}+\frac{1}{r}=\frac{1+2}{2r}$

$r_{eq}=\frac{2r}{3}$

Now redrawing the simplified circuit,


The upper and lower branches can be replaced by

$r^1=\frac{2r}{3}+\frac{2r}{3}+\frac{r}{2}$

$r^1=\frac{4r+4r+3r}{6}=\frac{11r}{6}$


The three resistances are in parallel about $A$ & $B$,

$\frac{1}{R_{eq}}=\frac{1}{\left(11r/6\right)}+\frac{1}{\left(11r/6\right)}+\frac{1}{\left(2r\right)}$

$\frac{1}{R_{eq}}=\frac{6}{11r}+\frac{6}{11r}+\frac{1}{2r}$

$\frac{1}{R_{eq}}=\frac{12+12+11}{22r}$

$\Rightarrow R_{eq}=\frac{22r}{35}$

$(\because r=1\Omega )$

$\Rightarrow R_{eq}=\frac{22}{35}\Omega$

Why this question? Tip: In problems involving complex circuit and resistances are arranged to form geometrical shape, then always think to identity the parallel or perpendicular axis of symmetry about the required terminal/end. This will help to remove unwanted junction, as well as gives ''insight to points with same potential''.