Question

What is the resistance of a copper wire of length \(1=10.9 \mathrm{~m}\) and diameter \(d=1.30 \mathrm{~mm}\) ? The resistivity of copper is \(1.72 \cdot 10^{-8} \Omega \mathrm{m}\).

Short answer

Answer: The resistance of the copper wire is 1.41 Ω.

Step-by-step solution

Write down the given values

We are given the following values: - Length of copper wire (\(L\)): \(10.9\) m - Diameter of copper wire (\(d\)): \(1.30\) mm (convert to meters) - Resistivity of copper (\(\rho\)): \(1.72 \cdot 10^{-8} \Omega \mathrm{~m}\)

Convert diameter to meters

We need to convert the diameter of the copper wire from millimeters to meters. To do this, divide the diameter by \(1000\). \(d = 1.30 \mathrm{~mm} \cdot \frac{1 \mathrm{~m}}{1000 \mathrm{~mm}} = 1.30 \cdot 10^{-3} \mathrm{~m}\)

Calculate the cross-sectional area of the wire

To calculate the cross-sectional area of the wire (\(A\)), we can use the formula for the area of a circle, which is \(A = \pi r^2\). The radius (\(r\)) is half of the diameter. So, let's first find the radius: \(r = \frac{d}{2} = \frac{1.30 \cdot 10^{-3} \mathrm{~m}}{2} = 0.65 \cdot 10^{-3} \mathrm{~m}\) Now we can find the area: \(A = \pi r^2 = \pi (0.65 \cdot 10^{-3} \mathrm{~m})^2 = 1.327 \cdot 10^{-6} \mathrm{~m^2}\)

Use the resistivity formula to find the resistance

Now we can use the resistivity formula to compute the resistance (\(R\)) of the copper wire. The formula is \(R = \frac{\rho L}{A}\). Using the given values, we can find the resistance: \(R = \frac{1.72 \cdot 10^{-8} \Omega \mathrm{~m} \cdot 10.9 \mathrm{~m}}{1.327 \cdot 10^{-6}\mathrm{~m^2}} = 1.41 \Omega\)

Write down the answer

The resistance of the copper wire with length \(10.9 \mathrm{~m}\) and diameter \(1.30 \mathrm{~mm}\) is \(1.41 \Omega\).