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.

Verified step by step guidance

1. Calculate the cross-sectional area, A, of the wire using the formula for the area of a circle: A = \pi (d/2)^2, where d is the diameter of the wire.
2. Find the resistivity, \( \rho \), of copper from a reference table. Resistivity is a material-specific property that quantifies how strongly a material opposes the flow of electric current.
3. Use the formula for resistance, R, of a uniform cylindrical conductor: R = \( \rho \frac{L}{A} \), where L is the length of the wire and A is the cross-sectional area calculated in step 1.
4. Substitute the values of \( \rho \), L, and A into the resistance formula to calculate the resistance of the wire.
5. Ensure the units are consistent when performing the calculations (e.g., converting diameters from millimeters to meters if necessary).

Key Concepts

Resistance

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

Resistivity

Resistivity is a fundamental property of materials that quantifies how strongly they resist electric current. It is denoted by the symbol ρ and is measured in ohm-meters (Ω·m). Different materials have different resistivities; for example, copper has a low resistivity, making it an excellent conductor for electrical applications.

Cross-sectional Area

The cross-sectional area of a wire is the area of a slice taken perpendicular to its length. For a circular wire, it can be calculated using the formula A = π(d/2)², where d is the diameter. The cross-sectional area is crucial in determining the resistance of the wire, as a larger area allows more current to flow, reducing resistance.