Question

Calculate the resistance of 2 km long copper wire of radius 2 mm. (Resistivity of copper = 1.72 x 1 0-8)

Answer 1

$\sf{\bf{Given}}\begin {cases} \sf{ The \:Length \:of\:copper\:wire\:is\:2\:km}& \\\\ \sf{ Radius \:of\:Copper\:wire\:is\:2\:mm}&\\\\ \sf{Resistivity \:of\:Copper \:is\:1.72\times 10^{8}}\end{cases}$

Need To Find : The Resistance.

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⠀⠀⠀⠀⠀⠀⠀⠀❍ Converting Units : ⠀⠀

$\sf{\bf{Converting\:units}}\begin {cases} \sf{ The \:Length \:of\:copper\:wire\:is\:2\:km\:or\:2000\:m\:or\:2 \times 10^{3} \:m.}& \\\\ \sf{ Radius \:of\:Copper\:wire\:is\:2\:mm\:or\:2 \times 10^{-3}}&\\\\ \sf{Resistivity \:of\:Copper \:is\:1.72\times 10^{8}}\end{cases}$

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⠀⠀⠀⠀⠀⠀⠀⠀$\underline{\frak{As\:We\:know\:that\::}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀$\pink{\boxed { \sf{Resistance = \dfrac{ Resistivity \:of\:Copper\:wire\: \times Length}{Area}}}}$

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⠀⠀⠀⠀⠀⠀$\underline {\sf{\bf{\star\:Now \: By \: Substituting \: the \: Given \: Values \::}}}$

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⠀⠀⠀⠀⠀⠀⠀⠀⠀$:\implies { \sf{Resistance = \dfrac{ Resistivity \:of\:Copper\:wire\: \times Length}{Area}}}$

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As , We know that :

• Area of Wire = $\pi\:\times Radius^{2}$

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⠀⠀⠀⠀⠀⠀⠀⠀$:\implies { \sf{Resistance = \dfrac{ Resistivity \:of\:Copper\:wire\: \times Length}{\pi\times Radius^{2}}}}$

⠀⠀⠀⠀⠀⠀⠀⠀$:\implies { \sf{Resistance = \dfrac{ 1.72 \times10^{-8}\times 2 \times 10^{3} \: }{\pi\times \big(2 \times 10^{-3}\big)^{2}}}}$

⠀⠀⠀⠀⠀⠀⠀⠀$:\implies { \sf{Resistance = \dfrac{ 1.72 \times10^{-8}\times 2 \times 10^{3} \: }{3.14 \times 4 \times 10^{-6}}}}$

⠀⠀⠀⠀⠀⠀⠀⠀$:\implies { \sf{Resistance = \dfrac{ 1.72 \times10^{-8}\times 2 \times 10^{3} \: }{12.56 \times 10^{-6}}}}$

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☆ Now By Using Law's of Exponents :

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⠀⠀⠀⠀⠀⠀⠀⠀$:\implies { \sf{Resistance = \dfrac{ 1.72 \times \cancel {10^{-8}} \times 2 \times \cancel {10^{3}} \: }{12.56 \times \cancel {10^{-6}}}}}$

⠀⠀⠀⠀⠀⠀⠀⠀[ 10 $^{(-8 + 3 + 6)}$ = 10 $^{-5 + 6}$= 10 $^{-1}$= 1 ]

⠀⠀⠀⠀⠀⠀⠀$:\implies { \sf{Resistance = \dfrac{ 1.72 \times 2 \: }{12.56 } }}$

⠀⠀⠀⠀⠀⠀⠀$:\implies { \sf{Resistance = \dfrac{ \cancel {34.4}\: }{\cancel {12.56} } }}$

⠀⠀⠀⠀⠀⠀⠀$:\pink{\boxed { \sf{Resistance = 2.7388 \:ohm}}}$

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Therefore,

⠀⠀⠀⠀⠀$\underline {\therefore\:{\pink{ \mathrm {The\:Resistance\:of\: Copper\:wire\:is\: = 2.7388 \:ohm  }}}}$

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Answer 2

Answer:

$\begin{gathered} \rho = 1.72 \times {10}^{-8} \\ r = 2mm \; or \; 2 \times {10}^{-3}\: m \\ l = 2km \; or\; 2 \times {10}^{3}\: m \\ We \: know \: that, \\ \qquad \qquad \qquad \boxed{R = \frac{\rho l}{A}} \\ \\ \Rightarrow \qquad R = \frac{1.72 \times {10}^{-8} \times 2 \times {10}^{3}}{\pi {r}^{2}} \\ \\ \Rightarrow \qquad R = \frac{3.44 \times {10}^{-5}}{12.56 \times {10}^{-6}} \\ \\ \Rightarrow \qquad R = 0.273 \times {10}^{(-5+6)} \\ \\ \mathfrak{Answer \: :} \quad \mathtt{\red{2.73 \; \text{\O}mega}} \end{gathered}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀$\begin{gathered}\blue{\boxed { \sf{ \dfrac{2.73}{ohm}}}}\\\end{gathered}$