Question

The wire has linear resistance $p($in $\Omega \text{/m})$. The resistance between points $\text{A}$ and $\text{B}$ if the side of the big square is $d$.

• A. $\frac{pd}{\sqrt{2}}$
• B. $\sqrt{2}pd$
• C. $2pd$
• D. $3pd$

Answer 1

The correct option is A $\frac{pd}{\sqrt{2}}$


The given circuit is symmetrical about diagonal $\text{AB}$.

Therefore, points $(2,8),(3,7)$ and $(4,6)$ are at same potential.

Each half side of bigger square has resistance $r=\left(\frac{pd}{2}\right)$

Hence, the circuit is equivalent to


Further, the reduced form of circuit will be


So, equivalent resistance will be

$\therefore R_{AB}=\frac{1}{2}[2r+\frac{2r\times \sqrt{2}r}{2r+\sqrt{2}r}]=\sqrt{2}r$

$\Rightarrow R_{AB}=\sqrt{2}\times \frac{pd}{2}=\frac{pd}{\sqrt{2}}$

Hence, option $\left(a\right)$ is correct.
$$\begin{array}{c}\text{Why this question ?}\\ \text{Key Idea-This question helps student to}\\ \text{get more firm on concept of symmetry}\\ \text{of resistors network.}\end{array}$$