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Chapter 25: Problem 30

What is the resistance of a copper wire of length \(l=\) \(10.9 \mathrm{~m}\) and diameter \(d=1.3 \mathrm{~mm} ?\) The resistivity of copper is \(1.72 \cdot 10^{-8} \Omega \mathrm{m}\)

Short Answer

Expert verified
Question: Calculate the resistance of a 10.9 meters long copper wire with a diameter of 1.3 mm and resistivity of 1.72 x 10^-8 Ωm. Answer: The resistance of the copper wire is approximately 1.414 x 10^-3 Ω.

Step by step solution

01

Convert diameter to meters

Given diameter \(d = 1.3\) mm, we need to convert it to meters. Since there are 1000mm in 1m, the conversion is: \(d = 1.3 \times 10^{-3} \mathrm{~m}\)
02

Calculate cross-sectional area

Now we need to find the cross-sectional area of the wire. The formula for the area of a circle is: \(A = \pi \cdot (\frac{d}{2})^2\) Using the diameter in meters we found in step 1: \(A = \pi \cdot (\frac{1.3 \times 10^{-3}}{2})^2\) \(A \approx 1.325 \times 10^{-6} \mathrm{~m}^2\)
03

Calculate resistance

Now that we have the area, we can use the formula for resistance: \(R = \frac{\rho \cdot L}{A}\) Plug in the given values for resistivity (\(\rho = 1.72 \times 10^{-8} \Omega \mathrm{m}\)) and length (\(L = 10.9 \mathrm{~m}\)) and the area we calculated: \(R = \frac{(1.72 \times 10^{-8} \Omega \mathrm{m}) \cdot (10.9 \mathrm{~m})}{1.325 \times 10^{-6} \mathrm{~m}^2}\) \(R \approx 1.414 \times 10^{-3} \Omega\) Thus, the resistance of the copper wire is approximately \(1.414 \times 10^{-3} \Omega\).

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