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The resistance of wire is \[20\Omega \]. The wire is stretched to three times its length. Now the resistance will be ?

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Last updated date: 20th Jul 2024
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Answer
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Hint:Think of what will happen to the length and area of the wire when it is stretched to three times its length and look for the relation between the resistance and the length of the wire.Resistance in simple words is a measure of how much the current is slowed down. The bigger the resistance, the smaller the current. It’s S.I unit is ohms \[\left( \Omega \right)\].

Complete step by step answer:
The resistance of a wire depends on its length, its cross-sectional area and the resistivity of the material. Resistivity is the resistance of a material of unit length and unit cross-sectional area. It is the characteristic property of the material and is independent of its length and area of cross section. Mathematically, Resistance of a wire is given by:
\[R=\rho \dfrac{l}{A}\]
where $R$ is the resistance, $ρ$ is the resistivity, l is the length, $A$ is the cross sectional area.

Area of the cross section for the wire is \[\pi {{r}^{2}}\], where $r$ is the radius of the cross section of the wire. When a wire is stretched to three times its length, then the length changes but the volume of the wire still remains constant as it is the same wire. Therefore, Final volume should be the same as original volume and volume we know is area multiplied by length. We already know the relation between the final length and the initial length \[{{l}_{f}}=3{{l}_{i}}\] .

Therefore, equating final and initial volumes, we get
\[{{V}_{initial}}={{V}_{final}} \\
\Rightarrow {{A}_{i}}\times {{l}_{i}}={{A}_{f}}\times {{l}_{f}} \\
\Rightarrow {{A}_{i}}\times {{l}_{i}}={{A}_{f}}\times 3{{l}_{i}} \\
\Rightarrow \dfrac{{{A}_{f}}}{{{A}_{i}}}=\dfrac{{{l}_{i}}}{3{{l}_{i}}} \\
\Rightarrow \dfrac{{{A}_{f}}}{{{A}_{i}}}=\dfrac{1}{3} \\ \]
Now, let’s calculate the final resistance
\[{{R}_{f}}=\rho \dfrac{3{{l}_{i}}}{\dfrac{{{A}_{i}}}{3}} \\
\Rightarrow {{R}_{f}}=\rho \dfrac{{{l}_{i}}}{{{A}_{i}}}\times 9 \\
\therefore {{R}_{f}}=20\times 9=180\Omega \\ \]

Hence, the new resistance will be $180\,\Omega$.

Note:Resistivity is a qualitative measurement of a material’s ability to resist flowing electric current.Insulators will have a higher value of resistivity than that of conductors.Resistance of a conductor is qualitative quantity means it depends on length and cross-sectional area of a conductor.