Question

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

A

$\frac{1}{25}$

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B

$\frac{1}{5}$

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C

$5$

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D

$25$

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Solution

The correct option is D $25$Step 1. Given data It is given that a piece of wire of resistance $\mathrm{R}$ is cut into five equal parts. These parts are then connected in parallel.The equivalent resistance of the combination is $\mathrm{R}\text{'}$.Step 2: Formula usedResistance is the opposition that an electrical device has to the flow of electrical current. It is represented by symbol $\mathrm{R}$.The formula to find the resistance is,$\mathrm{R}=\mathrm{\rho }\frac{\mathcal{l}}{\mathrm{A}}$Here, $\mathrm{\rho }$ is the resistivity, $\mathcal{l}$ is the length of wire and $\mathrm{A}$ is the cross sectional area.Step 3. Find the ratio of $\frac{\mathrm{R}}{\mathrm{R}\text{'}}$.The resistance $\mathrm{R}$ is directly proportional to length of the wire.Thus, the resistance of each five parts after cutting the wire is, ${\mathrm{R}}_{1}={\mathrm{R}}_{2}={\mathrm{R}}_{3}={\mathrm{R}}_{4}={\mathrm{R}}_{5}=\frac{\mathrm{R}}{5}$When all the resistances are connected in parallel,Let us consider, $\mathrm{R}\text{'}$ is the equivalent resistance.Therefore, $\frac{1}{\mathrm{R}\text{'}}=\frac{1}{{\mathrm{R}}_{1}}+\frac{1}{{\mathrm{R}}_{2}}+\frac{1}{{\mathrm{R}}_{3}}+\frac{1}{{\mathrm{R}}_{4}}+\frac{1}{{\mathrm{R}}_{5}}$ $=\frac{1}{\frac{\mathrm{R}}{5}}+\frac{1}{\frac{\mathrm{R}}{5}}+\frac{1}{\frac{\mathrm{R}}{5}}+\frac{1}{\frac{\mathrm{R}}{5}}+\frac{1}{\frac{\mathrm{R}}{5}}$ $=\frac{5}{\mathrm{R}}+\frac{5}{\mathrm{R}}+\frac{5}{\mathrm{R}}+\frac{5}{\mathrm{R}}+\frac{5}{\mathrm{R}}$ ${R}^{\text{'}}=\frac{\mathrm{R}}{25}$Now, we will find the ratio of $\frac{\mathrm{R}}{\mathrm{R}\text{'}}$.We will find the ratio of $\frac{\mathrm{R}}{\mathrm{R}\text{'}}$.$⇒$ $\frac{\mathrm{R}}{\mathrm{R}\text{'}}=\frac{\mathrm{R}}{\frac{\mathrm{R}}{25}}$ $=\frac{25\overline{)\mathrm{R}}}{\overline{)\mathrm{R}}}$ $=25$The ratio of $\frac{\mathrm{R}}{\mathrm{R}\text{'}}$ is $25$.Hence, option $\mathrm{D}$ is correct answer.

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