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Ch 27: Current and Resistance
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 27: Current and ResistanceProblema 57b
Capitolo 27, Problema 57b

The resistivity of a metal increases slightly with increased temperature. This can be expressed as ρ=ρ₀[1+α(T−T₀)] , where T₀ is a reference temperature, usually 20°C, and α is the temperature coefficient of resistivity. For copper, α=3.9×10−3 °C−1. Suppose a 2.5-m-long, 0.40-mm-diameter copper wire is connected across the terminals of a 1.5 V ideal battery. What is the current in the wire at 20°C?

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Step 1: Start by identifying the given values in the problem. The resistivity equation is ρ = ρ₀[1 + α(T − T₀)], where T = 20°C, T₀ = 20°C, and α = 3.9×10⁻³ °C⁻¹. Since T = T₀, the term α(T − T₀) becomes zero, so ρ = ρ₀. For copper, the resistivity at 20°C (ρ₀) is approximately 1.68×10⁻⁸ Ω·m.
Step 2: Calculate the cross-sectional area of the wire. The diameter of the wire is given as 0.40 mm, so the radius r = 0.40 mm / 2 = 0.20 mm = 0.20 × 10⁻³ m. The cross-sectional area A of the wire is given by A = πr². Substitute the radius into this formula to find A.
Step 3: Use the formula for resistance R = ρL/A, where L is the length of the wire (2.5 m), ρ is the resistivity (1.68×10⁻⁸ Ω·m), and A is the cross-sectional area calculated in Step 2. Substitute the values to calculate the resistance R of the wire.
Step 4: Apply Ohm's Law to find the current in the wire. Ohm's Law states that I = V/R, where V is the voltage (1.5 V) and R is the resistance calculated in Step 3. Substitute the values to calculate the current I.
Step 5: Verify the units and ensure all calculations are consistent. The final current I will be in amperes (A), as the voltage is in volts (V) and resistance is in ohms (Ω).

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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 and is influenced by factors such as temperature. In metals, resistivity typically increases with temperature due to increased atomic vibrations that impede electron flow.
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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. It is mathematically expressed as V = IR. This fundamental principle is essential for calculating current in electrical circuits.
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Temperature Coefficient of Resistivity

The temperature coefficient of resistivity (α) is a parameter that quantifies how much the resistivity of a material changes with temperature. For copper, α is approximately 3.9×10⁻³ °C⁻¹, indicating that for each degree Celsius increase in temperature, the resistivity increases by this fraction. This concept is crucial for understanding how temperature variations affect electrical properties in conductors.
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