An engineer cuts a 1.0-m-long, 0.33-mm-diameter piece of wire, connects it across a 1.5 V battery, and finds that the current in the wire is 8.0 A. Of what material is the wire made?

검증된 단계별 안내

Determine the resistivity of the material using Ohm's Law and the formula for resistance. Start with Ohm's Law: \( V = IR \), where \( V \) is the voltage, \( I \) is the current, and \( R \) is the resistance. Rearrange to find \( R \): \( R = \frac{V}{I} \). Substitute \( V = 1.5 \ \text{V} \) and \( I = 8.0 \ \text{A} \) to calculate \( R \).

Use the formula for resistance in terms of resistivity: \( R = \rho \frac{L}{A} \), where \( \rho \) is the resistivity, \( L \) is the length of the wire, and \( A \) is the cross-sectional area. Rearrange to solve for \( \rho \): \( \rho = R \frac{A}{L} \).

Calculate the cross-sectional area \( A \) of the wire using the formula for the area of a circle: \( A = \pi r^2 \), where \( r \) is the radius of the wire. The diameter is given as \( 0.33 \ \text{mm} \), so \( r = \frac{0.33}{2} \ \text{mm} \). Convert this to meters before substituting into the formula.

Substitute the values for \( R \), \( A \), and \( L \) into the formula for \( \rho \) to calculate the resistivity of the material. \( L \) is given as \( 1.0 \ \text{m} \).

Compare the calculated resistivity \( \rho \) with known resistivity values for various materials (e.g., copper, aluminum, etc.) to identify the material of the wire.

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주요 개념

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

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. This relationship is expressed mathematically as V = I * R. Understanding this law is crucial for analyzing electrical circuits and determining the properties of materials based on their response to applied voltage.

Resistivity

Resistivity is a material property that quantifies how strongly a given material opposes the flow of electric current. It is denoted by the symbol ρ (rho) and is measured in ohm-meters (Ω·m). The resistivity of a material, combined with its dimensions, allows for the calculation of resistance using the formula R = ρ * (L/A), where L is the length and A is the cross-sectional area of the wire. This concept is essential for identifying the material based on the measured current and voltage.

Cross-sectional Area

The cross-sectional area of a wire is the area of its circular end face, which affects its resistance and current-carrying capacity. It is calculated using the formula A = π * (d/2)², where d is the diameter of the wire. A larger cross-sectional area results in lower resistance, allowing more current to flow for a given voltage. Understanding this concept is vital for determining how the wire's dimensions influence its electrical properties.