Question

What is the resistance of a 20.0 m -long piece of 12 -gauge copper wire having a 2.053 mm diameter?

Short answer

The resistance of the copper wire is obtained as:$103.8\mathrm{m\Omega }$ .

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 $L$and cross-sectional area symbolized as $A$) and on the resistivity $\rho$of the material of which it is made:

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

The given data

• The length of the wire is: $\mathrm{L}=20.0\hspace{0.17em}\mathrm{m}$.
• The diameter of the wire is: $\mathrm{D}=2.053\mathrm{mm}\times \frac{1\mathrm{m}}{1000\mathrm{mm}}=2.053\times {10}^{-3}\mathrm{m}$.
• The radius of the wire is: $r=D/2=1.0256\times {10}^{-3}\mathrm{m}$.
• The resistivity of the copper is:$\rho =1.72\times {10}^{-8}\mathrm{\Omega }·\mathrm{m}$ .

Evaluation of resistance of the copper wire

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

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

The value of $A=\pi r^2$, which is the cross-sectional area of the wire:

$R=\rho \frac{L}{\pi r^2}$

Entering the values, we obtain:

$$\begin{gathered} R=1.72\times {10}^{-8}\mathrm{\Omega }·\mathrm{m}\times \frac{20.0}{\pi {\left(1.0265\times {10}^{-3}\mathrm{m}\right)}^2} \\ =0.1038\mathrm{\Omega } \\ =0.1038\mathrm{\Omega }\times \frac{1000\mathrm{m\Omega }}{1\mathrm{\Omega }} \\ =103.8\mathrm{m\Omega } \end{gathered}$$

Therefore, the resistance of the copper wire is: $103.8\mathrm{m\Omega }$.