Question

# A uniform wire of resistance $50\mathrm{\Omega }$ is cut into five equal parts. These parts are now connected in parallel. Calculate the equivalent resistance of the combination.

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Solution

## Step 1: Given informationResistance, $\mathrm{R}=50\mathrm{\Omega }$Wire cuts in five equal parts.Step 2: To findWe have to find the equivalent resistance of the combination.Step 3: Calculate each resistance of the combinationWe know the relation,$⇒\mathrm{R}\propto \mathrm{l}$Where,$\mathrm{l}$ is the length of the conductor and is the resistance of the conductor. Whenever the wire is split into five equal sections, the length of each portion decreases by one-fifth, as does the resistance of each part.$R=\frac{50}{5}\phantom{\rule{0ex}{0ex}}=10\mathrm{\Omega }$Step 3: Calculate the equivalent resistance of the combinationUse the expression for equivalent resistors in parallel,$\frac{1}{{\mathrm{R}}_{\mathrm{eq}}}=\frac{1}{{\mathrm{R}}_{1}}+\frac{1}{{\mathrm{R}}_{2}}+\frac{1}{{\mathrm{R}}_{3}}+\frac{1}{{\mathrm{R}}_{4}}+\frac{1}{{\mathrm{R}}_{5}}\phantom{\rule{0ex}{0ex}}\frac{1}{{\mathrm{R}}_{\mathrm{eq}}}=\frac{1}{\mathrm{R}}+\frac{1}{\mathrm{R}}+\frac{1}{\mathrm{R}}+\frac{1}{\mathrm{R}}+\frac{1}{\mathrm{R}}\phantom{\rule{0ex}{0ex}}\frac{1}{{\mathrm{R}}_{\mathrm{eq}}}=\frac{5}{\mathrm{R}}\phantom{\rule{0ex}{0ex}}{\mathrm{R}}_{\mathrm{eq}}=\frac{\mathrm{R}}{5}$Equivalent resistance, ${R}_{eq}=\frac{R}{5}$ $=\frac{10}{5}\phantom{\rule{0ex}{0ex}}=2\mathrm{\Omega }$Therefore, the equivalent resistance is $2\mathrm{\Omega }$.

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