Question

Eight identical resistances $r$ are connected as shown in figure. Find equivalent resistance between $A$ and $D$.

• A. $\frac{7r}{16}$
• B. $\frac{2r}{9}$
• C. $\frac{8r}{15}$
• D. $\frac{7r}{19}$

Answer 1

The correct option is C $\frac{8r}{15}$

To find the equivalent resistance across $A$ and $D$, let us connect a $100\text{V}$ battery and distribute potentials as shown in figure below.


When a battery is connected across terminals $A$ and $D$, the circuit becomes symmetric about a plane passing through mid-points of edges $BC$ and $AD$ and vertex $O$, so the potential for $O$ is taken as $50\text{V}$.

To calculate unknown potential we write $\text{KCL}$ equation for $x$ as

$\frac{x-0}{r}+\frac{x-50}{r}+\frac{x-(100-x)}{r}=0$

$\Rightarrow 4x=150$

$\Rightarrow x=\frac{75}{2}\text{V}$

Current through battery is given as

$I_{battery}=\frac{x}{r}+\frac{100}{r}+\frac{50}{r}=\frac{75}{2r}+\frac{150}{r}$

$\Rightarrow I_{battery}=\frac{375}{2r}$

Equivalent resistance across terminals $A$ and $D$ is given as

$R_{eq}=\frac{V_{battery}}{I_{battery}}=\frac{100\times 2r}{375}=\frac{8r}{15}$

Hence, option (c) is the correct answer.