Question

The diameter of 0-gauge copper wire is 8.252 m m. Find the resistance of a 1.00 kmlength of such wire used for power transmission.

Short answer

The resistance of the copper wire is obtained as:$\mathrm{R}=321\mathrm{m\Omega }.$

Step-by-step solution

Define Resistance

In an electrical circuit, resistance is a measure of the resistance to current flow. The Greek letter omega $\left(\Omega \right)$is used to represent resistance in ohms.

Concepts and Principles

The electrical resistance of the value R of a resistive circuit element depends on its geometric structure (its length symbolized as Land cross-sectional area symbolized as A) and on the resistivity pof the material of which it is made:

$R=P=\frac{L}{A}$

The given data

$$\begin{gathered} •\mathrm{The}\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{wire}\mathrm{is}:L=1.00\left[1\mathrm{km}\times \frac{1000\mathrm{m}}{1\mathrm{km}}\right]=1000\mathrm{m}. \\ •\mathrm{The}\mathrm{diameter}\mathrm{of}\mathrm{the}\mathrm{wire}\mathrm{is}:D=(8.252).\left[1\mathrm{mm}\times \frac{1\mathrm{m}}{1000\mathrm{mm}}\right]=8.252\times {10}^{-3}\mathrm{m}. \\ •\mathrm{The}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{wire}\mathrm{is}:r=D/2=4.126\times {10}^{-3}\mathrm{m}. \\ •\mathrm{The}\mathrm{resistivity}\mathrm{of}\mathrm{the}\mathrm{copper}\mathrm{is}:p=1.72\times {10}^{-8}\mathrm{\Omega }.\mathrm{m} \end{gathered}$$

Evaluation of resistance of the copper wire

The resistance of the copper wire is found from the above equation as:

$$\begin{gathered} R=p\frac{L}{A} \\ \mathrm{The}\mathrm{value}\mathrm{of}A={\mathrm{\pi r}}^2\mathrm{which}\mathrm{is}\mathrm{the}\mathrm{cross}-\mathrm{sectional}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{wire}: \\ R=p\frac{L}{\pi r^2} \\ \mathrm{Entering}\mathrm{the}\mathrm{values},\mathrm{we}\mathrm{obtain}: \\ \mathrm{R}=1.72\times {10}^{-8}\mathrm{\Omega }.\mathrm{m}\times \frac{1000\mathrm{m}}{\mathrm{\pi }\times {(4.126\times {10}^{-3}\mathrm{m})}^2} \\ =0.321\mathrm{\Omega } \\ =(0.321).(1\mathrm{\Omega }\times \frac{1000\mathrm{m\Omega }}{1\mathrm{\Omega }}) \\ =321\mathrm{m\Omega } \\ \mathrm{Therefore},\mathrm{the}\mathrm{resistance}\mathrm{of}\mathrm{the}\mathrm{copper}\mathrm{wire}\mathrm{is}:R=321\mathrm{m\Omega } \end{gathered}$$