In household wiring, copper wire 2.052.05 mm in diameter is often used. Find the resistance of a 24.0 24.0-m length of this wire.

Ch 25: Current, Resistance, and EMF

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 25: Current, Resistance, and EMF

문제 19

25장, 문제 19

In household wiring, copper wire $2.05$ mm in diameter is often used. Find the resistance of a $24.0$ -m length of this wire.

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First, identify the formula for resistance in a wire, which is given by $ R = \frac{\rho L}{A} $, where $ R $ is the resistance, $ \rho $ is the resistivity of the material, $ L $ is the length of the wire, and $ A $ is the cross-sectional area of the wire.

Determine the resistivity $ \rho $ of copper. This is a known value and can be found in tables of material properties. For copper, $ \rho \approx 1.68 \times 10^{-8} \text{ ohm meters} $.

Calculate the cross-sectional area $ A $ of the wire. Since the wire is cylindrical, use the formula for the area of a circle: $ A = \pi r^2 $, where $ r $ is the radius of the wire. Convert the diameter to radius by dividing by 2: $ r = \frac{2.05 \text{ mm}}{2} = 1.025 \text{ mm} $. Convert this to meters: $ r = 1.025 \times 10^{-3} \text{ m} $.

Substitute the radius into the area formula: $ A = \pi (1.025 \times 10^{-3})^2 \text{ m}^2 $. Calculate this value to find the cross-sectional area.

Finally, substitute the values for $ \rho $, $ L $, and $ A $ into the resistance formula: $ R = \frac{1.68 \times 10^{-8} \times 24.0}{A} $. Calculate $ R $ using the computed area from the previous step.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Resistance

Resistance is a measure of the opposition to the flow of electric current through a conductor. It is determined by the material's properties, length, and cross-sectional area. The formula for resistance is R = ρL/A, where ρ is the resistivity, L is the length, and A is the cross-sectional area.

Resistivity

Resistivity is a fundamental property of materials that quantifies how strongly a material opposes the flow of electric current. It is denoted by ρ and is measured in ohm-meters (Ω·m). Copper has a low resistivity, making it an excellent conductor for household wiring.

Cross-sectional Area

The cross-sectional area of a wire affects its resistance. For a cylindrical wire, the area A can be calculated using the formula A = π(d/2)^2, where d is the diameter. A larger cross-sectional area results in lower resistance, allowing more current to flow through the wire.