Question

A piece of wire of resistance R is cut into five equal parts. These parts are then connected in parallel. If the equivalent resistance of this combination is $R^{{}^{\prime }}$, then the ratio $\frac{R^{{}^{\prime }}}{R}$ is:

• A. $\frac{1}{25}$
• B. $\frac{1}{5}$
• C. $5$
• D. $25$

Answer 1

The correct option is A $\frac{1}{25}$

Given the wire is cut into 5 equal parts.
Let the initial resistance of the wire be $R$
where $R=\rho \frac{l}{A}\to R\propto l\left(lengthofthewire\right)$
then, resistance of each one of the five parts $=\left(\frac{R}{5}\right)$
Equivalent resistance (R') of resistors connected in parallel is given by $\frac{1}{R^{\prime }}=\frac{1}{R_1}+\frac{1}{R_2}..\frac{1}{R_n}$
Equivalent resistance (R') of five parts connected in parallel is given by:
$\frac{1}{R^{\prime }}=\frac{1}{\left(\frac{R}{5}\right)}+\frac{1}{\left(\frac{R}{5}\right)}+\frac{1}{\left(\frac{R}{5}\right)}+\frac{1}{\left(\frac{R}{5}\right)}+\frac{1}{\left(\frac{R}{5}\right)}{R}^{\prime }=\frac{R}{25}$
Therefore, $\frac{R}{R^{{}^{\prime }}}=25$
or
$\frac{R^{{}^{\prime }}}{R}=\frac{1}{25}$