Physics - Current Electricity Question with Solution | TestHub

Given the total resistance of the wire is $R$. If the polygon has $n$ numbers of side, then the resistance $\left(r\right)$, as shown in the following figure, of each side can be written as

$r= \frac{R}{n}.................\left(1\right)$

Among all the resistances, $\left(n-1\right)$ resistances are connected in series combination, whereas, their equivalent resistance is connected in parallel with the $n^{\mathrm{th}}$ resistance. The equivalent circuit diagram for the above set of resistances can be depicted as follows:

From the above equivalent circuit diagram, the net equivalent resistance $\left(R_{eq}\right)$ for the entire combination can be calculated as follows-

$\begin{array}{rcl}\frac{1}{R_{eq}}& =& \frac{1}{\left(n-1\right){\displaystyle r}}+\frac{1}{r}\\ & =& \frac{1}{\left(n-1\right){\displaystyle \frac{R}{n}}}+\frac{1}{{\displaystyle \frac{R}{n}}}\\ & =& \frac{n}{\left(n-1\right){\displaystyle R}}+\frac{n}{R}\\ & =& \frac{n^2}{\left(n-1\right){\displaystyle R}}\\ R_{eq}& =& \frac{\left(n-1\right)R}{n^2}\end{array}$