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 $R / R^{\prime}$ is
(1) $\frac{1}{25}$
(2) $\frac{1}{5}$
(3) 5
(4) 25
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 $R / R^{\prime}$ is
(1) $\frac{1}{25}$
(2) $\frac{1}{5}$
(3) 5
(4) 25
solution:
Resistance of each one of the five parts $=\frac{\mathrm{R}}{5}$ Resistance of five parts connected in parallel is
given by
$\frac{1}{\mathrm{R}^{\prime}}=\frac{1}{\mathrm{R} / 5}+\frac{1}{\mathrm{R} / 5}+\frac{1}{\mathrm{R} / 5}+\frac{1}{\mathrm{R} / 5}+\frac{1}{\mathrm{R} / 5}$
or $\frac{1}{R^{\prime}}=\frac{5}{R}+\frac{5}{R}+\frac{5}{R}+\frac{5}{R}+\frac{5}{R}=\frac{25}{R}$
or $\frac{R}{R^{\prime}}=25$
Thus, (4) is the correct answer.
Resistance of each one of the five parts $=\frac{\mathrm{R}}{5}$ Resistance of five parts connected in parallel is
given by
$\frac{1}{\mathrm{R}^{\prime}}=\frac{1}{\mathrm{R} / 5}+\frac{1}{\mathrm{R} / 5}+\frac{1}{\mathrm{R} / 5}+\frac{1}{\mathrm{R} / 5}+\frac{1}{\mathrm{R} / 5}$
or $\frac{1}{R^{\prime}}=\frac{5}{R}+\frac{5}{R}+\frac{5}{R}+\frac{5}{R}+\frac{5}{R}=\frac{25}{R}$
or $\frac{R}{R^{\prime}}=25$
Thus, (4) is the correct answer.
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