A house wire is to carry a current of 30 A while dissipating...

A house wire is to carry a current of 30 A while dissipating no more than $1.40 \mathrm{~W}$ of heat per meter of its length. What is the minimum diameter of the wire if its resistivity is $1.68 \times 10^{-8} \Omega \cdot \mathrm{m}$ ?

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Key Concepts

Joule Heating

Joule heating refers to the process by which the passage of an electric current through a conductor releases heat. The power dissipated as heat in a conductor is given by the formula P = I²R, where P is the power, I is the current, and R is the resistance. This concept is essential in ensuring that electrical components and wiring are designed to limit unwanted heat generation, thereby maintaining safety and efficiency.

Electrical Resistivity

Electrical resistivity is a fundamental property of a material that quantifies how strongly the material opposes the flow of electric current. It is used in calculating the resistance of a conductor through the formula R = ?(L/A), where R is resistance, ? is resistivity, L is the length, and A is the cross-sectional area. This concept is crucial in selecting materials for electrical wiring based on how much energy is lost as heat.

Geometric Considerations in Electrical Conductors

In the context of electrical wires, the cross-sectional area plays a critical role in determining the wire's resistance. For cylindrical conductors, the cross-sectional area is calculated using the formula A = ?(d/2)², where d is the diameter of the wire. Understanding this relationship is important for designing wires that can adequately handle electrical current without exceeding specified thermal limits.

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